WO2024035444A2 - Static metamaterial design methods - Google Patents

Abstract

The disclosure provides a system and methods for the design of devices containing metamaterials.

Description

PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) STATIC METAMATERIAL DESIGN METHODS CROSS-REFERENCE TO RELATED APPLICATIONS [0001] This application claims the benefit of U.S. Provisional App. No. 63/397,693, filed August 12, 2022, for “STATIC METAMATERIALS USING FAST INVERSE-PROBLEM SOLVING ALGORITHMS,” which is incorporated herein by reference. FIELD OF THE INVENTION [0002] The present application relates to the field of metamaterials. More particularly, the present application relates to systems and methods for designing, controlling, and manufacturing devices containing metamaterials. BACKGROUND OF THE INVENTION [0003] The inverse problem, as it relates to the design of metamaterials to achieve a specific field distribution, currently lacks a well-defined unique solution. The inverse problem leads to a large (sometimes exponentially large) number of solutions with comparable figures of merits (FoM), among which the globally best solution is not necessarily well-distinguished. In this context, local and quasi-global optimization algorithms are used, which perform a huge number of evaluations of the FoM. The speed of existing algorithms is typically limited by two factors: (a) the speed of FoM evaluations, and (b) the speed-up due to the use of sensitivity (gradient) techniques. [0004] Thus, there remains a need for an improved method whereby solving the inverse problem in the design of metamaterials is accelerated. SUMMARY OF THE INVENTION [0005] It has been discovered that the design of metamaterial or devices containing the metamaterial in a complex structure can be accelerated by using the disclosed method and system. This discovery has been exploited to provide the present disclosure, which, in part is PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) directed to system and methods for designing, controlling, and manufacturing devices containing metamaterials. [0006] In one aspect, a system is provided for designing a metamaterial structure. The system includes one or more processors and a computer-readable medium comprising instructions stored therein, which, when executed by the one or more processors, instruct the one or more processors to: access a numerical representation of the geometry of a device and a region of the device containing a metamaterial structure under design. [0007] The instructions instruct the one or more processors to generate a discretized representation of linear partial differential equations describing a field interaction with the device for a set of values of a parameter vector sufficient to identify parameter-dependent and parameter-independent components of a linear system matrix, wherein the discretized representation x is in a form of the linear system matrix A(p) and a source vector b. [0008] The instructions also instruct the one or more processors to decompose, based on either or both the parameter-dependent components and the parameter-independent components of the linear system matrix, an original vector space of discretized fields of the discretized representation into a subspace ^^ ( ^^) representing a new reduced-dimension vector space and another subspace complementary to the new reduced-dimension vector space; and form an objective function ^^⁽ ^^, ^^⁾ that evaluates, for the parameter vector ^^, at least one figure of merit from a vector of fields. [0009] Additionally, the instructions instruct the one or more processors to select a set of initial estimates of the parameter vector of the metamaterial structure for generating a target output field pattern. [0010] The instructions also instruct the one or more processors to transform the set of initial estimates of the parameter vector into a set of reduced parameter vectors that is reduced with respect to an original dimension of the parameter space containing the initial estimate of the parameter vector. [0011] The instructions additionally instruct the one or more processors to apply an optimization algorithm to the objective function working in the reduced-dimension space of reduced parameter vectors, with the set of initial estimates of the reduced parameter vectors ^^^′ to generate a set of compact refined estimates of the parameter vector. The reduced parameter vector p’ denotes a parameter vector in the reduced-dimension parameter space. The reduced parameter vector p’ may be used as a new initial estimate or a compact refined estimate in the reduced-dimension parameter space. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [0012] In addition, the instructions instruct the one or more processors to transform the compact refined estimate of the parameter vector to the original parameter space containing the initial estimate of the parameter vector to generate a refined estimate of the parameter vector. The parameter vector p denotes a parameter vector in the original parameter space. The parameter vector p may be used as an initial estimate or a refined estimate. [0013] The instructions additionally instruct the one or more processors to and selectively identify one or more design characteristics for the metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector and record a numerical representation of the one or more design characteristics. [0014] In an example, which may be combined with any of the examples disclosed herein, the instructions further instructs the one or more processors to decompose, based on either or both the parameter-dependent components and the parameter-independent components of the linear system matrix, an original vector space of discretized fields of the discretized representation into a subspace C̃(p) representing a new reduced-dimension vector space and another subspace complementary to the new reduced-dimension vector space. [0015] In an example, which may be combined with any of the examples disclosed herein, the vector of fields is found by solving the linear problem in the reduced-dimension vector space and then transforming the solution back into the original vector space. [0016] In another example, which may be combined with any of the examples disclosed herein, the solution to the discretized representation of linear PDEs in a form ^^( ^^) ^^ = ^^ is produced by finding the basis in the field vector space which, together with its complementary basis, decomposes the field vector space into two subspaces such that the transformed linear system matrix A = U A U^† , restricted onto the second subspace, is independent of the parameter vector p; solving a reduced-dimension linear system in the first subspace for a given value of the parameter vector p; and using a precomputed matrix factorization of the transformed linear system matrix restricted onto the second subspace, which is independent of the parameter vector. [0017] In another example, which may be combined with any of the examples disclosed herein, the finding of the basis is performed using a singular value decomposition of linear system matrix that is used to identify the basis that maximizes the dimension of the second subspace, in which the linear system matrix is independent of the parameter vector. [0018] In yet another example, which may be combined with any of the examples disclosed herein, the singular value decomposition of the complex-valued linear system matrix and the PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) real-valued basis vectors of the corresponding singular value basis are computed as the real- valued singular value decomposition of a rectangular, real-valued matrix of size N-by- (N*2*N_p), where N is the dimension of the linear system, and N_p is the dimension of the original parameter vector (p), which is formed by first decomposing Np complex-valued square matrices A(p_n), where p_n is the n-th component of the parameter vector, into 2* N_p real-valued square matrices, and then stacking them to form a rectangular matrix. [0019] In yet another example, which may be combined with any of the examples disclosed herein, the optimization algorithm constructs a surrogate objective function based on previously computed values of the objective function and optimizes the surrogate objective function. [0020] In still another example, which may be combined with any of the examples disclosed herein, the surrogate objective function is constructed using the radial basis function method. [0021] In an additional example, which may be combined with any of the examples disclosed herein, the surrogate objective function is constructed using a surrogate model trained by a machine learning algorithm using previously computed values of the objective function. [0022] In another example, which may be combined with any of the examples disclosed herein, the machine-learning algorithm is one of the following: polynomial response surfaces, kriging, generalized Bayesian approaches, gradient-enhanced kriging (GEK), radial basis function, support vector machines, space mapping, artificial neural networks, deep neural networks, Bayesian networks, Fourier surrogate modeling, or random forests. [0023] In yet another example, which may be combined with any of the examples disclosed herein, the objective function contains a reduced-dimension linear system matrix, a reduced- dimension source vector, and matrices for converting a reduced-dimension vector of fields to the original vector space. [0024] In still another example, which may be combined with any of the examples disclosed herein, the vector of fields is found by solving the linear problem in the reduced-dimension vector space and then transforming the solution back into the original vector space. [0025] In an additional example, which may be combined with any of the examples disclosed herein, wherein the objective function generates a linear system matrix and a source vector for a specific value of the set of values of the parameter vector. [0026] In another example, which may be combined with any of the examples disclosed herein, wherein the objective function generates the linear system matrix and the source vector for a given value of the parameter vector based on an algorithm that generates the discretized PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) representation of the linear partial differential equations describing the field interaction with the device including the static metamaterial structure. [0027] In still another example, which may be combined with any of the examples disclosed herein, the objective function generates a linear system matrix and a source vector for a given value of a parameter vector based on an evaluation of a component of a power series expansion of the linear system matrix and the source vector expanded as functions of the parameter vector at a predefined initial value of the parameter vector, wherein parameters of the power series expansion are computed during the evaluation of the objective function and reusing the factorization of the linear system matrix computed for the evaluation of the evaluation function; and the optimization algorithm uses the parameters of the power series expansion to reduce the number of new computations of the objective function at new values of the parameter vector. [0028] In yet another example, which may be combined with any of the examples disclosed herein, the one or more processors transform the initial estimate of the parameter vector into the reduced parameter vector agnostic as to any figure of merit, and the same transformation is used to optimize multiple figures of merit. [0029] In another example, which may be combined with any of the examples disclosed herein, the one or more processors transform the initial estimate of the parameter vector into the reduced parameter vector by projecting the initial estimate of the parameter vector onto a linear subspace that spans a reduced number of singular vectors in comparison to the parameter vector. [0030] In another example, which may be combined with any of the examples disclosed herein, the optimization algorithm comprises a quadratic problem algorithm. [0031] In an additional example, which may be combined with any of the examples disclosed herein, the optimization algorithm comprises a sequential quadratic problem algorithm. [0032] In yet another example, which may be combined with any of the examples disclosed herein, the optimization algorithm comprises a gradient-assisted optimization algorithm, and the objective function further includes an algorithm to compute the gradient using a first order adjoint algorithm. [0033] In still another example, which may be combined with any of the examples disclosed herein, the optimization algorithm comprises a Hessian-assisted optimization algorithm, and the objective function further includes an algorithm to compute the Hessian using a second order adjoint algorithm. [0034] In yet another example, which may be combined with any of the examples disclosed herein, the optimization algorithm comprises a Hessian-assisted optimization algorithm, and the PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) objective function further includes an algorithm to compute the projection of the Hessian onto a set of parameter vectors, as needed for the optimization algorithm, using a second order adjoint algorithm. [0035] In still another example, which may be combined with any of the examples disclosed herein, the one or more processors: evaluate the refined estimate of the parameter vector and corresponding partial derivate constraints of figures of merit associated with the static metamaterial structure in relation to a convergence criterion; and identify the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector in response to an evaluation that the refined estimate of the parameter vector and the corresponding partial derivative constraints meeting the convergence criterion. [0036] In an additional example, which may be combined with any of the examples disclosed herein, the one or more processors, in response to a determination that the refined estimate of the parameter vector and the corresponding partial derivative constraints fail to meet the convergence criterion, iteratively, determine a new initial estimate of the parameter vector; transform the new initial estimate of the parameter vector into a new reduced parameter vector with respect to an original dimension space of the new initial estimate of the parameter vector; apply the local optimization to the new reduced parameter vector to generate a new compact refined estimate of the parameter vector in a reduced dimension space of the new reduced parameter vector; transform the new compact refined estimate of the parameter vector to the original dimension space of the new initial estimate of the parameter vector to generate a new refined estimate of the parameter vector; and selectively identify the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the new refined estimate of the parameter vector. [0037] In yet another example, which may be combined with any of the examples disclosed herein, the one or more processors: determine a field associated with the static metamaterial structure generating the target output field pattern for the parameter vector of the static metamaterial structure; transform the field into a reduced field with respect to an original dimension space of the field; and selectively identify the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on both the refined estimate of the parameter vector and the reduced field. [0038] In still another example, which may be combined with any of the examples disclosed herein, a size of the original dimensions space of the initial estimate of the parameter vector is PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) a number of figures of merit (FOM) associated with the static metamaterial structure in generating the target output field pattern. [0039] In yet another example, which may be combined with any of the examples disclosed herein, the static metamaterial structure comprises a static metamaterial. [0040] In still another example, which may be combined with any of the examples disclosed herein, the parameter vector comprises parameters related to a static structure design and/or one or more material characteristics of the static metamaterial. [0041] In another aspect, a method is provided for designing a metamaterial structure. The method includes accessing a numerical representation of the geometry of the device and a region of the device containing a static metamaterial structure under design. [0042] The method also includes generating a discretized representation of linear partial differential equations describing a field interaction with the device for a set of values of a parameter vector sufficient to identify parameter-dependent and parameter-independent components of the linear system matrix, wherein the discretized representation is in a form of a linear system matrix A(p) and the at least one source vector b. [0043] The method also includes identifying at least one quantitative figure of merits that corresponds to a desirable performance characteristic of the device. [0044] In addition, the method includes forming an objective function that evaluates, for the parameter vector, at least one figure of merit from a vector of fields from the at least one vector of fields, wherein the vector(s) of fields x is a solution to the discretized representation of linear partial differential equations in a form A(p) x = b. [0045] The method additionally includes selecting a set of initial values of the parameter vector of the static metamaterial structure for generating a target output field pattern. [0046] Further, the method includes transforming the set of initial values of the parameter vector into a set of reduced parameter vectors that is reduced with respect to an original dimension of the parameter space containing the initial values of the parameter vector. [0047] Additionally, the method includes applying an optimization algorithm to the objective function working in the reduced-dimension space of reduced parameter vectors, with the set of initial values of the reduced parameter vectors to generate a set of compact refined estimates of the parameter vector. [0048] The method also includes transforming the compact refined estimate of the parameter vector to the original parameter space containing the initial values of the parameter vector to generate a refined estimate of the parameter vector. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [0049] In addition, the method includes selectively identifying one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector. [0050] The method further includes recording a numerical representation of the one or more design characteristics. [0051] In an example, which may be combined with any of the examples disclosed herein, the method further includes decomposing, based on either or both the parameter-dependent components and the parameter-independent components of the linear system matrix, an original vector space of discretized fields of the discretized representation into a subspace representing a new reduced-dimension vector space and another subspace complementary to the new reduced-dimension vector space. [0052] In an example, which may be combined with any of the examples disclosed herein, the method further includes evaluating the refined estimate of the parameter vector and corresponding partial derivate constraints of figures of merit associated with the static metamaterial structure in relation to a convergence criterion; and identifying the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector in response to an evaluation that the refined estimate of the parameter vector and the corresponding partial derivative constraints meeting the convergence criterion. [0053] In another example, which may be combined with any of the examples disclosed herein, in response to a determination that the refined estimate of the parameter vector and the corresponding partial derivative constraints fail to meet the convergence criterion, the method includes: determining a new initial estimate of the parameter vector; transforming the new initial estimate of the parameter vector into a new reduced parameter vector with respect to an original dimension space of the new initial estimate of the parameter vector; applying the local optimization to the new reduced parameter vector to generate a new compact refined estimate of the new reduced parameter vector in a reduced dimension space; transforming the new compact refined estimate of the new reduced parameter vector to the original dimension space of the new initial estimate of the parameter vector to generate a new refined estimate of the parameter vector; and selectively identifying the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the new refined estimate of the parameter vector. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [0054] In yet another example, which may be combined with any of the examples disclosed herein, the method further includes: determining a field associated with the static metamaterial structure generating the target output field pattern for the parameter vector of the static metamaterial structure; transforming the field into a reduced field with respect to an original dimension space of the field; and selectively identifying the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on both the refined estimate of the parameter vector and the reduced field. [0055] In a further aspect, the disclosure provides a non-transitory computer-readable storage medium including instructions stored therein, which, when executed by one or more processors, instruct the one or more processors to perform the above-described method. [0056] In other aspects, methods of operating and manufacturing devices containing a metamaterial structure are provided using the methods described above. [0057] Additional embodiments and features are outlined in part in the description that follows and will become apparent to those skilled in the art upon examination of the specification or may be learned by the practice of the disclosed subject matter. A further understanding of the nature and advantages of the disclosure may be realized by reference to the remaining portions of the specification and the drawings, which form a part of this disclosure. DESCRIPTION OF THE DRAWINGS [0058] The foregoing and other objects of the present disclosure, the various features thereof, as well as the disclosure itself, may be more fully understood from the following description, when read together with the accompanying drawings in which: [0059] FIG. 1 is a schematic representation of a system for fast inverse problem-solving in metamaterial designs in accordance with some aspects of the disclosed technology; [0060] FIG. 2 is a diagrammatic representation of a flowchart of an example method of designing a device containing a metamaterial structure in accordance with some aspects of the disclosed technology; [0061] FIG. 3 is a diagrammatic representation of a flowchart of an example method of operating a device containing a metamaterial structure in accordance with some aspects of the disclosed technology; PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [0062] FIG. 4 is a diagrammatic representation of a flowchart of an example method of manufacturing a device containing a metamaterial structure in accordance with some aspects of the disclosed technology. [0063] FIG. 5 is a schematic representation depicting an example of a computing system in accordance with some aspects of the disclosed technology; [0064] FIG.6 is a schematic representation depicting an example environment for providing engineered frequency dispersion in manipulating wave fields through a dynamic wave- processing device including a static metamaterial in accordance with some aspects of the disclosed technology; and [0065] FIG.7 is a schematic representation depicting an example dynamic wave-processing device including a dynamic metamaterial in accordance with some aspects of the disclosed technology. DESCRIPTION [0066] Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs. The initial definition provided for a group or term herein applies to that group or term throughout the present specification individually or as part of another group unless otherwise indicated. [0067] Definitions [0068] “Artificially structured materials” are materials whose electromagnetic, acoustic or other physical properties are derived from their structural configurations, rather than or in addition to their material composition. [0069] “Metamaterials” are a type of artificially structured material that includes subwavelength elements. Subwavelength elements can include structural elements with portions having spatial length scales smaller than an operating wavelength of the metamaterial. Further, the subwavelength elements have a collective response to waves or radiation that corresponds to an effective continuous medium response. For example, in the case of electromagnetic metamaterials, the collective response may be characterized by an effective permittivity, an effective permeability, an effective magnetoelectric coefficient, or any combination thereof. For example, electromagnetic radiation may induce charges and/or currents in the subwavelength elements, and the subwavelength elements can acquire nonzero electric and/or magnetic dipole PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) moments. Some metamaterials provide an artificial magnetic response. For example, split-ring resonators (SRRs) and other plasmonic resonators can exhibit an effective magnetic permeability. Some metamaterials have “hybrid” electromagnetic properties that emerge partially from the structural characteristics of the metamaterial, and partially from the intrinsic properties of the constituent materials. For example, a metamaterial consisting of a wire array embedded in a nonconducting ferrimagnetic host medium can exhibit effects of both the wire array and the host medium. [0070] “Metamaterials” can be designed and fabricated to exhibit selected permittivity, permeability, and/or magnetoelectric coefficient values that depend upon material properties of the constituent materials as well as shapes, chirality, configurations, position, orientations, and couplings between the subwavelength elements. The selected permittivity, permeabilities, and/or magnetoelectric coefficients values can be positive or negative, complex (having loss or gain), anisotropic, variable in space (as in a gradient index lens), variable in time (e.g., in response to an external or feedback signal), variable in frequency (e.g., in the vicinity of a resonant frequency of the metamaterial), or any combination thereof. The selected electromagnetic properties can be provided at wavelengths that range from radio wavelengths to visible wavelengths and beyond. [0071] “Metamaterials” can include either or both discrete elements or structures and non- discrete elements or structures. For example, a metamaterial may include discrete structures, such as split-ring resonators. In another example, a metamaterial may include non-discrete elements that are inclusions, exclusions, layers, or other variations along with some continuous structure. [0072] Further, “metamaterials” can include extended structures having distributed electromagnetic responses, such as distributed inductive responses, distributed capacitive responses, and distributed inductive-capacitive responses. For example, metamaterials can include structures consisting of loaded and/or interconnected transmission lines, artificial ground plane structures, and/or interconnected/extended nanostructures. [0073] A “metasurface” is a thin layer of a metamaterial. A thin layer of a metamaterial can include a subset of the total volume of the metamaterial. A metasurface can be approximated as an infinitely thin sheet having a surface impedance, or surface impedances for anisotropic responses. When approximated as an infinitely thin sheet the metasurface can lack a refractive index, as waves do not propagate or refract "inside" of the metasurface. Instead, the metasurface can act as a discontinuity in space. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [0074] A “static metamaterial” is the metamaterial in which the degrees of freedom are typically structural or material choices. The properties of the static metamaterial remain unchanged. [0075] A “dynamic metamaterial” is the metamaterial in which the degrees of freedom can be modulated using externally controlled physical stimuli. The properties of the dynamic metamaterial may be tunable. [0076] A “linear metamaterial” is a metamaterial whose response to a physical field is linear in that field, i.e., any induced field in the metamaterial is linearly proportional to an incident field. Consequently, a linear metamaterial that can be modeled using linear partial differential equations (PDEs) with constant (field-independent) coefficients. [0077] A “design characteristic” can include electromagnetic and/or acoustic characteristics of elements of a metamaterial. [0078] A “figure of merit” is a numerical expression representing the performance or efficiency of a given device, material, or procedure. [0079] “Components” of a field include both orientation and polarization of the field. [0080] As used herein, the articles “a” and “an” refer to one or more than one (i.e., to at least one) of the grammatical object of the article. By way of example, “an element” means one element or more than one element. Furthermore, the use of the term “including” as well as other forms, such as “include,” “includes,” and “included,” is not limiting. [0081] As used herein, the term “about” will be understood by persons of ordinary skill in the art and will vary to some extent on the context in which it is used. As used herein when referring to a measurable value such as an amount, a temporal duration, and the like, the term “about” is meant to encompass variations of ±20% or ±10%, including ±5%, ±1%, and ±0.1% from the specified value, as such variations are appropriate to perform the disclosed methods. [0082] Reference will now be made to specific examples illustrating the disclosure. It is to be understood that the examples are provided to illustrate exemplary embodiments and that no limitation to the scope of the disclosure is intended thereby. [0083] The disclosed methods apply to the design of highly complex metamaterials, i.e. structures having a large number of degrees of freedom (DoF). The metamaterials may include static metamaterials, where DoFs are typically structural or material choices. The metamaterial may also include dynamic (or tunable) metamaterials, where DoFs can be modulated using externally controlled physical stimuli. The disclosed methods are especially advantageous for the design of aperiodic, including quasi-periodic and chaotic, metamaterials, and metasurfaces. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) Volumetric (thick) metamaterials and metamaterials with a large period (or unit cell area) are also examples of extraordinarily complex structures. [0084] The disclosure provides an algorithm that offers various speedups compared to the known approaches. The disclosed inverse problem solver includes (1) forward problem speed- up (fast FoM evaluation) due to dimensionality reduction in the forward problem, (2) fast second-order adjoint-solution method producing the entire Hessian matrix (second derivatives) in addition to first-order derivatives of the objective (cost) function, at a cost comparable to the computation of the FoM itself, and (3) reduced dimensionality of the inverse problem, which leads to a much faster exploration of the parameter space than in the known approaches. [0085] FIG.1 is a schematic representation of a system 100 for fast inverse problem-solving in metamaterial designs in accordance with some aspects of the disclosed technology. In one embodiment, the system 100 includes a field pattern generator 102 containing a metamaterial 104 that can generate a desired target output field pattern 106. System 100 also includes computing device 108, which may implement, in any suitable combination of software, hardware, and/or firmware, an inverse problem solver 110 as described hereafter. The inverse problem solver 110 may be used to rapidly design the metamaterial 104 to create the desired target output field pattern 106. The target output field pattern 106 may be as used as the input for the computing device 108 to replace the initial guess parameters 120 if the target output field pattern 106 is tunable. [0086] A user may specify the desired field pattern. For example, the algorithm may ask the user “What do you want to achieve?” The desired target field pattern can be specified by a user in the far field as a function of two coordinates on the sphere in the far field, for example, the two spherical angles. The desired target field pattern can also be specified by a user in the near field as a function of two coordinates on a predefined surface, such as a sphere, a plane, and so on. The desired target field pattern can also be selected by a user from a list of predefined field patterns that are frequently used to design devices with a particular function. For example, predefined field patterns may include (1) a beam with a selectable beam direction and beam width or beam divergence angle; (2) a Gaussian beam with a selectable beam direction and Gaussian beam parameters; (3) a beam with a rectangular far-field pattern of selectable angular width and mean direction, among others. [0087] In some aspects, the field pattern generator 102 may be a device. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [0088] In some aspects, the inverse problem solver 110 may be a software package implementing an algorithm that can speed up the inverse problem solving in the metamaterial design. [0089] In some aspects, the metamaterial 104 may be a static metamaterial, i.e., a metamaterial in which the degrees of freedom are typically structural or material choices. The properties of the static metamaterial remain unchanged. In other aspects, the metamaterial 104 may be a dynamic metamaterial, i.e., a metamaterial in which the degrees of freedom can be modulated using externally controlled physical stimuli. The properties of the dynamic metamaterial may be tunable. [0090] The inverse problem solver 110 according to aspects of the present disclosure builds upon established Finite Element Method solvers, which are available commercially in the public domain. The disclosed inverse problem solver 110 also takes advantage of the special form of the inverse problem in metamaterial design, e.g., the linearity of the forward problem with respect to both the fields and the design parameters. [0091] The disclosed inverse problem solver 110 is applicable in the context of the PDE (partial differential equation)-constrained optimization problem, which is stated as follows: Find parameter vector ^^ to minimize objective function ^^( ^^, ^^) subject to PDE constraints ^^ ( ^^) ^^ = ^^ and bounds ^^( ^^) ≤ 0, where ^^ is a linear operator acting on the field x, and p is either a vector or a field of parameters. The field x is then discretized as a vector in a finite-dimensional vector space (of dimension N), and the PDE constraints are reduced to linear-algebraic constraints as follows: ^^( ^^) ^^ = ^^( ^^) Eq. (1) where A( ^^) is a square matrix of rank N and is a linear system matrix, b(p) is the source vector, and x is the vector of discretized fields. The source vector corresponds to the “incident” or “external” fields that excite the metamaterial system. The fields produced in the system (or metamaterial) are thus linearly proportional to the source vector. The discretization algorithms for specific physical models of a device or a component are described in the literature. Examples of such algorithms include the Finite Element Method, Boundary Element Method, or Integral Equation Method, their variations, and hybrids. A particularly general tool for assembling the discretized matrices ^^( ^^) is COMSOL Multiphysics, available from COMSOL Inc. of Stockholm, Sweden. [0092] The dimension of the parameter vector ^^ is called the number of DoF and denoted n going forward. Solving Eq. (1) for the field vector x with a given value of parameter vector p is PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) called the forward problem. Evaluating ^^⁽ ^^⁾ = ^^⁽ ^^, ^^⁾ with an x that is a solution to the forward problem is called evaluating the figure of merit (FoM) on the parameter vector p. [0093] In some embodiments, the inverse problem solver 110 includes three elements or functions. Each of the three elements may contribute to the speedup of the solution process. [0094] A first element of the inverse problem solver 110 is field space dimensionality reducer 112 to implement a suitable dimensionality reduction in the field space. The field space (of dimension N) can be split into two subspaces, one of which corresponding to the components such that the matrix A( ^^) is independent of the parameters. After an appropriate parameter ordering, _{^^( ^^) = [} ^{^^( ^^) ^^( ^^)} ^_{^( ^^) ^^} ^{] Eq. (2)} matrices ^^( ^^) and ^^( ^^) are square (of size ^^ _^^ and ^^₀, respectively), and matrices

