CN109165416A - Conjugate gradient iteration hard -threshold radiation energy method for solving based on backtracking - Google Patents
Conjugate gradient iteration hard -threshold radiation energy method for solving based on backtracking Download PDFInfo
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Abstract
The invention belongs to the Preliminary design fields of pellet target chamber in inertial confinement fusion test, and extensive and ultra-large nonlinear radiative energy equation solution, are related to a kind of conjugate gradient iteration hard -threshold radiation energy method for solving based on backtracking.The present invention mainly consists of two parts, first part is the target chamber and pellet radiation energy sparse representation method for designing different topology structure, and nonlinear radiative energy-balance equation model of the building about sparse coefficient, second part is to propose that a kind of conjugate gradient iteration hard -threshold based on backtracking solves the energy-balance equation.The present invention reduces the number of required energy-balance equation on a large scale, reduces the solution time of model, to greatly improve actinomorphy assessment efficiency；Information useful in previous iteration is taken full advantage of, the selection strategy of supported collection is optimized, it is ensured that supported collection is quick and precisely found；It is suitable for the solution of non-linear sparse model, and solution efficiency and precision can be protected.
Description
Technical field
The invention belongs to the Preliminary design field of pellet target chamber in inertial confinement fusion (ICF) test, and it is extensive and
The solution of ultra-large nonlinear radiative energy equation changes more particularly, to a kind of conjugate gradient based on backtracking
For hard -threshold radiation energy method for solving.
Background technique
Realize that controllable nuclear fusion is a kind of important channel that the mankind solve energy problem, inertial confinement fusion (ICF) is real
One of the best mode of existing controlled thermonuclear fusion.The driving successful key of ICF igniting experiments is fuel pellet surface emissivity indirectly
Energy distribution is symmetrical enough, and the principal element for influencing pellet actinomorphy is target chamber geometric parameter, assessment pellet radiation pair
Title property needs to solve the radiation energy on pellet surface, since the energy exchange processes in the intracavitary portion of target is one extremely complex non-
Linear process, it is directly highly difficult with Analytic Method, with discrete viewing factor method, the applicability of mathematical model is become more
Well, the form of energy-balance equation is simpler.It is to realize that the radiation energy symmetry on fuel pellet surface, which reaches 99% or more,
The successful key factor of ICF igniting experiments, pellet radiation energy symmetrical analysis is the process of a complicated and time consumption, using number
Value emulation mode can provide reference and estimation for actual experiment, improve actual experiment efficiency, reduce experimental cost.
Using based on discrete viewing factor mathematical model assess radiation energy when, need to carry out pellet and target chamber from
Dispersion processing, i.e. grid division need to solve large-scale even ultra-large type Nonlinear System of Equations when grid dividing is very fine
(energy-balance equation) acquires pellet radiation energy, if solved with traditional method, when huge calculation amount will lead to simulation
Between greatly prolong, seriously affect ICF simulation time.
Model (TDEBM) and time non-dependent model are relied on to the main having time of ICF actinomorphy assessment models at present
(TIEBM), TDEBM is nonlinear, and TIEBM is linear.The method for solving of numerical model can be mainly divided into two classes, and one
Class is solved with conventional method, and another kind of solved with compressive sampling method.It is main for linear model in conventional method
Method for solving has LU factorization, Cholesky decomposition method etc., and for nonlinear model, method for solving has Newton iteration method, conjugation
Gradient method, preconditioning conjugate gradient based on GPU concurrent operation etc..In compressive sampling method, have just for linear model
It hands over matching pursuit algorithm (Orthogonal Matching Pursuit, OMP), for nonlinear model, there are no preferable
Method.
The advantages of conventional method solution large-scale nonlinear radiation energy equilibrium equation is that precision is higher, is suitable for small rule
Modular equation, but as ICF experiment is higher and higher to pellet and target chamber Preliminary design required precision, the face element size of division must
Fixed smaller and smaller, this results in face element quantity to sharply increase, and face element total amount may even exceed 107, is such as asked with conventional method
The so large-scale Nonlinear System of Equations of solution is a job that is very time-consuming and consuming memory, on common laptop very
It is unacceptable for the scientific research personnel for being engaged in this field to not being available.