^^( ^^) and ^^( ^^) are generally rectangular, of dimension ^^ _^^-by- ^^₀ and ^^₀-by- ^^ _^^, respectively, and ^^₀ + ^^ _^^ = ^^. This decomposition is possible because, in a typical metamaterial design problem, adjustable parameters (p) are localized in a portion of the volume of the modeling domain. [0095] To explain the concept of field space dimensionality reduction, let us assume that the matrix ^^ of the form in Eq. (2) can be factorized as follows: ^^ = ^^ ^^ ^^, Eq. (3) where the matrix ^^ = diag[ ^^ − ^^ ^^⁻¹ ^^, ^^] Eq. (3a) is a diagonal matrix, and the upper and lower triangular matrices ^^, ^^ are given by: ^^ = [^{1 ^^ ^^−1} ], ^^ = [ ^{1 0} ^_{^−1 ^^} . Eq. (3b)

It is evident from Eq. (3a,b) that factorization (3) exists whenever ^^⁻¹ exists, i.e., the constant matrix ^^ is invertible. The general case where it is not invertible

discussed below in [0085]. With this factorization, the solution to the forward problem can be represented as follows: _^^ ^{^^1 ( ^^ − ^^ ^^−1 ^^)−1( ^^1 − ^^ ^^−1 ^^2)} _{], Eq. (4)}

PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) w_{here [} ^{^^1} ^^₂] = ^^. [0096]

is constant, its inverse or any suitable factorization can be pre- computed and does not need to be repeated at every evaluation of the FoM. The only matrix to be inverted or factorized at every evaluation of the FoM is the matrix ^^ ( ^^) as follows: ^^ ( ^^ = ^^( ^^) − ^^( ^^) ^^⁻¹ ^^( ^^), Eq. (5) whose dimension is only ^^ < ^^. This leads to an overa ³ ^_^ ll speedup by the factor of ( ^^/ ^^ _^^) , which is typically a large number. The matrix ^^ ( ^^) effectively replaces the matrix ^^( ^^) in the forward problem solver and can also be referred to as the reduced-dimension linear system matrix. [0097] In some aspects, ^^⁽ ^^⁾ in Eq. ⁽2⁾ maybe simplified, which leads to an extra speedup. The simplification can be achieved if the field space decomposition can be achieved where the coupling matrices E and F are also independent of parameter vector p. The simplified matrix A is as follows: _{^^( ^^) = [} ^{^^( ^^) ^^}]. Eq. (2a) ^^ ^^

where the matrix ^^ ^^⁻¹ ^^ is entirely constant and can thus be precomputed, and the constant matrices ^^ ^^⁻¹ and ^^⁻¹ ^^ are present in Eq. (4). [0098] In some aspects, a decomposition of the field space may be based on the approximate independence of a group of matrix elements of matrix A upon the parameters. That is, Eq. (2) is replaced with approximate equations, such that _{^^( ^^) = [} ^{^^( ^^) ^^( ^^) ^^( ^^) ^^( ^^)} ^_{^ ^^} ^≈

small number ^^. The decomposition is specific to the choice of ^^ and is chosen to maximize the number of “approximately constant” dimensions, ^^₀. [0099] Likewise, there is an approximate variant of Eq. (2a), where the matrices D, E, and F are approximately constant. [00100] In some aspects, the matrix ^^ in Eq. (2) may be non-invertible. This situation is handled as follows. First, the rank of the rectangular matrix [ ^^( ^^), ^^] is evaluated. There are only two, mutually exclusive, possibilities: (a) this matrix is full rank, i.e., all of its rows are PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) linearly independent, or (b) this matrix is rank-deficient, i.e., some of its rows are linearly dependent upon the other. If (a) is the case, a permutation of the columns of the matrix ^^( ^^) is performed, which amounts to permuting the elements of the field vector (x), such that in the new basis the matrix ^^ is full-rank and hence invertible. If (b) is the case, first a selection of ^{rows in the matrix [ ^^( ^^), ^^] is moved to the matrix [ ^^( ^^), ^^( ^^)], such that the new matrix} _{[ ^^’( ^^), ^^’] of a smaller height is a full-rank matrix. Then, the situation is identical to case (a)} and is handled accordingly. A matrix of size M-by-N is called “full rank” if its rank is equal to min(M,N), which is the maximum possible rank for a matrix of that size. [00101] An alternative algorithm for creating a full-rank square matrix ^^ can be based on a more general linear matrix transformation of ^^( ^^) than the column permutation described in the preceding paragraph. Any linear transformation of Eq.(2) of the form ^^^′ = ^^ ^^ ^^ Eq. (0085a) with some unitary matrices ^^, ^^ may be suitable for this purpose, as long as it preserves parameter-independence of the ^^ block, i.e., the lower-right block of ^^( ^^) as defined by Eq.(3). For example, if ^^ and ^^ have the following block representation, ^^ = [ ^{^^} 0^{11 ^^} ^_^ ¹² 2₂], ^^ = [ ^{^^} ^_^ ^{11 0} 2_{1 ^^22}], Eq.(0085b) is again parameter-independent. Even more

general linear transformations than those given by Eq.(0085a,b) are possible if the matrix ^^( ^^) and/or matrix ^^( ^^) are parameter-independent. [00102] Among all such transformations, the most suitable may be selected in accordance with some criterion. For example, minimizing the condition number of the matrix ^^’ is a good criterion that leads to higher numerical accuracy and stability. In another example, if the matrix ^^ is sparse (i.e., has a small fill factor), then choosing ^^, ^^ to minimize the fill factor of ^^′⁻¹ is a good condition that reduces computational time for evaluating expressions in Eq.(4). [00103] Dimensionality reduction in the field space is useful since the computational complexity of solving the forward problem via matrix factorization (which is a step for computing the gradients) is ^^( ^^³), and reducing the dimension of the forward problem from N to ^^ _^^ is typically a very significant speedup, even if the effective dimension is reduced only by a relatively small factor. [00104] The choice of the basis functions leading to the discretization A(p) of the operator ^^ may affect the maximum dimensionality reduction that can be achieved with this technique. In general, choosing the basis functions to be more localized, and/or avoiding nonlocal boundary PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) conditions and constraints leads to the greater benefit of this technique. The most extreme example of basis function localization is the point dipole discretization, or Discrete Dipole Approximation, where the basis functions are localized to points. The basis functions in this context are the basis functions in which the electromagnetic fields as a function of spatial coordinates are written in. For example, if the Maxwell’s equations in frequency domain are stated as ^^ ( ^^) ^^ ( ^^) = ^^( ^^), Eq. (0087a) discretization implies choosing a finite set of functions known as the basis functions, such that ^^ _^^ ⁽ ^^⁾ = ^∑ _^^ ^^ _{^^, ^^} ^^ _{^^, ^^}( ^^) , ^^ = 1,2,3. is replaced with a linear problem