It is existing to be solved using compression sampling technology model TIEBM (linear equation) non-dependent to the time, such as with just
Matching pursuit algorithm (Orthogonal Matching Pursuit, OMP) solution efficiency is handed over to be significantly better than conventional method, still
For Time Dependent model TDEBM, there are no a kind of efficient compression sampling modes to be solved, wherein main ask
Topic is: first, needing the geometry according to its inner surface Energy distribution feature and target chamber for the target chamber of different topology structure
Mechanical feature designs different sparse representation methods, constructs the nonlinear radiative energy-balance equation based on rarefaction representation；Its
Two, need to design a kind of reconstructing method of high-efficiency high-accuracy to solve nonlinear radiative energy rarefaction representation equilibrium equation.
Summary of the invention
The present invention in order to overcome at least one of the drawbacks of the prior art described above, provides a kind of conjugation ladder based on backtracking
Iteration hard -threshold radiation energy method for solving is spent,.
In order to solve the above technical problems, the technical solution adopted by the present invention is that: a kind of conjugate gradient iteration based on backtracking
Hard -threshold radiation energy method for solving, comprising the following steps:
S1. the radiation energy equilibrium equation model based on discrete viewing factor is established；
S2. it designs suitable sparse basis and rarefaction representation is carried out to pellet, target chamber, and construct the radiation about sparse coefficient
Energy-balance equation；
S3. compression sampling is carried out to the radiation energy equilibrium equation about sparse coefficient；
S4. the conjugate gradient iteration hard -threshold based on backtracking solves the radiation energy equilibrium equation about sparse coefficient；
S5. according to required sparse coefficient reverse radiation energy.
Further, the S1 step specifically includes:
S11. it establishes ICF and is based on discrete viewing factor nonlinear energy equilibrium equation:
In formula, BiIndicate radiation energy to be asked, EiIndicate right after i-th of face element primary incident light source is converted into soft x light
The DIRECT ENERGY of other face elements, AjIndicate the area of j-th of face element, FjiIt indicates viewing factor matrix, is a symmetrical matrix,
λiIt is the albedo of i-th of face element；
Wherein, EiAnd λiIt is acquired with following calculation formula:
In formula,Indicate the energy of primary incident light source, C, α and β are constants, respectively equal to 4.87,8/13,16/13；
S12. ICF is based on discrete viewing factor nonlinear energy equilibrium equation and is write as matrix form:
Enable y=E, then:
In formula, I indicates unit matrix；V is indicated by FijThe viewing factor matrix of composition, i.e. V={ Fij}；E is indicated by EiStructure
At vector, i.e. E=[E1,E2,…En]T；Each element does exponential depth with b in a ο b expression vector a.
Further, since ideal pellet is a sphere, the radiation energy on pellet is only few on ball harmonics domain
The several big coefficients of number, for other coefficients all close to 0, i.e. radiation energy has good sparsity therefore can on pellet
Use the humorous multinomial of ball as the sparse basis of radiation energy on pellet.Similar, for the target chamber of cylindrical cavity shape, Fu can be used
In leaf sum of series Legnedre polynomial two-dimensional quadrature multinomial as sparse basis (hereinafter referred to as cylinder multinomial).It is right
In the radiation energy of cylinder resonator end surface, Zernike multinomial can be used as sparse basis.
The S2 step specifically includes:
S21. radiation energy is indicated with sparse basis are as follows:
B=ψ x
S22., the energy-balance equation of S12 step is rewritten as to the equation about sparse coefficient
In formula, wherein ψ is sparse basis, and above formula is a Nonlinear System of Equations, its first order Taylor expansion can be expressed as:
F (x)=f (xn)+f'(xn)(xn+1-xn)
Wherein, f'(xn)=J × Basis, J are energy-balance equation in S22 stepJacobian matrix, Basis
For sparse basis；
Enable f'(xn)=Φ, f (x)-f (xn)+f'(xn)xn=y, then, first order Taylor may be expressed as:
Y=Φ x.
Further, the S3 step specifically includes:
S31. it is sent out using Latin Hypercube Sampling and compression sampling is carried out to y=Φ x, number of samples calculates according to the following formula:
m≥k log(N)
In formula, m is hits, and k is degree of rarefication of the radiation energy in sparse basis, and N is total face element quantity；
S32. the y=Φ x after sampling is one with l0Norm is the optimization problem of constraint, is indicated with following formula:
In formula, A is the matrix after sampling to Φ.
Further, the S4 step acquires approximate solution first with the method that supported collection is recalled in every step iteration,
Then approximate solution is updated using improved conjugate gradient method.