= where the matrix elements of the matrix ^^ (and the source vector ^^) are constructed from the operator field ^^ ( ^^) (and vector field of sources, ^^( ^^)) given the basis functions ^^ _{^^, ^^}( ^^). This process is known as the “Galerkin method”, and its in-depth discussion can be found in textbooks on Finite Element Method. The finite-dimensional vector ^^ = { ^^ _{^^, ^^}} is the r^{epresentation of the electromagnetic fields in the linear space of functions spanning the basis} { _{^^ ^^, ^^( ^^)}, and the dimension of this vector is called the field space dimensionality in the} discussion above. Note that in the discussion prior to this paragraph, “ ^^” is referring to such vector of unknown fields in the given basis, and not the spatial coordinates. Eq. (0087b) is therefore the same as Eq. (1), with ^^ → ^^, ^^ → ^^. The choice of basis functions affects the dimension and form of matrix ^^( ^^) and the source vector ^^( ^^). [00105] A second element of the disclosed inverse problem solver 110 is a derivative calculator 114 for producing an efficient computation of the first- and second-order derivatives using a generalization of the adjoint solution method. The addition of the second-order derivative information enables the use of quickly converging local optimization algorithms, such as the known Quadratic Problem (QP) and Sequential Quadratic Problem (SQP) solvers. Such solvers can converge to a local minimum in one or two iterations with full Hessian evaluation. [00106] Additionally, a significant speedup in the evaluation of the first-order and second- order derivatives can be obtained in the special case where the matrix A(p) depends linearly on parameter vector p. Such is the case with all linear metamaterials because the linear metamaterials can be modeled using linear PDEs with constant (field-independent) coefficients. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) These constant coefficients relate linearly to the material properties and can be used as the optimization parameter vector p which is also referred to as parameters (p). [00107] In some cases, it is not suitable to choose local material properties as parameters (p), for example, due to additional nonlocal design constraints. In that case, a linear or nonlinear variable transformation from the parameters (p) and the PDE coefficients can be established, and the first and second-order derivatives can be inexpensively converted from one basis to the other using the Jacobian matrix of the variable transformation. [00108] A third element of the inverse problem solver 110 is a parameter dimensionality reducer 116 to implement dimensionality reduction in the parameter p that is applied at the linear system matrix level and is agnostic about the cost function (FoM). This is in contrast with various other dimensionality reduction techniques (e.g., in Machine Learning algorithms) that rely on the knowledge of the FoM. There may be many diverse FoMs, each leading to a different landscape in the p-space. The technique is applied once regardless of the number of FoMs in the multi-objective problem. [00109] In some embodiments, the complex-valued matrix A(p) depends linearly upon the real-valued parameter vector p. In those scenarios, dimension reduction of the parameter space can be performed as follows. First, the complex-valued matrix A is split into the real and imaginary parts, and each part is transformed into a real-valued vector, either of length ^^ = ^^² for a general asymmetric matrix A, or a vector of length ^^ = ^^( ^^ + 1)/2, for a symmetric matrix A. The new vector function is then written as ^^⁽ ^^⁾ = [ ^^_{0 ^^} + ^^ _^^ ^^, ^^_{0 ^^} + ^^ _^^ ^^], which is a real vector of length 2M, where each of ^^ _^^ and ^^ _^^ is a real-valued matrix of dimension M- by-n, and n is the dimension of the real-valued parameter vector p. Finally, a real-valued matrix of dimension 2M-by-n is formed by stacking the two matrices ^^ _^^ and ^^ _^^, i.e., ^^ = [ ^^ _^^; ^^ _^^]. [00110] In this representation, a singular value decomposition of the rectangular matrix G is given in Eq. (6) as follows: ^^ = ^^ ^^ ^^⁺, Eq. (6) where U is a real-valued, unitary square matrix of size 2 ^^, ^^ is a real-valued, rectangular diagonal matrix of size 2M-by-n, V is a real-valued, unitary square matrix of size n, and ^^⁺ is the conjugate transpose of V. The rank of the diagonal matrix ^^ does not exceed the smallest of 2M and n, i.e., r = min(2M, n). The first r diagonal entries in ^^ are the singular values of the matrix G, and the remainder of the diagonal of ^^ is zero. Singular values are real-valued and non-negative; zero singular values are possible. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [00111] To compute the SVD (singular value decomposition) of ^^ per Eq.6 in another way, another version of an SVD algorithm described below can be used. The number of rows, ^^ ^^ = ^^ ^^ ^{^^}, of ^^ is larger than the number of its columns, n. The large matrix ^^ (of size ^^ ^^ ^{^^} ≫ ^^) is not used for subsequent calculations. The modified SVD algorithm computes ^^, ^^⁺ by the following steps: (a) a Q-less QR decomposition of matrix ^^ is computed, which results in an upper-rectangular matrix ^^ of the same size as ^^; they are related as ^^ = ^^ ^^, where a large unitary matrix ^^ is not explicitly computed; (b) the upper-rectangular matrix ^^ is truncated to a square matrix ^^ _{^^ ^^ ^^ ^^} of size ^^; such matrix is called the economy form of ^^; (c) an SVD of the square matrix ^{^^} _{^^ ^^ ^^ ^^} is computed, yielding matrices ^{^^} _{^^ ^^ ^^ ^^} ^{, ^^+} that satisfy ^{^^} _{^^ ^^ ^^ ^^} ^{= ^^} _^^ ^{^^} _{^^ ^^ ^^ ^^} ^{^^+ with} some unitary matrix ^{^^} _^^, which is not explicitly used. The square, unitary matrix ^{^^+} is also the useful matrix ^^⁺ of Eq.(6). The square, real-valued, non-negative definite, matrix ^^ _{^^ ^^ ^^ ^^} is the economy form of ^^ from Eq.(6). [00112] Assume that the singular values in ^^ are sorted in descending order. To achieve dimensionality reduction, the diagonal matrix ^^ is then truncated at the first singular value that is below a pre-selected threshold value, to produce a reduced-dimension diagonal square matrix ^^ ′. The threshold value can be arbitrarily small, or zero. The dimension of ^^ ′ is now ^^’ < ^^. [00113] The unitary matrix of left singular vectors (U) is then truncated to obtain U’, a rectangular matrix of size 2M-by- ^^’. Similarly, the unitary matrix of right singular vectors (V) is truncated to V’, a rectangular matrix of size n-by-n’; this makes the conjugated matrix ^^′⁺to be of size ^^’-by-n. [00114] Finally, an approximate representation of ^^^′( ^^′) is obtained as follows: ^^^′( ^^′) = ^^^′Γ^′ ^^^′, Eq. (7a) ^^^′ = ^^′⁺ ^^. Eq. (7b) [00115] Note that ^^^′( ^^′) is represented by a real vector of length 2 ^^, and therefore can be back-transformed into its representation by a complex-valued square matrix, of the same dimension and kind as the original ^^( ^^). However, the new parameter vector ^^^′is of lower dimension, ^^’ < ^^; thus, dimensionality reduction has been achieved. The vector ^^^′ is also referred to as reduced parameter vector. Equations (6,7) describe the transformation from the original parameter space to the reduced-dimension parameter space, and the inverse transformation from the reduced-dimension parameter to the original parameter space. The PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) original parameter vector p is in the original space while the reduced-dimension parameter vector p’ is in reduced-dimension space. [00116] The inverse transformation ^^′ → ^^ can be done using the original unitary matrix V as follows: ^^ = ^^ ^^′′, Eq. (8) where ^^′′ is ^^′ padded with zeros to the length ^^. Note that, since V and ^^⁺ are real-valued, both ^^ and ^^’ vectors are real-valued, too. [00117] The intuitive interpretation of the third element is that the parameter vector is projected onto a linear subspace that spans a reduced number of singular vectors. The reason that this projection operation has only a negligible effect on the linear operator ^^( ^^) is that the complementary subspace corresponds to zero or negligible singular values. [00118] The disclosed three elements can be arranged into the disclosed inverse problem solver 110 for solving the inverse problem for a broad class of metamaterials. [00119] The disclosed inverse problem solver 110 may perform a process including the steps as follows: step (1) assembling and precomputing all objects needed for subsequent steps; step (2) choosing an integer n’ between 1 and n to represent a reduced dimension,; step (3) choosing a list of initial guesses for the parameter vector; step (4) transforming all necessary objects, including the parameter vector to the reduced-dimension parameter space as described in the third element; step (5) feeding the initial guesses and precomputed objects to an optimization solver or algorithm that utilizes first-order and second-order derivatives computed as described in the second element—the inverse problem solver 110 produces one or more optimized parameter vectors ^^’; step (6) back-transforming the one or more optimized reduced-dimension parameter vector(s) ^^’ into the original parameter space using Eq. (8), and step (7) evaluating FoM and the first-order derivative using the original matrix ^^( ^^) and parameter vector(s) from step 6. If the FoM value on at least one of the new parameter vectors is better than a pre-selected satisfactory value of FoM, or if some other convergence criterion is satisfied, the process is finished. [00120] Otherwise, the process includes (1) either decreasing the value of the threshold parameter that is used to reduce the dimension of parameter space (this decrease results in a larger ^^’), or otherwise increasing the integer n’ by a positive increment, (2) taking the value of ^^ obtained so far, and going to step 3, while adding the newly obtained (refined) estimate of parameter p to the list of initial guesses for the parameter vector used in step 3. The process PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) stops looping when all non-zero singular values of G are included, i.e., ^^’ = ^^, or when another termination criterion is satisfied, whichever happens sooner. Examples of other termination criteria include: (1) total elapsed time exceeds a threshold; (2) computational time per last execution of the loop (steps 3-7) exceeds a threshold; (3) improvement in the best value of FoM from the previous iteration is less than a threshold. [00121] When the process is complete, the optimized parameter vector ^^ corresponding to the best-found value of FoM may be used to generate a set of design parameters 118, which may be stored/recorded in, for example, a non-transitory computer-readable medium within the computing device 108. The design parameters 118 may then be used, for example, to design the metamaterial 104 (in the case of a static metamaterial) to have design characteristics that produce the target output field pattern 106. [00122] In some aspects, the components of the parameter vector correspond to the geometric dimensions of the structure comprising the metamaterial. This approach is known as geometry parameterization. A parameterized geometry is then evaluated with a specific value of the parameter vector, namely, the optimized parameter vector computed using the described steps, and the evaluated geometry is saved onto a memory device as a file in one of the standard CAD formats. [00123] In some aspects, the components of the parameter vector correspond to parameters of one or more parameterized, closed surfaces, which enclose domains of the same material, or separate domains of different materials. Air, or absence of any material, is modeled as a material with the dielectric constant of one and is treated in the same fashion as any actual materials. This approach is sometimes called “topology optimization”, since it allows, in general, for a variable (parameter-dependent) topology of the same-material domains. The set of closed surfaces computed for a particular value of the parameter vector is then discretized using conventional surface meshing, combined into a single solid geometry, and saved onto a memory device as a file in one of the standard CAD formats. [00124] In some aspects, a combination of geometry parameterization and topology optimization is used, such that some components of the parameter vector correspond to dimensions of some fixed-topology structural elements of the metamaterial structure, and some other components of the parameter vector correspond to the parameters of one or more parameterized surfaces. [00125] A file containing the digital description of the geometry in one of the standard CAD formats is then transferred to a computer-aided manufacturing system. The system, which PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) includes a computer system that reads files and electronically controls the manipulations inside the apparatus, uses the data from such file to produce the structure whose geometry is described in that file. [00126] Examples of computer-aided manufacturing systems that may accept a CAD file and produce a structure include (1) For microwave metamaterials: CNC machining, laser cutting, 3D Printing and (2) For THz and optical metamaterials: Micro-3D printing (such as micro- Stereolithography), Photolithography, Direct Last Writing, X-Ray lithography, Electron Beam Lithography (EBL), Focused Ion Beam (FIB) lithography, and so on. [00127] Computer-aided manufacturing systems generally use the data from a CAD file to control the movement, positioning and orientation of mechanized tools integrated into the system. In some instances, they also use the data to control the intensity of the generated stimulus, to turn it on or off, and to coordinate these actions with the movements of the mechanized tools in the system. [00128] FIG. 2 illustrates an example method 200 for designing a metamaterial structure by identifying one or more design characteristics for the metamaterial structure. Although the example method 200 depicts a particular sequence of operations, the sequence may be altered without departing from the scope of the present disclosure. For example, some of the operations depicted may be performed in parallel or in a different sequence that does not materially affect the function of the method 200. In other examples, different components of an example device or system that implements the method 200 may perform functions at substantially the same time or in a specific sequence. [00129] According to some examples, method 200 may include accessing a numerical representation of the geometry of the device and a region of the device containing a metamaterial structure under design at block 205. For example, the computing device 108 as illustrated in FIG.1 may access a numerical representation of the geometry of the device and a region of the device containing a metamaterial structure under design. [00130] According to some examples, method 200 may include generating a discretized representation of linear partial differential equations describing a field interaction with the device for a set of values of a parameter vector sufficient to identify parameter-dependent and parameter-independent components of the linear system matrix at block 210. For example, the computing device 108 as illustrated in FIG.1 may generate a discretized representation of linear partial differential equations describing a field interaction with the device for a set of values of a parameter vector sufficient to identify parameter-dependent and parameter-independent PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) components of the linear system matrix. The discretized representation is in a form of a linear system matrix and a source vector. [00131] According to some examples, method 200 may include decomposing, based on either or both the parameter-dependent components and the parameter-independent components of the linear system matrix, an original vector space of discretized fields of the discretized representation into a subspace representing a new reduced-dimension vector space and another subspace complementary to the new reduced-dimension vector space at block 215. For example, the computing device 108 as illustrated in FIG.1 may decompose, based on either or both the parameter-dependent components and the parameter-independent components of the linear system matrix, an original vector space of discretized fields of the discretized representation into a subspace representing a new reduced-dimension vector space and another subspace complementary to the new reduced-dimension vector space. [00132] According to some examples, method 200 may include forming an objective function that evaluates, for the parameter vector, at least one figure of merit from a vector of fields at block 220. For example, the computing device 108 as illustrated in FIG. 1 may form an objective function that evaluates, for the parameter vector, at least one figure of merit from a vector of fields. [00133] According to some examples, method 200 may include selecting a set of initial estimates of the parameter vector of the metamaterial structure for generating a target output field pattern at block 225. For example, the computing device 108 as illustrated in FIG.1 may select a set of initial estimates of the parameter vector of the metamaterial structure for generating a target output field pattern. [00134] According to some examples, method 200 may include transforming the set of initial estimates of the parameter vector into a set of reduced parameter vectors that is reduced with respect to an original dimension of the parameter space containing the initial estimate of the parameter vector at block 230. For example, the computing device 108 as illustrated in FIG. 1 may transform the set of initial estimates of the parameter vector into a set of reduced parameter vectors that is reduced with respect to an original dimension of the parameter space containing the initial estimate of the parameter vector. [00135] According to some examples, method 200 may include applying an optimization algorithm to the objective function working in the reduced-dimension space of reduced parameter vectors, with the set of initial estimates of the reduced parameter vectors to generate a set of compact refined estimates of the parameter vector at block 235. For example, the PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) computing device 108 as illustrated in FIG. 1 may apply an optimization algorithm to the objective function working in the reduced-dimension space of reduced parameter vectors, with the set of initial estimates of the reduced parameter vectors to generate a set of compact refined estimates of the parameter vector. [00136] According to some examples, method 200 may include transforming the compact refined estimate of the parameter vector to the original parameter space containing the initial estimate of the parameter vector to generate a refined estimate of the parameter vector at block 240. For example, the computing device 108 as illustrated in FIG.1 may transform the compact refined estimate of the parameter vector to the original parameter space containing the initial estimate of the parameter vector to generate a refined estimate of the parameter vector. [00137] These refined estimates are produced by the optimization algorithm. An optimization algorithm takes one or more initial values of parameter vectors and returns one or more parameter vectors corresponding to local optima of the objective function. The refined estimates are not the final values of the parameter vector that are returned by the algorithm, except or the last iteration, where the output is determined to be final. The refined estimates in each iteration, except the last one, are intermediate results, but not the best possible or final. Furthermore, the final output does not necessarily correspond to the best result possible. Changing the threshold for various termination criteria, or the sequence of reduced dimensions used in iterations, may produce different results, which can be either better or worse than the results obtained previously. In general, the global optimum of a specific design problem is not known, and its calculation typically takes an exponentially large computational time, which scales exponentially with the parameter dimension n. Therefore, it is typically unknown if the outputs of the algorithm achieve the best possible value of FoM. [00138] In other words, it is very difficult, in a reasonable amount of time, to positively confirm that the global optimum was found. It may be, however, possible to confirm the negative, i.e., the global optimum was not found, by choosing the threshold values of the algorithm that allow for a more thorough exploration of the parameter space, such as allowing the algorithm to run longer. If a new final output is better than the previously found best value, this confirms that the previously found best value is not the globally best value. However, each subsequent negative confirmation (by virtue of proving that a further improvement was possible) takes progressively longer computational time. For this reason, the algorithm is typically designed to terminate when a specified “timeout” is reached. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [00139] According to some examples, method 200 may include selectively identifying one or more design characteristics for the metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector at block 245. For example, the computing device 108 as illustrated in FIG. 1 may selectively identify one or more design characteristics for the metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector. [00140] According to some examples, method 200 may include recording a numerical representation of the one or more design characteristics at block 250. For example, the computing device 108 as illustrated in FIG.1 may record a numerical representation of the one or more design characteristics. [00141] According to some examples, the optimization algorithm additionally constructs a surrogate objective function based on previously computed values of the objective function and optimizes the surrogate objective function. The surrogate objective function may be constructed using the radial basis function method. Techniques for using the radial basis function method to construct a surrogate objective function are disclosed, for example, in Gutmann, H.-M., A radial basis function method for global optimization. Journal of Global Optimization 19, Issue 3, 2001, pp.201–227, https://doi.org/10.1023/A:1011255519438. [00142] According to some examples, the surrogate objective function is constructed using a surrogate model trained by a machine learning algorithm using previously computed values of the objective function. The machine-learning algorithm may include, without limitation, one or more of the following: polynomial response surfaces, kriging, generalized Bayesian approaches, gradient-enhanced kriging (GEK), radial basis function, support vector machines, space mapping; artificial neural networks, deep neural networks, Bayesian networks, Fourier surrogate modeling, and/or random forests. [00143] According to some examples, as shown in FIG. 3, the method 200 of FIG. 2 may include a further step of controlling tuning of the metamaterial structure (i.e., a dynamic component of the metamaterial structure) according to the design parameters in block 305. This step may include controlling operation of the field pattern generator 102 of FIG. 1 to tune the metamaterial 104 so that it produces the target field pattern 106. [00144] According to some examples, as shown in FIG. 4, the method 200 of FIG. 3 may include a further step of constructing the metamaterial structure according to the design parameters at block 405. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [00145] The disclosed inverse problem solver 110 or methods implemented thereby can be used in various software applications. The disclosed inverse problem solver 110 can be used to implement design automation software for static and dynamic metamaterials. The disclosed inverse problem solver 110 can also be used to control firmware for dynamic metamaterials, among others. [00146] The design of static and dynamic metamaterials and metasurfaces can be applied to various applications, including microwave or radio frequency devices, optical or electro-optical devices or components, and acoustic devices or components, among others. [00147] As applied to static or passive metamaterials, these techniques can be used to design the structure in the metamaterial. Typically, these are used as components in a larger system. For example, diffractive and other non-standard lenses and beamformers, beam splitters, beam redirectors, waveguide couplers, and fiber-optics couplers are all examples of components used across the electromagnetic spectrum in various systems. [00148] The disclosed inverse problem solver 110 or method implemented thereby can be used for devices containing metamaterials. As applied to dynamic, tunable, or active metamaterials, the disclosed inverse problem solver 110 can be used to compute the electronic control stimuli needed to properly optimize the performance of the devices. [00149] In some aspects, the devices containing dynamic metamaterials may include beamforming and beam steering antennas, such as those used in radar, 5G wireless, next- generation satellite communications, microwave or millimeter-wave imaging systems, machine vision, wireless power beaming, radio frequency jammers, among others. The devices containing dynamic metamaterials may also include optical devices such as lidar, free-space optical communications, fiber-optical communications, and directed energy weapons, among others. The devices containing dynamic metamaterials may include acoustic devices, such as directive sound speakers, directive microphones, ultrasound imaging, and ultrasonic haptics, among others. [00150] FIG. 5 shows an example of computing system 500, which can be for example any computing device making up any of the entities illustrated in FIG.1, for example, the computing device 108, or any component thereof in which the components of the system are in communication with each other using connection 505. Connection 505 can be a physical connection via a bus, or a direct connection into processor 510, such as in a chipset architecture. Connection 505 can also be a virtual, networked connection, or logical connection. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [00151] In some embodiments, computing system 500 is a distributed system in which the functions described in this disclosure can be distributed within a data center, multiple data centers, a peer network, etc. In some embodiments, one or more of the described system components represents many such components each performing some or all of the function for which the component is described. In some embodiments, the components can be physical or virtual devices. [00152] An example system 500 includes at least one processing unit (CPU or processor) 510 and connection 505 that couples various system components including system memory 515, such as read-only memory (ROM) 520 and random-access memory (RAM) 525 to processor 510. Computing system 500 can include a cache of high-speed memory 512 connected directly with, close to, or integrated as part of processor 510. [00153] Processor 510 can include any general-purpose processor and a hardware service or software service, such as services 532, 534, and 536 stored in storage device 530, configured to control processor 510 as well as a special-purpose processor where software instructions are incorporated into the actual processor design. Processor 510 may essentially be a completely self-contained computing system, containing multiple cores or processors, a bus, memory controller, cache, etc. A multi-core processor may be symmetric or asymmetric. [00154] To enable user interaction, computing system 500 includes an input device 545, which can represent any number of input mechanisms, such as a microphone for speech, a touch- sensitive screen for gesture or graphical input, keyboard, mouse, motion input, speech, etc. Computing system 500 can also include output device 535, which can be one or more of many output mechanisms known to those of skill in the art. In some instances, multimodal systems can enable a user to provide multiple types of input/output to communicate with computing system 500. Computing system 500 can include communications interface 540, which can generally govern and manage the user input and system output. There is no restriction on operating on any particular hardware arrangement, and therefore the basic features here may easily be substituted for improved hardware or firmware arrangements as they are developed. [00155] Storage device 530 can be a non-volatile memory device and can be a hard disk or other types of computer-readable media which can store data that are accessible by a computer, such as magnetic cassettes, flash memory cards, solid-state memory devices, digital versatile disks, cartridges, random-access memories (RAMs), read-only memory (ROM), and/or some combination of these devices. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [00156] The storage device 530 can include software services, servers, services, etc., and when the code that defines such software is executed by the processor 510, it causes (instructs) the system to perform a function. In some embodiments, a hardware service that performs a particular function can include the software component stored in a computer-readable medium in connection with the various hardware components, such as processor 510, connection 505, output device 535, etc., to carry out the function. [00157] For clarity of explanation, in some instances, the present technology may be presented as including individual functional blocks including functional blocks comprising devices, device components, steps or routines in a method embodied in software, or combinations of hardware and software. [00158] Any of the steps, operations, functions, or processes described herein may be performed or implemented by a combination of hardware and software services or services, alone or in combination with other devices. In some embodiments, a service can be software that resides in the memory of a client device and/or one or more servers of a content management system and perform one or more functions when a processor executes the software associated with the service. In some embodiments, a service is a program or a collection of programs that carry out a specific function. In some embodiments, a service can be considered a server. The memory can be a non-transitory computer-readable medium. [00159] In some embodiments, the computer-readable storage devices, mediums, and memories can include a cable or wireless signal containing a bitstream and the like. However, when mentioned, non-transitory computer-readable storage media expressly exclude media such as energy, carrier signals, electromagnetic waves, and signals per se. [00160] Methods according to the above-described examples can be implemented using computer-executable instructions that are stored or otherwise available from computer-readable media. Such instructions can comprise, for example, instructions and data which instruct or otherwise configure a general-purpose computer, special purpose computer, or special purpose processing device to perform a certain function or group of functions. Portions of computer resources used can be accessible over a network. The executable computer instructions may be, for example, binaries, intermediate format instructions such as assembly language, firmware, or source code. Examples of computer-readable media that may be used to store instructions, information used, and/or information created during methods according to described examples include magnetic or optical disks, solid-state memory devices, flash memory, and USB devices provided with non-volatile memory, networked storage devices, and so on. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [00161] Devices implementing methods according to these disclosures can comprise hardware, firmware, and/or software, and can take any of a variety of form factors. Typical examples of such form factors include servers, laptops, smartphones, small form factor personal computers, personal digital assistants, and so on. The functionality described herein also can be embodied in peripherals or add-in cards. Such functionality can also be implemented on a circuit board among different chips or different processes executing in a single device, by way of further example. [00162] The instructions, media for conveying such instructions, computing resources for executing them, and other structures for supporting such computing resources are means for providing the functions described in these disclosures. EXAMPLES [00163] The following examples are for illustration purposes only. It will be apparent to those skilled in the art that many modifications, both to materials and methods, may be practiced without departing from the scope of the disclosure. [0100] An example environment 600 for providing engineered frequency dispersion in manipulating wave fields through a dynamic wave-processing device including a static metamaterial is depicted in FIG. 6. The example environment 600 includes a dynamic wave- processing device 602. The dynamic wave-processing device 602 functions to manipulate wave fields to create an output field profile 604. Specifically and as will be discussed in greater detail later, the dynamic wave-processing device 602 can manipulate input waves to generate the output field profile 604. The output field profile 604 can be formed by manipulating a single wave or a plurality of waves. Further, while only a single output field profile 604 is shown in the example environment 600 in FIG.6, the dynamic wave-processing device 602 can generate multiple different output field profiles, e.g., either simultaneously or in sequence. [0101] In manipulating a wave field to create the output field profile 604, the dynamic wave- processing device 602 can be an acoustic wave-processing device. Specifically, the dynamic wave-processing device 602 can function as an acoustic wave-processing device by manipulating acoustic waves to create an acoustic output field profile. Further, in manipulating the wave field to create the output field profile 604, the dynamic wave-processing device 602 can be an electromagnetic wave-processing device. Specifically, the dynamic wave-processing device 602 can function as an electromagnetic wave-processing device by manipulating electromagnetic waves to create an electromagnetic output field profile. Additionally, in PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) manipulating wave field(s) to create the output field profile 604, the dynamic wave-processing device 602 can be an optical wave-processing device. Specifically, the dynamic wave- processing device 602 can function as an optical wave-processing device by manipulating optical waves to create an optical output field profile. [0102] The dynamic wave-processing device 602 includes an artificially structured material 606. The artificially structured material 606 can include an applicable material whose electromagnetic or acoustic properties are derived from their structural configurations, such as the previously described artificially structured materials. Specifically, the artificially structured material 606 can include a metamaterial. While reference is made throughout this disclosure to a dynamic wave-processing device that includes a metamaterial, the systems and methods described herein can be implemented using a dynamic wave-processing device that includes an artificially structured material. Further, while reference is made throughout this disclosure to a metamaterial, the metamaterial can include one type of metamaterial or a plurality of different types of metamaterials. Additionally, while reference is made throughout this disclosure to a metamaterial, as will be discussed in greater detail later, the metamaterial can include two separable parts, e.g., two separable metamaterial components or different types of metamaterials that are configured to perform separate operations during the operation of the dynamic-wave processing device 602. [0103] The artificially-structure material 606 of the dynamic wave-processing device 602 includes a static structure or static metamaterial 608. The static structure 608 can include at least a portion of the artificially structured material 606 that does not change during operation of the dynamic wave-processing device 602. Specifically, the static structure 608 can include a plurality of elements that do not change from a structural perspective during the operation of the dynamic wave-processing device 602. An element of the artificially structured material 606, as used herein, can include a micro-structured element of a plurality of micro-structured elements that are arranged to form the artificially structured material 606. For example, the static structure 608 can include a three-dimensional volumetric arrangement of micro-structured elements that do not change, with respect to the volumetric arrangement of the elements, during the operation of the dynamic wave-processing device. Further, the static structure 608 can include a plurality of elements that are not tuned or otherwise remain unchanged, from an element perspective, during the operation of the dynamic wave-processing device 602. For example, the static structure 608 can include a plurality of micro-structured elements whose electromagnetic or acoustic properties remain static during the operation of the dynamic wave- PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) processing device 602. In another example, the static structure 608 can include a plurality of elements coupled to power sources that remain unchanged during the operation of the dynamic wave-processing device 602. [0104] The static structure 608 in the artificially structured material 606 can function to manipulate wave fields to create the output field profile 604. Specifically, the static structure 608 can manipulate input waves to generate the output field profile 604. The static structure 608 of the artificially structured material 606 can function to generate, at least in part, a plurality of different output field profiles through the operation of the dynamic wave-processing device 602. The static structure 608 can generate a plurality of different output field profiles while a structure of elements forming the static structure 608 remains unchanged. For example, the static structure 608 can be configured to create a plurality of different output field profiles while the elements forming the static structure 608 remain stationary during the operation of the dynamic wave-processing device 602. Further, the static structure 608 can be configured to create a plurality of different output field profiles while individual elements of the static structure 608 remain unchanged or are otherwise not tuned during the operation of the dynamic wave-processing device 602. For example, the static structure 608 can be configured to generate a plurality of output field profiles while electromagnetic and/or acoustic properties of elements of the static structure 608 remain unchanged during the operation of the dynamic wave- processing device 602. [0105] In utilizing the static structure 608 to generate different output field profiles, the dynamic wave-processing device 602 can generate the different field profiles using a few or as little as one electronic component. Examples of electronic components include wave sources, e.g., transducers, switches, time-delay lines, and phase shifters. Specifically, as the static structure 608 can remain static in creating the different output field profiles, the dynamic wave- processing device 602 can generate the different output field profiles using fewer electronic components, e.g., as little as one electronic component, integrated as part of the dynamic wave- processing device 602. Fewer electronic components, when discussed with respect to the dynamic wave-processing device 602, can include fewer electronic components when compared to devices that tune individual artificially structured material elements to create various output field profiles. In turn, this can simplify control of the dynamic wave-processing device 602 in creating various output field profiles. Additionally, this can result in faster switching between the different output field profiles during the operation of the dynamic wave-processing device. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) [0106] The static structure 608 can be configured to enable a specific set of functional parameters at the dynamic wave-processing device 602 to create a plurality of output field profiles, including the output field profile 604. Functional parameters at the dynamic wave- processing device 602 include applicable parameters related to the functioning of the dynamic wave-processing device 602 in manipulating wave fields to generate output field profiles. An output field profile can be specific to one or more functional parameters and corresponding values of the one or more functional parameters. In turn, when the one or more functional parameters and the corresponding values of the parameters are enabled at the dynamic wave- processing device 602, e.g., through the static structure 608, then the dynamic wave-processing device 602 can reproduce the output field profile. For example, if a specific focal length is necessary for recreating a specific output field profile and the specific focal length is enabled at the dynamic wave-processing device 602 through the static structure 608, then the dynamic wave-processing device 602 is capable of manipulating an input wave field to generate the specific output field profile. [0107] The dynamic wave-processing device 602 can perform directional beamforming to generate one or more output field profiles based on the functional parameters enabled at the dynamic wave-processing device 602. Specifically, the dynamic wave-processing device 602 can manipulate one or more input wave fields to create a directive beam that is focused at either infinity or at a finite length to generate one or more output field profiles through directional beamforming. As will be discussed in greater detail later, the one or more output field profiles can be formed as a continuous trajectory, a quasi-continuous trajectory, or an unstructured point cloud. [0108] Functional parameters enabled at the dynamic wave-processing device 602 can correspond to one or more dimensions in space, e.g., a multidimensional space. Specifically, functional parameters enabled at the dynamic wave-processing device 602 through the static structure 608 can include a set of directions in either two dimensions or three dimensions. More specifically, the functional parameters can include a set of directions, in either two dimensions or three dimensions, for generating one or more output field profiles through directional beamforming. Further, parameters enabled at the dynamic wave-processing device 602 through the static structure 608 can include a set of focal lengths. Specifically, the functional parameters can include a set of focal lengths for generating one or more output field profiles through directional beamforming. Additionally, parameters enabled at the dynamic wave-processing device 602 through the static structure 608 can include one or more sets of direction and focal PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) length pairs. Specifically, the functional parameters can include a set of direction and focal length pairs for generating one or more output field profiles through directional beamforming. [0109] Further, functional parameters enabled at the dynamic wave-processing device 602 through the static structure 608 can span a multidimensional space formed across the functional parameters. Specifically, a functional parameter of a set of functional parameters enabled at the dynamic wave-processing device 602 can correspond to one or more dimensions in a multidimensional space formed by at least a portion of the set of functional parameters. For example, a first functional parameter can correspond to a focal length in a multidimensional space formed by a set of functional parameters enabled at the dynamic wave-processing device 602. Further, in the example, a second functional parameter can correspond to a direction in the multidimensional space formed by the set of functional parameters enabled at the dynamic wave-processing device 602. In another example, first and second functional parameters can correspond to first and second direction angles in a multidimensional space formed by a set of functional parameters enabled at the dynamic wave-processing device 602. [0110] Functional parameters can be simultaneously provided to the dynamic wave- processing device 602, e.g., through the static structure 608. A specific set of functional parameters can be provided to the dynamic wave-processing device 602 so that the functional parameters are dynamically enabled/retrievable/selectable at the dynamic wave-processing device 602 during the operation of the dynamic wave-processing device 602. More specifically, one or more functional parameters in the specific set of parameters can be dynamically retrieved at the dynamic wave-processing device to selectively generate one or more specific output field profiles during the operation of the dynamic wave-processing device 602. In being simultaneously provided to the dynamic wave-processing device 602, the functional parameters can each be achievable at the dynamic wave-processing device 602 during the operation of the wave-processing device 602. [0111] One or more sets of functional parameters can be dynamically enabled at the dynamic wave-processing device 602 through the static structure 608 as part of providing, e.g., simultaneously providing, the one or more sets of functional parameters to the dynamic wave- processing device 602. In being dynamically enabled, the sets of functional parameters can be selected/retrieved at the dynamic wave-processing device 602 during the operation of the dynamic wave-processing device 602 to generate specific output field profiles. Specifically, operations of the dynamic wave-processing device 602, e.g., through external sources, can be PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) controlled to select specific functional parameters of the functional parameters enabled at the dynamic wave-processing device 602 to generate specific output field profiles. [0112] Operations of the dynamic wave-processing device 602 can be controlled to iteratively select specific functional parameters at the dynamic wave-processing device 602, e.g., as part of dynamically selecting the specific functional parameters. In particular, the specific functional parameters can be iteratively selected to create one or more specific output field profiles through the dynamic wave-processing device 602, e.g., through directional beamforming. For example, an external source can be controlled to enable/select a first set of functional parameters at the dynamic wave-processing device 602 to generate a first output field profile. Further, in the example, the external source can be controlled to switch from the first set of functional parameters and enable/select a second set of functional parameters to generate a second output field profile. Still further in the example, both the first set of functional parameters and the second set of functional parameters can already be provided to the dynamic wave-processing device 602 before they are selected at the dynamic wave-processing device 602. [0113] Functional parameters enabled at the dynamic wave-processing device 602 can be specific to one or more operational frequencies of the dynamic wave-processing device 602, e.g., frequency-encoded at the dynamic wave-processing device 602. In particular, the dynamic wave-processing device 602 can be controlled to operate at a specific operational frequency corresponding to a specific set of functional parameters to select/retrieve/enable the specific set of functional parameters at the dynamic wave-processing device 602. One or more operational frequencies specific to one or more functional parameters of the dynamic wave-processing device 602 can be a subset of a plurality of operational frequencies achievable at the dynamic wave-processing device 602. In turn, the operational frequency of the dynamic wave-processing device 602 can be varied to select or otherwise dynamically enable different functional parameters/sets of functional parameters based on the operational frequencies corresponding to the functional parameters/sets of functional parameters. As follows, different output field profiles corresponding to the different functional parameters/sets of functional parameters can be generated by the dynamic wave-processing device 602, e.g., through directional beamforming, by varying the operating frequency of the wave-processing device 602. For example, the dynamic wave-processing device 602 can be controlled to operate at a first operational frequency corresponding to a first set of functional parameters to dynamically enable the first set of functional parameters at the dynamic wave-processing device 602. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) Further, in the example, the dynamic wave-processing device 602 can be controlled to switch to a second operational frequency corresponding to a second set of functional parameters to dynamically enable and switch to the second set of functional parameters at the dynamic wave- processing device 602. Still further in the example, different output field profiles corresponding to the first and second sets of functional parameters can be selectively output by the dynamic wave-processing device 602 by switching between the first and second operational frequencies of the dynamic wave-processing device 602 in response to varying the operational frequency. [0114] As an example of dynamically enabling functional parameters at the dynamic wave- processing device 602 according to an operational frequency of the dynamic wave-processing device 602, an oscillatory refractive index n2(f) can be implemented through the artificially structured material 606. Specifically, the artificially structured material 606 can include elements, e.g., micro-structured elements, having a series of resonant frequencies f1, f2, etc. More specifically, the artificially structured material 606 can include a first set of elements having a resonant frequency at or near f1 and a second set of elements having a resonant frequency at or near f2, etc. As follows, the artificially structured material 606 can provide a refractive index that oscillates as the operating frequency is advanced through each of the successive resonant frequencies f1, f2, etc. Further, the artificially structured material 606 can also implement a non-oscillatory refractive index n1(f) through elements having resonant frequencies f’ that are all above or below the set of resonant frequencies f1, f2, etc. Accordingly, n1 can change monotonically as the operating frequency is advanced through the set of resonant frequencies f1, f2, etc. While this example is discussed with respect to a single artificially structured material, a plurality of artificially structured materials can implement the example technique of dynamically enabling functional parameters according to operational frequency. [0115] An operational frequency of the dynamic wave-processing device 602 can be controlled or otherwise varied through an applicable technique for controlling an operational frequency of the dynamic wave-processing device 602. Specifically, the operational frequency of the dynamic wave-processing device 602 can be varied by varying a carrier frequency of illumination patterns incident on the dynamic wave-processing device 602. As follows, one or more functional parameters can be dynamically enabled at the dynamic wave-processing device 602 by varying the carrier frequency of illumination patterns incident on the dynamic wave- processing device 602. Specifically, the carrier frequency of the illumination patterns incident on the dynamic wave-processing device 602 can be varied to select/enable specific functional PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) parameters at the dynamic wave-processing device 602 and create one or more specific output field profiles. [0116] Additionally, functional parameters can be spatially encoded at the dynamic wave- processing device 602. Specifically, the functional parameters can be specific to incident illumination patterns based on one or more spatial relationships between the at least a portion of the artificially structured material 606 encoding the functional parameters and the incident illumination patterns. For example, a functional parameter can be specific to illumination patterns incident to the static structure 608 at a 45° angle. In being spatially encoded at the dynamic wave-processing device 602, the functional parameters can be selected, or otherwise dynamically enabled, by varying the spatial interactions of incident illumination patterns with respect to the dynamic wave-processing device 602. In turn, one or more output field profiles can be generated through the dynamic wave-processing device 602 by varying the spatial interactions of the incident illumination patterns at the dynamic wave-processing device 602. For example, an incidence angle of illumination patterns at the static structure 608 can be varied to dynamically enable different functional parameters at the dynamic wave-processing device 602. In turn, different output field profiles can be generated through the dynamic wave- processing device 602 by varying the incidence angle of the illumination patterns. [0117] Spatial interactions of incident illumination patterns at the dynamic wave-processing device 602 can be varied through an applicable technique. For example, the dynamic wave- processing device can be mechanically manipulated with respect to a source of incident illumination patterns to vary spatial interactions between the incident illumination patterns and the dynamic wave-processing device 602. In another example, the source of incident illumination patterns can be mechanically manipulated with respect to the dynamic wave- processing device 602 to vary the spatial interactions. Spatial interactions of incident illumination patterns at the dynamic wave-processing device 602 can be varied without individually controlling micro-structured elements of the static structure 608. For example, the entire static structure 608 can be rotated instead of rotating individual micro-structured elements of the static structure 608 to vary spatial interactions of incident illumination patterns at the dynamic wave-processing device 602. [0118] The example environment 600 shown in FIG. 1 includes field source(s) 610. The field source(s) 610 are configured to output one or more illumination patterns 612 incident on the dynamic wave-processing device 602. The incident illumination patterns 612 that are generated by the field source(s) 610 can include either or both acoustic field waves or PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) electromagnetic field waves with corresponding carrier frequencies. As follows, the operational frequency of the dynamic wave-processing device 602 can be modulated by varying the carrier frequencies of the incident illumination patterns 612 output by the field source(s) 610. Specifically and when the field source(s) 610 include a plurality of field sources, the carrier frequency of the incident illumination patterns 612 can be adjusted by selectively adjusting the amplitude of each of the plurality of field sources. Alternatively, the carrier frequency of the incident illumination patterns 612 can be adjusted by selectively adjusting a phase of each of the field sources of the plurality of field sources. Additionally, the carrier frequency of the incident illumination patterns 612 can be adjusted by selectively adjusting both a phase and an amplitude of each of the plurality of field sources. Alternatively, when the field source(s) 610 are a single field source, then the single field source can be selectively controlled to modulate the carrier frequency of the incident illumination patterns 612. [0119] The output field profile 604 created by the dynamic wave-processing device 602 can be an unstructured point cloud of radiation points. Specifically, the radiation points forming the output field profile 604 can be created by dynamically enabling specific functional parameters at the dynamic wave-processing device 602. More specifically, the operational frequency of the dynamic wave-processing device 602 can be varied to enable specific functional parameters at the dynamic wave-processing device 602 and generate the radiation points forming the unstructured point cloud. For example, the carrier frequency of the incident illumination patterns 612 can be adjusted to enable specific functional parameters at the dynamic wave- processing device 602 and form the radiation points in the unstructured point cloud. Further, spatial interactions of the incident illumination patterns 612 with the dynamic wave-processing device 602 can be adjusted to enable specific functional parameters at the dynamic wave- processing device 602 and form the radiation points in the unstructured point cloud. [0120] Additionally, the output field profile 604 created by the dynamic wave-processing device 602 can be a quasi-continuous trajectory of radiation points. A quasi-continuous trajectory of radiation points can include a plurality of radiation points that in combination cover a portion of the total trajectory. For example, a quasi-continuous trajectory of radiation points can include a plurality of radiation points spaced by 1 mm along a trajectory to form a quasi- continuous trajectory. The radiation points forming the quasi-continuous trajectory of the output field profile 604 can be created by dynamically enabling specific functional parameters at the dynamic wave-processing device 602. More specifically, the operational frequency of the dynamic wave-processing device 602 can be varied to enable specific functional parameters PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) at the dynamic wave-processing device 602 and generate the radiation points forming the quasi- continuous trajectory. For example, the carrier frequency of the incident illumination patterns 612 can be adjusted to enable specific functional parameters at the dynamic wave-processing device 602 and form the radiation points in the quasi-continuous trajectory. Further, spatial interactions of the incident illumination patterns 612 with the dynamic wave-processing device 602 can be adjusted to enable specific functional parameters at the dynamic wave-processing device 602 and form the radiation points in the quasi-continuous trajectory. [0121] A quasi-continuous trajectory of radiation points formed by the dynamic wave- processing device 602 can be formed through a meanderline sweeping one or more specific solid angles of a unit sphere. The meanderline can include a plurality of arms and spacing between consecutive arms can match an angle width of a directional beam controlled by the dynamic wave-processing device 602, e.g., through directional beamforming. Further, the quasi- continuous trajectory can be a spiral, e.g., formed by dynamically enabling functional parameters at the dynamic wave-processing device 602. Additionally, the quasi-continuous trajectory can be a Lissajous pattern, e.g., formed by e.g., formed by dynamically enabling functional parameters at the dynamic wave-processing device 602. Also, the quasi-continuous trajectory can be a flower pattern, e.g., formed by dynamically enabling functional parameters at the dynamic wave-processing device 602. [0122] The functional parameters enabled at the dynamic wave-processing device 602 can be enabled as holograms at the dynamic wave-processing device 602. Specifically, the functional parameters can be enabled as holograms at the artificially structured material 606, e.g., in the static structure 608 of the artificially structured material 606. As used herein, the term “hologram” refers to as a scattering and/or radiating medium, such as an artificially structured material, which generates a holographic projection when properly excited with a specific illumination pattern. The artificially structured material 606 can be multi-holographic and store multiple holograms for the functional parameters at the dynamic wave-processing device 602. [0123] Holograms stored at the dynamic wave-processing device 602 can be selectively retrieved to selectively enable functional parameters at the wave-processing device 602. Specifically, holograms stored at the dynamic wave-processing device 602 can be selectively retrieved to create one or more specific output field profiles, e.g., through directional beamforming. Holograms stored at the dynamic wave-processing device 602, e.g., at the artificially structured material 606, can be both frequency-encoded and spatially encoded. In PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) turn, the artificially structured material 606 can be either or both a spatially-encoded multi- holographic material and a frequency-encoded multi-holographic material. For example, holograms can be frequency-encoded at the dynamic wave-processing device 602 and retrieved, or otherwise dynamically enabled, by varying the operating frequency of the dynamic wave- processing device 602. In another example, holograms can be spatially-encoded at the dynamic wave-processing device 602 and retrieved, or otherwise dynamically enabled, by varying spatial interactions of incident illumination patterns with respect to the dynamic wave-processing device 602. [0124] A plurality of field sources can be utilized in retrieving holograms that are spatially encoded and holograms that are frequency-encoded at the dynamic wave-processing device 602. Specifically, a plurality of spatially localized field sources can be positioned at different locations with respect to the dynamic wave-processing device 602 to retrieve different spatially- encoded holograms. More specifically, the plurality of field sources, or switches, can provide a plurality of switchable illumination patterns. Each illumination pattern can retrieve different holograms stored at the dynamic wave-processing device 602 at different frequencies. This can increase the total number of independent holograms that can be dynamically enabled at the dynamic wave-processing device 602 to Nf-by-Ns, where Nf is the number of frequencies used and Ns is the number of spatially localized field sources or switches. [0125] Holograms stored at the dynamic wave-processing device 602 can be iteratively selected/retrieved to create one or more output field profiles. For example, holograms stored at the dynamic wave-processing device 602 can be iteratively selected to form corresponding radiation points in a trajectory of radiation points forming an output field profile. In another example, holograms stored at the dynamic wave-processing device 602 can be iteratively selected to form radiation points in an unstructured point cloud of radiation points forming an output field profile. [0126] Physical design parameters of the dynamic wave-processing device 602 can be determined using the techniques described in FIGs. 1-2, i.e., using the inverse problem solver 110, to provide/dynamically enable specific functional parameters at the dynamic wave- processing device 602. In particular, physical design parameters can be selected and implemented at the dynamic wave-processing device 602 to dynamically enable holograms corresponding to a specific set of functional parameters at the dynamic wave processing device 602. More specifically, physical design parameters can be selected to create holograms at the dynamic wave-processing device 602. In turn, the dynamic wave-processing device 602 can be PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) fabricated according to the selected physical design parameters to store, or otherwise dynamically enable, the holograms at the dynamic wave-processing device 602. As will be discussed in greater detail later, physical design parameters of the dynamic wave-processing device 602 can be selected according to one or more applicable techniques for selecting physical design parameters to dynamically enable one or more specific functional parameters at the dynamic wave-processing device 602. [0127] Physical design parameters can include applicable parameters of the dynamic wave- processing device 602 that remain unchanged during the operation of the dynamic wave- processing device 602. Specifically, physical design parameters at the dynamic wave- processing device 602 can include static design parameters of the static structure 608 of the artificially structured material 606 that remain unchanged during the operation of the dynamic wave-processing device 602. For example, physical design parameters of the dynamic wave- processing device 602 can include electromagnetic and/or acoustic characteristics of elements of the static structure 608 that remain unchanged during the operation of the dynamic wave- processing device 602. In another example, physical design parameters include sizes of elements of the static structure 608 and spacing between the elements of the static structure 608 that remain unchanged during the operation of the dynamic wave-processing device 602. In yet another example, physical design parameters can include locations of elements to form the static structure 608 of the artificially structured material 606. [0128] Further, physical design parameters can include adjustable control inputs for the dynamic wave-processing device 602. Specifically, physical design parameters can include adjustable control input for controlling the operation of the static structure 608 of the dynamic wave-processing device 602. In turn, the dynamic wave-processing device 602 can be controlled according to the adjustable control inputs defined by the physical design parameters to dynamically enable one or more functional parameters at the dynamic wave-processing device 602. Specifically, the dynamic wave-processing device 602 can be controlled according to the adjustable control inputs defined by the physical design parameters to store one or more holograms corresponding to one or more functional parameters at the dynamic wave-processing device 602. [0129] Physical design parameters can include design parameters of micro-structured elements forming the artificially structured material 606. Specifically, physical design parameters can include design parameters of micro-structured elements forming the static structure 608 of the artificially structured material 606. For example, the physical design PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) parameters of the micro-structured elements can define the physical locations of the micro- structured elements in forming the artificially structured material 606. The physical design parameters of the micro-structured elements forming the artificially structured material 606 can be selected and implemented to provide/dynamically enable specific functional parameters through the micro-structured elements. For example, micro-structured elements can be selectively positioned to enable one or more specific functional parameters at the artificially structured material 606. [0130] Two or more micro-structured elements forming the artificially structured material 606 can have different physical properties that distinguish the micro-structured elements from each other. Specifically, each of the micro-structured elements forming the artificially structured material 606 can have different physical properties that distinguish the micro- structured elements from each other. For example, the micro-structured elements can have different structural shapes that distinguish the micro-structured elements from each other. In another example, the micro-structured elements can have different electromagnetic and/or acoustic properties that distinguish the micro-structured elements from each other. [0131] Each of the micro-structured elements can correspond to a hologram of a plurality of holograms dynamically enabled at the dynamic wave-processing device 602. In turn, the operation of the dynamic wave-processing device 602 can be controlled to dynamically enable specific holograms corresponding to each of the micro-structured elements. Physical design parameters can be selected for each of the micro-structured elements to dynamically enable holograms corresponding to the micro-structured elements at the dynamic wave-processing device 602. For example, electromagnetic properties of a micro-structured element can be selected and implemented to dynamically enable a specific hologram through the micro- structured element at the dynamic wave-processing device 602. [0132] An example of a dynamic wave-processing device 700 including a dynamic metamaterial is depicted in FIG. 7. The dynamic wave-processing device 700 can function according to an applicable device for generating one or more specific output field profiles, such as the dynamic wave-processing device 602 in the example environment 600 shown in FIG.1. In particular, the dynamic wave-processing device 700 can include a plurality of functional parameters enabled at the dynamic wave-processing device 700. As follows, the functional parameters can be dynamically enabled/selected during the operation of the dynamic wave- processing device 700 to generate one or more specific output field profiles corresponding to the functional parameters. More specifically, the physical design parameters of the dynamic PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) wave-processing device 700 can be selected and implemented to enable the creation of one or more specific output field profiles through the dynamic wave-processing device 700. [0133] The dynamic wave-processing device 700 includes a first artificially structured material 702 and a second artificially structured material 704. Both the first and second artificially structured materials 702 and 704 can have functional parameters, e.g., different functional parameters, enabled at the artificially structured materials 702 and 704. Specifically, both the first and second artificially structured materials 702 and 704 can include corresponding static structures that dynamically enable specific functional parameters at the first and second artificially structured materials 702 and 704. More specifically, both the first and second artificially structured materials 702 and 704 can be formed through micro-structured elements that dynamically enable one or more functional parameters, e.g., store holograms, at the first and second artificially structured materials 702 and 704. [0134] Functional parameters can be enabled at the first and second artificially structured materials 702 and 704 based on physical design parameters for the dynamic wave-processing device 700, e.g., physical design parameters for the first and second artificially structured materials 702 and 704. Specifically, both the first and second artificially structured materials 702 and 704 can have engineered frequency dispersions based on selected physical design parameters to provide a specific set of functional parameters at the dynamic wave-processing device 700. For example, static electromagnetic and/or acoustic properties of micro-structured elements forming the first and second artificially structured materials 702 and 704 can be selected and implemented to provide a specific set of functional parameters at the dynamic wave-processing device 700. [0135] Additionally, functional parameters enabled at the first and second artificially structured materials or dynamic metamaterial or dynamic structure 702 and 704 can be selected, e.g., dynamically enabled, during the operation of the dynamic wave-processing device 700 through an applicable technique. Specifically, functional parameters enabled at the first and second artificially structured materials 702 and 704 can be selected by varying an operating frequency of the dynamic wave-processing device 700. For example, a carrier frequency of illumination patterns incident to the dynamic wave-processing device 700 can be varied to dynamically enable one or more functional parameters in either or both the first artificially structured material 702 and the second artificially structured material 704. Further, functional parameters enabled at the first and second artificially structured materials 702 and 704 can be PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) selected by varying spatial interactions of incident illumination patterns with either or both the first and second artificially structured materials 702 and 704. [0136] While the first and second artificially structured materials 702 and 704 are shown as conceptually separate in the dynamic wave-processing device 700, the first and second artificially structured materials 702 and 704 can be disposed within the dynamic wave- processing device 700 such that the materials 702 and 704 interact with each other. Specifically, both the first and second artificially structured materials 702 and 704 can be formed by two layers of materials that are disposed within the dynamic wave-processing device 700 to form stacked artificially structured material layers. In interacting with each other, the first and second artificially structured materials 702 and 704 can function together to process one or more wave fields and generate one or more specific output field profiles. In particular, specific functional parameters can be dynamically enabled at the first and second artificially structured materials 702 and 704 to generate one or more specific output field profiles from one or more wave fields interacting with both the first and second artificially structured materials 702 and 704. [0137] The first and second artificially structured materials 702 and 704 can be implemented as separate prisms, e.g., prism layers. Specifically, the first and second artificially structured materials 702 and 704 can be implemented as prisms having different refractive characteristics. For example, the first artificially structured material 702 can be a metamaterial prism with a spatially-uniform effective refractive index n1(f). n1(f) can be a monotonic function of frequency, gradually increasing (or decreasing) over an entire allocated frequency band, e.g., a frequency band associated with the dynamic wave-processing device 700. Further, in the example, the second artificially structured material 704 can be a different metamaterial prism with a spatially-uniform effective refractive index n2(f). n2(f) can be an oscillatory (but not necessarily periodic) function of frequency, having multiple maxima and minima over the entire allocated frequency band. Still further in the example, the first and second artificially structured materials 702 and 704 can provide angular deflection in two orthogonal planes. As a result, quasi-continuous beam steering can be achieved at the dynamic wave-processing device with nearly full control over the two angular degrees of freedom of the beam. EQUIVALENTS [00164] Those skilled in the art will recognize, or be able to ascertain, using no more than routine experimentation, numerous equivalents to the specific embodiments described PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) specifically herein. Such equivalents are intended to be encompassed in the scope of the following claims.