Further, due toIt is a underdetermined system of equations, solving it is a NP
Difficult problem, but if matrix A meets certain limited equidistant condition, following algorithm Accurate Reconstruction x can be passed through；Its algorithm
Specific steps include:
In following steps,Indicate empty set, Hs(α) is a nonlinear operator, indicates s that take maximum absolute value in α
Element, μ are iteration step length, ΓnIndicating the index set of nth iteration, ∪ indicates union, and inner product is sought in <, > expression,
Supp () indicates that vector nonzero element index, ε indicate iteration stopping error threshold；
S41. x is initialized0=0, residual error r0=y, initial support collectionInitial Gradient: g0=r0, initial optimizing side
To d0=g0, step-lengthx1=Hs(x0+μ0d0), Γ1=supp (x1)；
S42. iterative process:
S421. gradient direction g is updatedn=AT(y-Ax)；
S422. judge whether Γn=Γn-1, if so, S424 is gone to step, if not, going to step S423；
S423. gradient descent method:
S4231. material calculation
S4232. iteration an=Hs(xn-1+μngn)；
S424. conjugate gradient method:
S4241. direction factor is calculated
S4242. search direction d is updatedn=gn+βndn-1；
S4243. step-length is updated
S4244. iteration an=Hs(xn-1+μndn)；
S425. recall:
S4251. merge supported collection Γn=supp (an+1)∪supp(xn)；
S4252.
S426. residual error r is updatedn+1=y-Axn+1；
S43. judge whether | | rn+1||2< ε, if so, iteration stopping, otherwise returns to S422 step and continue iteration.
Iteration hard -threshold class convergence speed of the algorithm depends primarily on two aspects, is on the one hand how algorithm finds correctly
Supported collection, be on the other hand how to approach optimal solution after finding correct supported collection.IHT algorithm in every step iteration due to inciting somebody to action
Signal Residual projection, with its most matched s vector, acquires approximation using the linear combination of this s vector into observing matrix
The shortcomings that solution, then continuous iteration, this method, is possible to that certain supports is caused to be repeated selection, so as to cause IHT algorithm
It finds correct supported collection and needs more the number of iterations.The thought for introducing backtracking can reduce support quilt to a certain extent
The probability for repeating selection, so that accelerating algorithm is quickly found out correct support.But made using the gradient descent method of fixed step size
For optimizing strategy, and it is that convergence rate is slow the shortcomings that gradient descent method, replaces gradient direction to improve this with conjugate gradient and lack
Point, but in the actual process, since the column atom of observing matrix is not exclusively orthogonal, lead to matrixAΓnIt takes on morbit forms existing
As affecting convergence speed of the algorithm to influence the conjugacy of search direction in the iteration of front and back.
BCGIHT algorithm as being quickly found out the strategy of correct supported collection, and is adopted using backtracking thought on optimization method
It is alternately tactful with conjugate direction with gradient direction.By using the comparison of front and back supported collection, to determine search direction, automatically
Substitute gradient descent direction and conjugated gradient direction, overcomes conjugate gradient iteration hard threshold algorithm and use conjugate gradient merely
The shortcomings that direction, can further speed up algorithmic statement.If front and back supported collection is equal, i.e. Γn=Γn-1, illustrate by front and back branch
The matrix of the column composition of the corresponding observing matrix of support collection is identical, i.e.,ThereforeIt at this moment can be after
It is continuous to carry out next step optimizing with conjugated gradient direction；And when currently rear support collection is unequal, above-mentioned two symmetric positive definite matrixWithIt is no longer equal, front and back search direction dnAnd dn-1It is no longer conjugation, therefore adjusting search direction is gradient
Descent direction.Simple compared to replacement uses gradient direction or the method for conjugate direction, the tactful energy of search direction replacement
Adjust automatically search direction in each iteration, so that accelerating algorithm restrains.
Further, the S4 step are as follows: after the completion of the solution of radiation energy sparse representation model, obtain x, then
The radiation energy on required pellet and target chamber can be obtained by B=ψ x.