Claims

PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) CLAIMS What is claimed is: 1. A system for designing a device containing a metamaterial structure, the system comprising: one or more processors; and a computer-readable medium comprising instructions stored therein, which, when executed by the one or more processors, instruct the one or more processors to: access a numerical representation of a geometry of a device and a region of the device containing a static metamaterial structure under design; generate a discretized representation of linear partial differential equations (PDEs) describing a field interaction with the device for a set of values of a parameter vector sufficient to identify parameter-dependent and parameter- independent components of a linear system matrix, wherein a discretized representation x is in a form of a linear system matrix A(p) and at least one source vector b; identify at least one quantitative figure of merits that corresponds to a desirable performance characteristic of the device; form an objective function ^^⁽ ^^, ^^⁾ that evaluates, for the parameter vector ^^, at least one figure of merit from at least one vector of fields, wherein the vector(s) of fields x is a solution to the discretized representation of linear partial differential equations in a form A(p) x = b; select a set of initial estimates of the parameter vector of the static metamaterial structure for generating a target output field pattern; transform the set of initial estimates of a parameter vector p into a set of reduced parameter vectors p’ that is reduced with respect to an original parameter space containing the initial estimate of the parameter vector; apply an optimization algorithm to the objective function working in a reduced-dimension space of reduced parameter vectors, with the set of initial estimates of the reduced parameter vectors ^^^′ to generate a set of compact refined estimates of the reduced parameter vector; transform a compact refined estimate of the reduced parameter vector p’ to the original parameter space containing the initial estimate of the parameter vector to generate a refined estimate of the parameter vector; PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) selectively identify one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector; and record or transmit a numerical representation of the one or more design characteristics. 2. The system of claim 1, wherein the computer-readable medium comprising instructions stored therein, which, when executed by the one or more processors, instruct the one or more processors to decompose, based on either or both the parameter-dependent components and the parameter-independent components of the linear system matrix, an original vector space of discretized fields of the discretized representation into a subspace ^^ ( ^^) representing a new reduced-dimension vector space and another subspace complementary to the new reduced- dimension vector space. 3. The system of claim 1, wherein the computer-readable medium comprises instructions stored therein, which, when executed by the one or more processors, instruct the one or more processors to find the vector of fields by solving the linear problem in the reduced-dimension vector space and then transforming the solution back into the original vector space. 4. The system of claim 1, wherein the computer-readable medium comprises instructions stored therein, which, when executed by the one or more processors, instruct the one or more processors to produce the solution to the discretized representation of linear PDEs in a form ^^( ^^) ^^ = ^^ by: finding the basis in the field vector space which, together with its complementary basis, decomposes the field vector space into two subspaces such that the transformed linear system matrix A = U A U^† , restricted onto the second subspace, is independent of the parameter vector p; solving a reduced-dimension linear system in the first subspace for a given value of the parameter vector p; and using a precomputed matrix factorization of the transformed linear system matrix restricted onto the second subspace, which is independent of the parameter vector. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 5. The system of claim 4, wherein the finding of the basis is performed using a singular value decomposition of linear system matrix that is used to identify the basis that maximizes the dimension of the second subspace, in which the linear system matrix is independent of the parameter vector. 6. The system of claim 5, wherein the computer-readable medium comprises instructions stored therein, which, when executed by the one or more processors, instruct the one or more processors to compute the singular value decomposition of the complex-valued linear system matrix and the real-valued basis vectors of the corresponding singular value basis as the real- valued singular value decomposition of a rectangular, real-valued matrix of size N-by- (N*2*N_p), where N is the dimension of the linear system, and N_p is the dimension of the original parameter vector (p), which is formed by first decomposing Np complex-valued square matrices A(p_n), where p_n is the n-th component of the parameter vector, into 2* N_p real-valued square matrices, and then stacking them to form a rectangular matrix. 7. The system of claim 4, wherein the computer-readable medium comprises instructions stored therein, which, when executed by the one or more processors, instruct the one or more processors to use the optimization algorithm to construct a surrogate objective function based on previously computed values of the objective function and optimizes the surrogate objective function. 8. The system of claim 7, wherein the computer-readable medium comprises instructions stored therein, which, when executed by the one or more processors, instruct the one or more processors to construct the surrogate objective function using the radial basis function method. 9. The system of claim 8, wherein the computer-readable medium comprises instructions stored therein, which, when executed by the one or more processors, instruct the one or more processors to construct the surrogate objective function using a surrogate model trained by a machine learning algorithm using previously computed values of the objective function. 10. The system of claim 9, wherein the machine-learning algorithm is one of the following: polynomial response surfaces, kriging, generalized Bayesian approaches, gradient-enhanced kriging (GEK), radial basis function, support vector machines, space mapping, artificial neural PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) networks, deep neural networks, Bayesian networks, Fourier surrogate modeling, or random forests. 11. The system of claim 1, wherein the objective function contains a reduced-dimension linear system matrix, a reduced-dimension source vector, and matrices for converting a reduced- dimension vector of fields to the original vector space. 12. The system of claim 1, wherein the vector of fields is found by solving the linear problem in the reduced-dimension vector space and then transforming the solution back into the original vector space. 13. The system of claim 1, wherein the objective function generates a linear system matrix and a source vector for a specific value of the set of values of the parameter vector. 14. The system of claim 1, wherein the objective function generates the linear system matrix and the source vector for a given value of the parameter vector based on an algorithm that generates the discretized representation of the linear partial differential equations describing the field interaction with the device including the static metamaterial structure. 15. The system of claim 1, wherein: the objective function generates a linear system matrix and a source vector for a given value of a parameter vector based on an evaluation of a component of a power series expansion of the linear system matrix and the source vector expanded as functions of the parameter vector at a predefined initial value of the parameter vector, wherein parameters of the power series expansion are computed during the evaluation of the objective function and reusing the factorization of the linear system matrix computed for the evaluation of the evaluation function; and the optimization algorithm uses the parameters of the power series expansion to reduce the number of new computations of the objective function at new values of the parameter vector. 16. The system of claim 1, wherein the one or more processors transform the initial estimate of the parameter vector into the reduced parameter vector agnostic as to any figure of merit, and PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) the same transformation is used to optimize multiple figures of merit. 17. The system of claim 1, wherein the one or more processors transform the initial estimate of the parameter vector into the reduced parameter vector by projecting the initial estimate of the parameter vector onto a linear subspace that spans a reduced number of singular vectors in comparison to the parameter vector. 18. The system of claim 3, wherein the optimization algorithm comprises a quadratic problem algorithm. 19. The system of claim 3, wherein the optimization algorithm comprises a sequential quadratic problem algorithm. 20. The system of claim 3, wherein the optimization algorithm comprises a gradient-assisted optimization algorithm, and the objective function further includes an algorithm to compute the gradient using a first order adjoint algorithm. 21. The system of claim 20, wherein the optimization algorithm comprises a Hessian- assisted optimization algorithm, and the objective function further includes an algorithm to compute the Hessian using a second order adjoint algorithm. 22. The system of claim 20, wherein the optimization algorithm comprises a Hessian-assisted optimization algorithm, and the objective function further includes an algorithm to compute the projection of the Hessian onto a set of parameter vectors, as needed for the optimization algorithm, using a second order adjoint algorithm. 23. The system of claim 1, wherein the one or more processors: evaluate the refined estimate of the parameter vector and corresponding partial derivate constraints of figures of merit associated with the static metamaterial structure in relation to a convergence criterion; and identify the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector in response to an evaluation that the refined PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) estimate of the parameter vector and the corresponding partial derivative constraints meeting the convergence criterion. 24. The system of claim 23, wherein in response to a determination that the refined estimate of the parameter vector and the corresponding partial derivative constraints fail to meet the convergence criterion, iteratively, the one or more processors: determine a new initial estimate of the parameter vector; transform the new initial estimate of the parameter vector into a new reduced parameter vector with respect to an original dimension space of the new initial estimate of the parameter vector; apply the local optimization to the new reduced parameter vector to generate a new compact refined estimate of the parameter vector in a reduced dimension space of the new reduced parameter vector; transform the new compact refined estimate of the parameter vector to the original dimension space of the new initial estimate of the parameter vector to generate a new refined estimate of the parameter vector; and selectively identify the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the new refined estimate of the parameter vector. 25. The system of claim 1, wherein the one or more processors: determine a field associated with the static metamaterial structure generating the target output field pattern for the parameter vector of the static metamaterial structure; transform the field into a reduced field with respect to an original dimension space of the field; and selectively identify the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on both the refined estimate of the parameter vector and the reduced field. 26. The system of claim 1, wherein a size of the original dimensions space of the initial estimate of the parameter vector is a number of figures of merit (FOM) associated with the static metamaterial structure in generating the target output field pattern. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 27. The system of claim 1, wherein the static metamaterial structure comprises a static metamaterial. 28. The system of claim 27, wherein the parameter vector comprises parameters related to a static structure design and/or one or more material characteristics of the static metamaterial. 29. A system for designing a device containing a metamaterial structure, the system comprising: one or more processors; and a computer-readable medium comprising instructions stored therein, which, when executed by the one or more processors, instruct the one or more processors to: access a numerical representation of a geometry of the device and a region of the device containing a static metamaterial structure under design; generate a discretized representation of linear partial differential equations describing a field interaction with the device for a set of values of a parameter vector sufficient to identify parameter-dependent and parameter- independent components of a linear system matrix, wherein a discretized representation x is in a form of a linear system matrix A(p) and at least one source vector b; decompose, based on either or both the parameter-dependent components and the parameter-independent components of the linear system matrix, an original vector space of discretized fields of the discretized representation into a subspace ^^ ( ^^) representing a new reduced-dimension vector space and another subspace complementary to the new reduced-dimension vector space; identify at least one quantitative figure of merits that corresponds to a desirable performance characteristic of the device; form an objective function ^^⁽ ^^, ^^⁾ that evaluates, for the parameter vector ^^, at least one figure of merit from at least one vector of fields, wherein the vector(s) of fields x is a solution to the discretized representation of linear partial differential equations in a form A(p) x = b, wherein said linear system is solved using a decomposition of field vector space into the reduced-dimension vector space ^^ ( ^^) and the vector space complementary to ^^ ( ^^); PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) select a set of initial estimates of the parameter vector of the static metamaterial structure for generating a target output field pattern; apply an optimization algorithm to the objective function ^^⁽ ^^, ^^⁾ to generate a set of refined estimates of the parameter vector; selectively identify one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector; and record or transmit a numerical representation of the one or more design characteristics. 30. A system for designing a device containing a metamaterial structure, the system comprising: one or more processors; and a computer-readable medium comprising instructions stored therein, which, when executed by the one or more processors, instruct the one or more processors to: access a numerical representation of a geometry of the device and a region of the device containing a static metamaterial structure under design; generate a discretized representation of linear partial differential equations (PDEs) describing a field interaction with the device for a set of values of a parameter vector sufficient to identify parameter-dependent and parameter- independent components of a linear system matrix, wherein a discretized representation x is in a form of a linear system matrix A(p) and at least one source vector b, wherein the solution to the discretized representation of linear PDEs in a form ^^( ^^) ^^ = ^^ is produced by: finding the basis in the field vector space which, together with its complementary basis, decomposes the field vector space into two subspaces such that the transformed linear system matrix Ã=U A U^† , restricted onto the second subspace, is independent of the parameter vector p; solving a reduced-dimension linear system in the first subspace for a given value of the parameter vector p; and using a precomputed matrix factorization of the PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) transformed linear system matrix restricted onto the second subspace, which is independent of the parameter vector; identify at least one quantitative figure of merits that corresponds to a desirable performance characteristic of the device; form an objective function ^^⁽ ^^, ^^⁾ that evaluates, for the parameter vector ^^, at least one figure of merit from the at least one vector of fields, wherein the vector(s) of fields x is a solution to the discretized representation of linear partial differential equations in a form A(p) x = b; select a set of initial estimates of the parameter vector of the static metamaterial structure for generating a target output field pattern; applying an optimization algorithm to the objective function working in the reduced-dimension space of reduced parameter vectors, with the set of initial values of the reduced parameter vectors to generate a set of compact refined estimates of the parameter vector; identify one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector; and record or transmit a numerical representation of the one or more design characteristics. 31. A method for designing a metamaterial structure, comprising: accessing a numerical representation of a device’s geometry and a region of the device containing a static metamaterial structure under design; generating a discretized representation of linear partial differential equations describing a field interaction with the device for a set of values of a parameter vector sufficient to identify parameter-dependent and parameter-independent components of a linear system matrix, wherein the discretized representation is in a form of a linear system matrix A(p) and at least one source vector b; identifying at least one quantitative figure of merits that corresponds to a desirable performance characteristic of the device; forming an objective function that evaluates, for the parameter vector, at least one figure of merit from a vector of fields from the at least one vector of fields, wherein the vector(s) of fields x is a solution to the discretized representation of linear partial PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) differential equations in a form A(p) x = b; selecting a set of initial values of the parameter vector of the static metamaterial structure for generating a target output field pattern; transforming the set of initial values of the parameter vector into a set of reduced parameter vectors that is reduced with respect to an original dimension of a parameter space containing the initial values of the parameter vector; applying an optimization algorithm to the objective function working in the reduced-dimension space of reduced parameter vectors, with the set of initial values of the reduced parameter vectors to generate a set of compact refined estimates of the parameter vector; transforming the compact refined estimate of the parameter vector to the original parameter space containing the initial values of the parameter vector to generate a refined estimate of the parameter vector; selectively identifying one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector; and recording a numerical representation of the one or more design characteristics for designing the metamaterial structure. 32. The method of claim 31, wherein the vector of fields is found by solving the linear problem in the reduced-dimension vector space and then transforming the solution back into the original vector space. 33. The method of claim 31, wherein the solution to the discretized representation of linear PDEs in a form A(p)x=b is produced by: finding the basis in the field vector space which, together with its complementary basis, decomposes the field vector space into two subspaces such that the transformed linear system matrix A = U A U^† , restricted onto the second subspace, is independent of the parameter vector p; solving a reduced-dimension linear system in the first subspace for a given value of the parameter vector p; and using a precomputed matrix factorization of the transformed linear system matrix restricted onto the second subspace, which is independent of the parameter vector. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 34. The method of claim 32, wherein the finding of the basis is performed using a singular value decomposition of linear system matrix that is used to identify the basis that maximizes the dimension of the second subspace, in which the linear system matrix is independent of the parameter vector. 35. The method of claim 33, wherein the singular value decomposition of the complex- valued linear system matrix and the real-valued basis vectors of the corresponding singular value basis are computed as the real-valued singular value decomposition of a rectangular, real- valued matrix of size N-by-(N*2*Np), where N is the dimension of the linear system, and Np is the dimension of the original parameter vector (p), which is formed by first decomposing N_p complex-valued square matrices A(pn), where pn is the n-th component of the parameter vector, into 2* N_p real-valued square matrices, and then stacking them to form a rectangular matrix. 