Compared with prior art, beneficial effect is:
1. the present invention devises the sparse representation method of cylindrical target chamber radiation energy in ICF test, i.e., humorous multinomial using ball
Formula, Zernike multinomial and cylinder Polynomial combination are at a sparse basis, and with this sparse basis to ICF cylindrical cavity target model
Radiation energy carries out rarefaction representation, constructs the nonlinear radiative energy-balance equation about sparse coefficient, for compression sampling
Method solves radiation energy equilibrium equation and provides precondition；
2. method provided by the present invention as being quickly found out the strategy of correct supported collection, and is being sought using backtracking thought
It is alternately tactful using gradient direction and conjugate direction in excellent method.By using the comparison of front and back supported collection, to determine optimizing
Direction, it is automatic to substitute gradient descent direction and conjugated gradient direction, it overcomes conjugate gradient iteration hard threshold algorithm and uses merely
The shortcomings that conjugated gradient direction, can further speed up algorithmic statement.If front and back supported collection is equal, i.e. Γn=Γn-1, explanation
The matrix that the column of the observing matrix corresponding to the supported collection of front and back form is identical, i.e.,ThereforeThis
When can continue to carry out next step optimizing with conjugated gradient direction；And current rear support collection it is unequal when, it is above-mentioned two it is symmetrical just
Set matrixWithIt is no longer equal, front and back search direction dnAnd dn-1No longer it is conjugation, therefore adjusts search direction
For gradient descent direction.Simple compared to replacement uses gradient direction or the method for conjugate direction, search direction replacement
Strategy can adjust automatically search direction in each iteration, so that accelerating algorithm restrains；
3. invention provides a kind of compressed sensing based ICF test pellet target chamber radiation energy symmetry appraisal procedure,
This method can reduce the number of required energy-balance equation on a large scale, the solution time of model be reduced, to greatly improve
Actinomorphy assesses efficiency；
4. propose that a kind of new sparse restructing algorithm BCGIHT is used for the reconstruct of radiation energy sparse representation model,
The characteristics of BCGIHT algorithm, is that it introduces backtracking thought in each iteration, takes full advantage of letter useful in previous iteration
Breath, optimizes the selection strategy of supported collection, it is ensured that supported collection is quick and precisely found, secondly, optimizing strategy is designed as being conjugated
Adaptively alternate mode, this mode are suitable for the solution of non-linear sparse model, and solution efficiency for direction and gradient direction
It can be protected with precision.
Detailed description of the invention
Fig. 1 is flow chart of the method for the present invention.
Fig. 2 is ICF test pellet target chamber geometry exploded view of the present invention.
Fig. 3 is 16 3D renderings before the humorous multinomial of ball in the embodiment of the present invention.
Fig. 4 is 15 3D renderings before Zernike multinomial in the embodiment of the present invention.
Fig. 5 is 16 3D renderings before cylinder multinomial in the embodiment of the present invention.
Specific embodiment
Attached drawing only for illustration, is not considered as limiting the invention；In order to better illustrate this embodiment,
The certain components of attached drawing have omission, zoom in or out, and do not represent the size of actual product；Those skilled in the art are come
It says, the omitting of some known structures and their instructions in the attached drawings are understandable.Positional relationship is described in attached drawing to be only used for showing
Example property explanation, is not considered as limiting the invention.
As shown in Figure 1, a kind of conjugate gradient iteration hard -threshold radiation energy method for solving based on backtracking, including it is following
Step:
Step 1. establishes the radiation energy equilibrium equation model based on discrete viewing factor；
The geometry of pellet target chamber decompose as shown in Fig. 2, mainly pellet spherical as shown in 1., 2. shown in circle
Anchor ring and 3. shown in cylindrical side composition.
S11. it establishes ICF and is based on discrete viewing factor nonlinear energy equilibrium equation:
In formula, BiIndicate radiation energy to be asked, EiIndicate right after i-th of face element primary incident light source is converted into soft x light
The DIRECT ENERGY of other face elements, AjIndicate the area of j-th of face element, FjiIt indicates viewing factor matrix, is a symmetrical matrix,
λiIt is the albedo of i-th of face element；
Wherein, EiAnd λiIt is acquired with following calculation formula:
In formula,Indicate the energy of primary incident light source, C, α and β are constants, respectively equal to 4.87,8/13,16/13；
S12. ICF is based on discrete viewing factor nonlinear energy equilibrium equation and is write as matrix form:
Enable y=E, then:
In formula, I indicates unit matrix；V is indicated by FijThe viewing factor matrix of composition, i.e. V={ Fij}；E is indicated by EiStructure
At vector, i.e. E=[E1,E2,…En]T；Each element does exponential depth with b in a ο b expression vector a.
Step 2. designs suitable sparse basis and carries out rarefaction representation to pellet, target chamber, and constructs the spoke about sparse coefficient
Penetrate energy-balance equation；
Since ideal pellet is a sphere, only a few is big on ball harmonics domain for the radiation energy on pellet
Coefficient, other coefficients are all close to 0, i.e., radiation energy has good sparsity on pellet, therefore, can be humorous more with ball
Sparse basis of the item formula as radiation energy on pellet.Similar, for the target chamber of cylindrical cavity shape, Fourier space can be used
Two-dimensional quadrature multinomial with Legnedre polynomial is as sparse basis (hereinafter referred to as cylinder multinomial).For cylindrical cavity
The radiation energy of end face can use Zernike multinomial as sparse basis.The humorous multinomial of ball, Zernike multinomial and cylinder
Polynomial former rank 3-D images are as shown in Fig. 3, Fig. 4 and Fig. 5.
Step 2 step specifically includes:
S21. radiation energy is indicated with sparse basis are as follows:
B=ψ x (6)
S22., the energy-balance equation of S12 step is rewritten as to the equation about sparse coefficient
In formula, wherein ψ is sparse basis, and above formula (7) is a Nonlinear System of Equations, its first order Taylor expansion can indicate
At:
F (x)=f (xn)+f'(xn)(xn+1-xn) (8)
Wherein, f'(xn)=J × Basis, J are the Jacobian matrix of (7) formula, and Basis is sparse basis；
Enable f'(xn)=Φ, f (x)-f (xn)+f'(xn)xn=y, then, first order Taylor may be expressed as:
Y=Φ x. (9)
Step 3. carries out compression sampling to the radiation energy equilibrium equation about sparse coefficient；
S31. Taylor expansion after linear equation, sends out non-linear equation using Latin Hypercube Sampling to y=
Φ x carries out compression sampling, and number of samples calculates according to the following formula:
m≥k log(N) (10)
In formula, m is hits, and k is degree of rarefication of the radiation energy in sparse basis, and N is total face element quantity；
S32. the y=Φ x after sampling is one with l0Norm is the optimization problem of constraint, is indicated with following formula:
In formula, A is the matrix after sampling to Φ.
Step 4. solves the radiation energy balance side about sparse coefficient based on the conjugate gradient iteration hard -threshold of backtracking
Journey；
Due toIt is a underdetermined system of equations, solving it is a np hard problem, but
It is that can pass through some algorithm Accurate Reconstruction x, such as greedy calculation if matrix A meets certain limited equidistant condition
Method is based on l1Linear programming algorithm of norm etc..Quick optimizing is tracked in the thought of present invention combination supported collection backtracking and subspace
The advantages of, provide a kind of conjugate gradient iteration hard threshold algorithm based on backtracking, the algorithm in every step iteration first with
The method of supported collection backtracking acquires approximate solution, then updates approximate solution using improved conjugate gradient method.Specific algorithmic procedure
It is as follows:
In following steps,Show empty set, Hs(α) is a nonlinear operator, indicates s member for taking maximum absolute value in α
Element, μ are iteration step length, ΓnIndicating the index set of nth iteration, ∪ indicates union, and inner product is sought in <, > expression,
Supp () indicates that vector nonzero element index, ε indicate iteration stopping error threshold；
S41. x is initialized0=0, residual error r0=y, initial support collectionInitial Gradient: g0=r0, initial optimizing side
To d0=g0, step-lengthx1=Hs(x0+μ0d0), Γ1=supp (x1)；
S42. iterative process:
S421. gradient direction g is updatedn=AT(y-Ax)；
S422. judge whether Γn=Γn-1, if so, S424 is gone to step, if not, going to step S423；
S423. gradient descent method:
S4231. material calculation
S4232. iteration an=Hs(xn-1+μngn)；
S424. conjugate gradient method:
S4241. direction factor is calculated
S4242. search direction d is updatedn=gn+βndn-1；
S4243. step-length is updated
S4244. iteration an=Hs(xn-1+μndn)；
S425. recall:
S4251. merge supported collection Γn=supp (an+1)∪supp(xn)；
S4252.
S426. residual error r is updatedn+1=y-Axn+1；
S43. judge whether | | rn+1||2< ε, if so, iteration stopping, otherwise returns to S422 step and continue iteration.