36. The method of claim 31, wherein the optimization algorithm constructs a surrogate objective function based on previously computed values of the objective function and optimizes the surrogate objective function. 37. The method of claim 35, wherein the surrogate objective function is constructed using the radial basis function method. 38. The method of claim 36, wherein the surrogate objective function is constructed using a surrogate model trained by a machine learning algorithm using previously computed values of the objective function. 39. The method of claim 37, wherein the machine-learning algorithm is one of the following: polynomial response surfaces, kriging, generalized Bayesian approaches; gradient-enhanced kriging (GEK), radial basis function, support vector machines, space mapping; artificial neural networks, Bayesian networks, Fourier surrogate modeling, or random forests. 40. The method of claim 31, wherein the objective function contains a reduced-dimension linear system matrix, a reduced-dimension source vector, and matrices for converting a reduced- dimension vector of fields to the original vector space. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 41. The method of claim 31, wherein the vector of fields is found by solving the linear problem in the reduced-dimension vector space and then transforming the solution back into the original vector space. 42. The method of claim 31, wherein the objective function generates a linear system matrix and a source vector for a specific value of the set of values of the parameter vector. 43. The method of claim 31, wherein the objective function generates the linear system matrix and the source vector for a given value of the parameter vector based on an algorithm that generates the discretized representation of the linear partial differential equations describing the field interaction with the device including the static metamaterial structure. 44. The method of claim 31, wherein: the objective function generates a linear system matrix and a source vector for a given value of a parameter vector based on an evaluation of a component of a power series expansion of the linear system matrix and the source vector expanded as functions of the parameter vector at a predefined initial value of the parameter vector, wherein parameters of the power series expansion are computed during the evaluation of the objective function and reusing the factorization of the linear system matrix computed for the evaluation of the evaluation function; and the optimization algorithm uses the parameters of the power series expansion to reduce the number of new computations of the objective function at new values of the parameter vector. 45. The method of claim 31, wherein the initial estimate of the parameter vector is transformed into the reduced parameter vector agnostic as to the at least one figure of merit. 46. The method of claim 31, wherein the initial estimate of the parameter vector is transformed into the reduced parameter vector by projecting the initial estimate of the parameter vector onto a linear subspace that spans a reduced number of singular vectors in comparison to the parameter vector. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 47. The method of claim 31, wherein the optimization algorithm comprises a quadratic problem algorithm. 48. The method of claim 31, wherein the optimization algorithm comprises a sequential quadratic problem algorithm. 49. The method of claim 31, wherein the optimization algorithm comprises a gradient- assisted optimization algorithm, and the objective function further includes an algorithm to compute the gradient using a first order adjoint algorithm. 50. The method of claim 48, wherein the optimization algorithm comprises a Hessian- assisted optimization algorithm, and the objective function further includes an algorithm to compute the Hessian using a second order adjoint algorithm. 51. The method of claim 48, wherein the optimization algorithm comprises a Hessian- assisted optimization algorithm, and the objective function further includes an algorithm to compute the projection of the Hessian onto a set of parameter vectors, as needed for the optimization algorithm, using a second order adjoint algorithm. 52. The method of claim 31, further comprising decomposing, based on either or both the parameter-dependent components and the parameter-independent components of the linear system matrix, an original vector space of discretized fields of the discretized representation into a subspace representing a new reduced-dimension vector space and another subspace complementary to the new reduced-dimension vector space. 53. The method of claim 31, further comprising: evaluating the refined estimate of the parameter vector and corresponding partial derivate constraints of figures of merit associated with the static metamaterial structure in relation to a convergence criterion; and identifying the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector in response to an evaluation that the refined estimate of the parameter vector and the corresponding partial derivative constraints meeting the convergence PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) criterion. 54. The method of claim 53, wherein, in response to a determination that the refined estimate of the parameter vector and the corresponding partial derivative constraints fail to meet the convergence criterion, the method comprises: determining a new initial estimate of the parameter vector; transforming the new initial estimate of the parameter vector into a new reduced parameter vector with respect to an original dimension space of the new initial estimate of the parameter vector; applying the local optimization to the new reduced parameter vector to generate a new compact refined estimate of the new reduced parameter vector in a reduced dimension space; transforming the new compact refined estimate of the new reduced parameter vector to the original dimension space of the new initial estimate of the parameter vector to generate a new refined estimate of the parameter vector; and selectively identifying the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the new refined estimate of the parameter vector. 55. The method of claim 31, further comprising: determining a field associated with the static metamaterial structure generating the target output field pattern for the parameter vector of the static metamaterial structure; transforming the field into a reduced field with respect to an original dimension space of the field; and selectively identifying the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on both the refined estimate of the parameter vector and the reduced field. 56. The method of claim 31, wherein a size of the original dimensions space of the initial estimate of the parameter vector is a number of figures of merit (FOM) associated with the static metamaterial structure in generating the target output field pattern. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 57. The method of claim 31, wherein the static metamaterial structure comprises a static metamaterial. 58. The method of claim 57, wherein the parameter vector comprises parameters related to a static structure design and/or one or more material characteristics of the static metamaterial. 59. A non-transitory, computer-readable storage medium comprising instructions stored therein, the instructions, which when executed by one or more processors, instruct the one or more processors to: access a numerical representation of a device’s geometry and a region of the device containing a static metamaterial structure under design; generate a discretized representation of linear partial differential equations describing a field interaction with the device for a set of values of a parameter vector sufficient to identify parameter-dependent and parameter- independent components of a linear system matrix, wherein a discretized representation x is in a form of a linear system matrix A(p) and at least one source vector b; identify at least one quantitative figure of merits that corresponds to a desirable performance characteristic of the device; form an objective function ^^( ^^, ^^) that evaluates, for the parameter vector ^^, at least one figure of merit from the at least one vector of fields, wherein the vector(s) of fields x is a solution to the discretized representation of linear partial differential equations in a form A(p) x = b; select a set of initial estimates of the parameter vector of the static metamaterial structure for generating a target output field pattern; transform the set of initial estimates of the parameter vector into a set of reduced parameter vectors that is reduced with respect to an original dimension of a parameter space containing the initial estimate of the parameter vector; apply an optimization algorithm to the objective function working in the reduced-dimension space of reduced parameter vectors, with the set of initial values of the reduced parameter vectors, to generate a set of compact refined estimates of the parameter vector; transform a compact refined estimate of the reduced parameter vector to PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) the original parameter space containing the initial estimate of the parameter vector to generate a refined estimate of the parameter vector; selectively identify one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector; and record or transmit a numerical representation of the one or more design characteristics. 60. The non-transitory computer-readable storage medium of claim 59, wherein the instructions stored therein, which when executed by one or more processors, instruct the one or more processors to decompose, based on either or both the parameter-dependent components and the parameter-independent components of the linear system matrix, an original vector space of discretized fields of the discretized representation into a subspace representing a new reduced-dimension vector space and another subspace complementary to the new reduced- dimension vector space. 61. The non-transitory computer-readable storage medium of claim 59, wherein the vector of fields is found by solving the linear problem in the reduced-dimension vector space and then transforming the solution back into the original vector space. 62. The non-transitory computer-readable storage medium of claim 59, wherein the solution to the discretized representation of linear PDEs in a form A(p)x=b is produced by: finding the basis in the field vector space which, together with its complementary basis, decomposes the field vector space into two subspaces such that the transformed linear system matrix A = U A U^† , restricted onto the second subspace, is independent of the parameter vector p; solving a reduced-dimension linear system in the first subspace for a given value of the parameter vector p; and using a precomputed matrix factorization of the transformed linear system matrix restricted onto the second subspace, which is independent of the parameter vector. 63. The non-transitory computer-readable storage medium of claim 62, wherein the finding of the basis is performed using a singular value decomposition of linear system matrix that is PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) used to identify the basis that maximizes the dimension of the second subspace, in which the linear system matrix is independent of the parameter vector. 64. The non-transitory computer-readable storage medium of claim 63, wherein the singular value decomposition of the complex-valued linear system matrix and the real-valued basis vectors of the corresponding singular value basis are computed as the real-valued singular value decomposition of a rectangular, real-valued matrix of size N-by-(N*2*N_p), where N is the dimension of the linear system, and Np is the dimension of the original parameter vector (p), which is formed by first decomposing N_p complex-valued square matrices A(p_n), where p_n is the n-th component of the parameter vector, into 2* Np real-valued square matrices, and then stacking them to form a rectangular matrix. 65. The non-transitory computer-readable storage medium of claim 59, wherein the optimization algorithm constructs a surrogate objective function based on previously computed values of the objective function and optimizes the surrogate objective function. 66. The non-transitory computer-readable storage medium of claim 65, wherein the surrogate objective function is constructed using the radial basis function method. 67. The non-transitory computer-readable storage medium of claim 66, wherein the surrogate objective function is constructed using a surrogate model trained by a machine learning algorithm using previously computed values of the objective function. 68. The non-transitory computer-readable storage medium of claim 67, wherein the machine-learning algorithm is one of the following: polynomial response surfaces, kriging, generalized Bayesian approaches; gradient-enhanced kriging (GEK), radial basis function, support vector machines, space mapping; artificial neural networks, Bayesian networks, Fourier surrogate modeling, or random forests. 69. The non-transitory computer-readable storage medium of claim 59, wherein the objective function contains a reduced-dimension linear system matrix, a reduced-dimension source vector, and matrices for converting a reduced-dimension vector of fields to the original vector space. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 70. The non-transitory computer-readable storage medium of claim 59, wherein the vector of fields is found by solving the linear problem in the reduced-dimension vector space and then transforming the solution back into the original vector space. 71. The non-transitory computer-readable storage medium of claim 59, wherein the objective function generates a linear system matrix and a source vector for a specific value of the set of values of the parameter vector. 72. The non-transitory computer-readable storage medium of claim 59, wherein the objective function generates the linear system matrix and the source vector for a given value of the parameter vector based on an algorithm that generates the discretized representation of the linear partial differential equations describing the field interaction with the device including the static metamaterial structure. 73. The non-transitory computer-readable storage medium of claim 59, wherein the objective function generates a linear system matrix and a source vector for a given value of a parameter vector based on an evaluation of a component of a power series expansion of the linear system matrix and the source vector expanded as functions of the parameter vector at a predefined initial value of the parameter vector, wherein parameters of the power series expansion are computed once at the first evaluation of the objective function and stored in a computer-readable medium and read from that medium for any subsequent evaluation of the objective function. 74. The non-transitory computer-readable storage medium of claim 59, wherein the one or more processors transform the initial estimate of the parameter vector into the reduced parameter vector agnostic as to the at least one figure of merit. 75. The non-transitory computer-readable storage medium of claim 59, wherein the one or more processors transform the initial estimate of the parameter vector into the reduced parameter vector by projecting the initial estimate of the parameter vector onto a linear subspace that spans a reduced number of singular vectors in comparison to the parameter vector. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 76. The non-transitory computer-readable storage medium of claim 59, wherein the optimization algorithm comprises a quadratic problem algorithm. 77. The non-transitory computer-readable storage medium of claim 59, wherein the optimization algorithm comprises a sequential quadratic problem algorithm. 78. The non-transitory computer-readable storage medium of claim 59, wherein the optimization algorithm comprises a gradient-assisted optimization algorithm, and the objective function further includes an algorithm to compute the gradient using a first-order adjoint algorithm. 79. The non-transitory computer-readable storage medium of claim 78, wherein the optimization algorithm comprises a Hessian-assisted optimization algorithm, and the objective function further includes an algorithm to compute the Hessian using a second-order adjoint algorithm. 80. The non-transitory computer-readable storage medium of claim 78, wherein the optimization algorithm comprises a Hessian-assisted optimization algorithm, and the objective function further includes an algorithm to compute the projection of the Hessian onto a set of parameter vectors, as needed for the optimization algorithm, using a second order adjoint algorithm. 81. The non-transitory computer-readable storage medium of claim 59, wherein the instructions further instruct the one or more processors to: evaluate the refined estimate of the parameter vector and corresponding partial derivate constraints of figures of merit associated with the static metamaterial structure in relation to a convergence criterion; and identify the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector in response to an evaluation that the refined estimate of the parameter vector and the corresponding partial derivative constraints meeting the convergence criterion. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 82. The non-transitory computer-readable storage medium of claim 81, wherein the instructions, in response to a determination that the refined estimate of the parameter vector and the corresponding partial derivative constraints fail to meet the convergence criterion, further instruct the one or more processors to iteratively: determine a new initial estimate of the parameter vector; transform the new initial estimate of the parameter vector into a new reduced parameter vector with respect to an original dimension space of the new initial estimate of the parameter vector; apply the local optimization to the new reduced parameter vector to generate a new compact refined estimate of the parameter vector in a reduced dimension space of the new reduced parameter vector; transform the new compact refined estimate of the parameter vector to the original dimension space of the new initial estimate of the parameter vector to generate a new refined estimate of the parameter vector; and selectively identify the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the new refined estimate of the parameter vector. 83. The non-transitory computer-readable storage medium of claim 59, wherein the instructions further instruct the one or more processors to: determine a field associated with the static metamaterial structure generating the target output field pattern for the parameter vector of the static metamaterial structure; transform the field into a reduced field with respect to an original dimension space of the field; and selectively identify the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on both the refined estimate of the parameter vector and the reduced field. 84. The non-transitory computer-readable storage medium of claim 59, wherein a size of the original dimensions space of the initial estimate of the parameter vector is a number of figures of merit (FOM) associated with the static metamaterial structure in generating the target output field pattern. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 85. The non-transitory computer-readable storage medium of claim 59, wherein the static metamaterial structure comprises a static metamaterial. 86. The non-transitory computer-readable storage medium of claim 85, wherein the parameter vector comprises parameters related to a static structure design and/or one or more material characteristics of the static metamaterial. 87. A method of operating a device containing a metamaterial structure, the method comprising: accessing a numerical representation of a geometry of the device and a region of the device containing a static metamaterial structure under design; generating a discretized representation of linear partial differential equations describing a field interaction with the device for a set of values of a parameter vector sufficient to identify parameter-dependent and parameter-independent components of a linear system matrix, wherein the discretized representation is in a form of a linear system matrix A(p) and at least one source vector b; identifying at least one quantitative figure of merits that corresponds to a desirable performance characteristic of the device; forming an objective function that evaluates, for the parameter vector, at least one figure of merit from a vector of fields from the at least one vector of fields, wherein the vector(s) of fields x is a solution to the discretized representation of linear partial differential equations in a form A(p) x = b; selecting a set of initial values of the parameter vector of the static metamaterial structure for generating a target output field pattern; transforming the set of initial values of the parameter vector into a set of reduced parameter vectors that is reduced with respect to an original dimension of a parameter space containing the initial values of the parameter vector; applying an optimization algorithm to the objective function working in the reduced-dimension space of reduced parameter vectors, with the set of initial values of the reduced parameter vectors to generate a set of compact refined estimates of the parameter vector; transforming a compact refined estimate of the reduced parameter vector to the original parameter space containing the initial values of the parameter vector to generate PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) a refined estimate of the parameter vector; selectively identifying one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector; and recording a numerical representation of the one or more design characteristics. 88. The method of claim 87, the vector of fields is found by solving the linear problem in the reduced-dimension vector space and then transforming the solution back into the original vector space. 89. The method of claim 88, wherein the solution to the discretized representation of linear PDEs in a form A(p)x=b is produced by: finding the basis in the field vector space which, together with its complementary basis, decomposes the field vector space into two subspaces such that the transformed linear system matrix A = U A U^† , restricted onto the second subspace, is independent of the parameter vector p; solving a reduced-dimension linear system in the first subspace for a given value of the parameter vector p; and using a precomputed matrix factorization of the transformed linear system matrix restricted onto the second subspace, which is independent of the parameter vector. 90. The method of claim 89, wherein the finding of the basis is performed using a singular value decomposition of linear system matrix that is used to identify the basis that maximizes the dimension of the second subspace, in which the linear system matrix is independent of the parameter vector. 91. The method of claim 90, wherein the singular value decomposition of the complex- valued linear system matrix and the real-valued basis vectors of the corresponding singular value basis are computed as the real-valued singular value decomposition of a rectangular, real- valued matrix of size N-by-(N*2*Np), where N is the dimension of the linear system, and Np is the dimension of the original parameter vector (p), which is formed by first decomposing N_p complex-valued square matrices A(pn), where pn is the n-th component of the parameter vector, into 2* N_p real-valued square matrices, and then stacking them to form a rectangular matrix. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 92. The method of claim 87, wherein the optimization algorithm constructs a surrogate objective function based on previously computed values of the objective function and optimizes the surrogate objective function. 93. The method of claim 92, wherein the surrogate objective function is constructed using the radial basis function method. 94. The method of claim 93, wherein the surrogate objective function is constructed using a surrogate model trained by a machine learning algorithm using previously computed values of the objective function. 95. The method of claim 94, wherein the machine-learning algorithm is one of the following: polynomial response surfaces, kriging, generalized Bayesian approaches; gradient-enhanced kriging (GEK), radial basis function, support vector machines, space mapping; artificial neural networks, Bayesian networks, Fourier surrogate modeling, or random forests. 96. The method of claim 87, wherein the objective function contains a reduced-dimension linear system matrix, a reduced-dimension source vector, and matrices for converting a reduced- dimension vector of fields to the original vector space. 97. The method of claim 87, wherein the vector of fields is found by solving the linear problem in the reduced-dimension vector space and then transforming the solution back into the original vector space. 98. The method of claim 87, wherein the objective function generates a linear system matrix and a source vector for a specific value of the set of values of the parameter vector. 99. The method of claim 87, wherein the objective function generates the linear system matrix and the source vector for a given value of the parameter vector based on an algorithm that generates the discretized representation of the linear partial differential equations describing the field interaction with the device including the static metamaterial structure. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 100. The method of claim 87, wherein the objective function generates a linear system matrix and a source vector for a given value of a parameter vector based on an evaluation of a component of a power series expansion of the linear system matrix and the source vector expanded as functions of the parameter vector at a predefined initial value of the parameter vector, and wherein parameters of the power series expansion are computed once at the first evaluation of the objective function and stored in a computer-readable medium and read from that medium for any subsequent evaluation of the objective function. 101. The method of claim 87, wherein the initial estimate of the parameter vector is transformed into the reduced parameter vector agnostic as to the at least one figure of merit. 102. The method of claim 87, wherein the initial estimate of the parameter vector is transformed into the reduced parameter vector by projecting the initial estimate of the parameter vector onto a linear subspace that spans a reduced number of singular vectors in comparison to the parameter vector. 103. The method of claim 87, wherein the optimization algorithm comprises a quadratic problem algorithm. 104. The method of claim 87, wherein the optimization algorithm comprises a sequential quadratic problem algorithm. 105. The method of claim 87, wherein the optimization algorithm comprises a gradient- assisted optimization algorithm, and the objective function further includes an algorithm to compute the gradient using a first-order adjoint algorithm. 106. The method of claim 104, wherein the optimization algorithm comprises a Hessian- assisted optimization algorithm, and the objective function further includes an algorithm to compute the Hessian using a second-order adjoint algorithm. 107. The method of claim 104, wherein the optimization algorithm comprises a Hessian- assisted optimization algorithm, and the objective function further includes an algorithm to compute the projection of the Hessian onto a set of parameter vectors, as needed for the PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) optimization algorithm, using a second order adjoint algorithm. 108. The method of claim 87, further comprising decomposing, based on either or both the parameter-dependent components and the parameter-independent components of the linear system matrix, an original vector space of discretized fields of the discretized representation into a subspace representing a new reduced-dimension vector space and another subspace complementary to the new reduced-dimension vector space. 109. The method of claim 87, further comprising: evaluating the refined estimate of the parameter vector and corresponding partial derivate constraints of figures of merit associated with the static metamaterial structure in relation to a convergence criterion; and identifying the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector in response to an evaluation that the refined estimate of the parameter vector and the corresponding partial derivative constraints meeting the convergence criterion. 110. The method of claim 109, further comprising, in response to a determination that the refined estimate of the parameter vector and the corresponding partial derivative constraints fail to meet the convergence criterion, iteratively: determining a new initial estimate of the parameter vector; transforming the new initial estimate of the parameter vector into a new reduced parameter vector with respect to an original dimension space of the new initial estimate of the parameter vector; applying the local optimization to the new reduced parameter vector to generate a new compact refined estimate of the parameter vector in a reduced dimension space of the new reduced parameter vector; transforming the new compact refined estimate of the parameter vector to the original dimension space of the new initial estimate of the parameter vector to generate a new refined estimate of the parameter vector; and selectively identifying the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the new PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) refined estimate of the parameter vector. 111. The method of claim 87, further comprising: determining a field associated with the static metamaterial structure generating the target output field pattern for the parameter vector of the static metamaterial structure; transforming the field into a reduced field with respect to an original dimension space of the field; and selectively identifying the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on both the refined estimate of the parameter vector and the reduced field. 112. The method of claim 87, wherein a size of the original dimensions space of the initial estimate of the parameter vector is a number of figures of merit (FOM) associated with the static metamaterial structure in generating the target output field pattern. 113. The method of claim 87, wherein the static metamaterial structure comprises a static metamaterial. 114. The method of claim 87, wherein the parameter vector comprises parameters related to a static structure design and/or one or more material characteristics of the static metamaterial. 115. A method of manufacturing a device containing a metamaterial structure comprising: accessing a numerical representation of a geometry of the device and a region of the device containing a static metamaterial structure under design; generating a discretized representation of linear partial differential equations describing a field interaction with the device for a set of values of a parameter vector sufficient to identify parameter-dependent and parameter-independent components of a linear system matrix, wherein the discretized representation is in a form of a linear system matrix A(p) and at least one source vector b; identifying at least one quantitative figure of merits that corresponds to a desirable performance characteristic of the device; forming an objective function that evaluates, for the parameter vector, at least PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) one figure of merit from a vector of fields from the at least one vector of fields, wherein the vector(s) of fields x is a solution to the discretized representation of linear partial differential equations in a form A(p) x = b; selecting a set of initial values of the parameter vector of the static metamaterial structure for generating a target output field pattern; transforming the set of initial values of the parameter vector into a set of reduced parameter vectors that is reduced with respect to an original dimension of a parameter space containing the initial values of the parameter vector; applying an optimization algorithm to the objective function working in the reduced-dimension space of reduced parameter vectors, with the set of initial values of the reduced parameter vectors to generate a set of compact refined estimates of the parameter vector; transforming a compact refined estimate of the reduced parameter vector to the original parameter space containing the initial values of the parameter vector to generate a refined estimate of the parameter vector; and generating a machine-readable information file containing instructions for a computer-controlled manufacturing apparatus based on the refined estimate of the parameter vector. 116. The method of claim 115, further comprising decomposing, based on either or both the parameter-dependent components and the parameter-independent components of the linear system matrix, an original vector space of discretized fields of the discretized representation into a subspace representing a new reduced-dimension vector space and another subspace complementary to the new reduced-dimension vector space. 117. The method of claim 115, wherein the vector of fields is found by solving the linear problem in the reduced-dimension vector space and then transforming the solution back into the original vector space. 118. The method of claim 115, wherein the solution to the discretized representation of linear PDEs in a form A(p)x=b is produced by: finding the basis in the field vector space which, together with its complementary basis, decomposes the field vector space into two subspaces such that the transformed PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) linear system matrix A = U A U^† , restricted onto the second subspace, is independent of the parameter vector p; solving a reduced-dimension linear system in the first subspace for a given value of the parameter vector p; and using a precomputed matrix factorization of the transformed linear system matrix restricted onto the second subspace, which is independent of the parameter vector. 119. The method of claim 118, wherein the finding of the basis is performed using a singular value decomposition of linear system matrix that is used to identify the basis that maximizes the dimension of the second subspace, in which the linear system matrix is independent of the parameter vector. 120. The method of claim 119, wherein the singular value decomposition of the complex- valued linear system matrix and the real-valued basis vectors of the corresponding singular value basis are computed as the real-valued singular value decomposition of a rectangular, real- valued matrix of size N-by-(2*Np), where N is the dimension of the linear system, and Np is the dimension of the original parameter vector (p), which is formed by first decomposing N_p complex-valued square matrices A(pn), where pn is the n-th component of the parameter vector, into 2* N_p real-valued square matrices, and then stacking them to form a rectangular matrix. 121. The method of claim 115, wherein the optimization algorithm constructs a surrogate objective function based on previously computed values of the objective function and optimizes the surrogate objective function. 122. The method of claim 121, wherein the surrogate objective function is constructed using the radial basis function method. 123. The method of claim 122, wherein the surrogate objective function is constructed using a surrogate model trained by a machine learning algorithm using previously computed values of the objective function. 124. The method of claim 123, wherein the machine-learning algorithm is one of the following: polynomial response surfaces, kriging, generalized Bayesian approaches; gradient- PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) enhanced kriging (GEK), radial basis function, support vector machines, space mapping; artificial neural networks, Bayesian networks, Fourier surrogate modeling, or random forests. 125. The method of claim 115, wherein the objective function contains a reduced-dimension linear system matrix, a reduced-dimension source vector, and matrices for converting a reduced- dimension vector of fields to the original vector space. 126. The method of claim 115, wherein the vector of fields is found by solving the linear problem in the reduced-dimension vector space and then transforming the solution back into the original vector space. 127. The method of claim 115, wherein the objective function generates a linear system matrix and a source vector for a specific value of the set of values of the parameter vector. 128. The method of claim 115, wherein the objective function generates the linear system matrix and the source vector for a given value of the parameter vector based on an algorithm that generates the discretized representation of the linear partial differential equations describing the field interaction with the device including the static metamaterial structure. 129. The method of claim 115, wherein the objective function generates a linear system matrix and a source vector for a given value of a parameter vector based on an evaluation of a component of a power series expansion of the linear system matrix and the source vector expanded as functions of the parameter vector at a predefined initial value of the parameter vector, and wherein parameters of the power series expansion are computed once at the first evaluation of the objective function and stored in a computer-readable medium and read from that medium for any subsequent evaluation of the objective function. 130. The method of claim 115, wherein the initial estimate of the parameter vector is transformed into the reduced parameter vector agnostic as to the at least one figure of merit. 131. The method of claim 115, wherein the initial estimate of the parameter vector is transformed into the reduced parameter vector by projecting the initial estimate of the parameter PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) vector onto a linear subspace that spans a reduced number of singular vectors in comparison to the parameter vector. 132. The method of claim 115, wherein the optimization algorithm comprises a quadratic problem algorithm. 133. The method of claim 115, wherein the optimization algorithm comprises a sequential quadratic problem algorithm. 134. The method of claim 115, wherein the optimization algorithm comprises a gradient- assisted optimization algorithm, and the objective function further includes an algorithm to compute the gradient using a first-order adjoint algorithm. 135. The method of claim 134, wherein the optimization algorithm comprises a Hessian- assisted optimization algorithm, and the objective function further includes an algorithm to compute the Hessian using a second-order adjoint algorithm. 136. The method of claim 134, wherein the optimization algorithm comprises a Hessian- assisted optimization algorithm, and the objective function further includes an algorithm to compute the projection of the Hessian onto a set of parameter vectors, as needed for the optimization algorithm, using a second order adjoint algorithm. 137. The method of claim 115, further comprising: evaluating the refined estimate of the parameter vector and corresponding partial derivate constraints of figures of merit associated with the static metamaterial structure in relation to a convergence criterion; and identifying the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the refined estimate of the parameter vector in response to an evaluation that the refined estimate of the parameter vector and the corresponding partial derivative constraints meeting the convergence criterion. 138. The method of claim 115, further comprising, in response to a determination that the PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) refined estimate of the parameter vector and the corresponding partial derivative constraints fail to meet the convergence criterion, iteratively: determining a new initial estimate of the parameter vector; transforming the new initial estimate of the parameter vector into a new reduced parameter vector with respect to an original dimension space of the new initial estimate of the parameter vector; applying the local optimization to the new reduced parameter vector to generate a new compact refined estimate of the parameter vector in a reduced dimension space of the new reduced parameter vector; transforming the new compact refined estimate of the parameter vector to the original dimension space of the new initial estimate of the parameter vector to generate a new refined estimate of the parameter vector; and selectively identifying the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on the new refined estimate of the parameter vector. 139. The method of claim 115, further comprising: determining a field associated with the static metamaterial structure generating the target output field pattern for the parameter vector of the static metamaterial structure; transforming the field into a reduced field with respect to an original dimension space of the field; and selectively identifying the one or more design characteristics for the static metamaterial structure in generating the target output field pattern based on both the refined estimate of the parameter vector and the reduced field. 140. The method of claim 115, wherein a size of the original dimensions space of the initial estimate of the parameter vector is a number of figures of merit (FOM) associated with the static metamaterial structure in generating the target output field pattern. 141. The method of claim 115, wherein the metamaterial structure comprises a static metamaterial. PATENTS PLS-027PCT Attorney docket: MM020 (115222-740606) 142. The method of claim 139, wherein the parameter vector comprises parameters related to a static structure design and/or one or more material characteristics of the static metamaterial.