Iteration hard -threshold class convergence speed of the algorithm depends primarily on two aspects, is on the one hand how algorithm finds correctly
Supported collection, be on the other hand how to approach optimal solution after finding correct supported collection.IHT algorithm in every step iteration due to inciting somebody to action
Signal Residual projection, with its most matched s vector, acquires approximation using the linear combination of this s vector into observing matrix
The shortcomings that solution, then continuous iteration, this method, is possible to that certain supports is caused to be repeated selection, so as to cause IHT algorithm
It finds correct supported collection and needs more the number of iterations.The thought for introducing backtracking can reduce support quilt to a certain extent
The probability for repeating selection, so that accelerating algorithm is quickly found out correct support.But made using the gradient descent method of fixed step size
For optimizing strategy, and it is that convergence rate is slow the shortcomings that gradient descent method, replaces gradient direction to improve this with conjugate gradient and lack
Point, but in the actual process, since the column atom of observing matrix is not exclusively orthogonal, lead to matrixIt takes on morbit forms existing
As affecting convergence speed of the algorithm to influence the conjugacy of search direction in the iteration of front and back.
BCGIHT algorithm as being quickly found out the strategy of correct supported collection, and is adopted using backtracking thought on optimization method
It is alternately tactful with conjugate direction with gradient direction.By using the comparison of front and back supported collection, to determine search direction, automatically
Substitute gradient descent direction and conjugated gradient direction, overcomes conjugate gradient iteration hard threshold algorithm and use conjugate gradient merely
The shortcomings that direction, can further speed up algorithmic statement.If front and back supported collection is equal, i.e. Γn=Γn-1, illustrate by front and back branch
The matrix of the column composition of the corresponding observing matrix of support collection is identical, i.e.,ThereforeIt at this moment can be after
It is continuous to carry out next step optimizing with conjugated gradient direction；And when currently rear support collection is unequal, above-mentioned two symmetric positive definite matrixWithIt is no longer equal, front and back search direction dnAnd dn-1It is no longer conjugation, therefore adjusting search direction is gradient
Descent direction.Simple compared to replacement uses gradient direction or the method for conjugate direction, the tactful energy of search direction replacement
Adjust automatically search direction in each iteration, so that accelerating algorithm restrains.
Step 5. is according to required sparse coefficient reverse radiation energy；
After the completion of the solution of radiation energy sparse representation model, x is obtained, then can obtain required pellet by B=ψ x
With the radiation energy on target chamber.
Obviously, the above embodiment of the present invention be only to clearly illustrate example of the present invention, and not be pair
The restriction of embodiments of the present invention.For those of ordinary skill in the art, may be used also on the basis of the above description
To make other variations or changes in different ways.There is no necessity and possibility to exhaust all the enbodiments.It is all this
Made any modifications, equivalent replacements, and improvements etc., should be included in the claims in the present invention within the spirit and principle of invention
Protection scope within.
Claims (7)
1. a kind of conjugate gradient iteration hard -threshold radiation energy method for solving based on backtracking, which is characterized in that including following step
It is rapid:
S1. the radiation energy equilibrium equation model based on discrete viewing factor is established；
S2. it designs suitable sparse basis and rarefaction representation is carried out to pellet, target chamber, and construct flat about the radiation energy of sparse coefficient
Weigh equation；
S3. compression sampling is carried out to the radiation energy equilibrium equation about sparse coefficient；
S4. the conjugate gradient iteration hard -threshold based on backtracking solves the radiation energy equilibrium equation about sparse coefficient；
S5. according to required sparse coefficient reverse radiation energy.
2. the conjugate gradient iteration hard -threshold radiation energy method for solving according to claim 1 based on backtracking, feature
It is, the S1 step specifically includes:
S11. it establishes ICF and is based on discrete viewing factor nonlinear energy equilibrium equation:
In formula, BiIndicate radiation energy to be asked, EiIndicate that i-th of face element primary incident light source is converted into after soft x light to other faces
The DIRECT ENERGY of member, AjIndicate the area of j-th of face element, FjiIt indicates viewing factor matrix, is a symmetrical matrix, λiIt is i-th
The albedo of a face element；
Wherein, EiAnd λiIt is acquired with following calculation formula:
In formula,Indicate the energy of primary incident light source, C, α and β are constants；
S12. ICF is based on discrete viewing factor nonlinear energy equilibrium equation and is write as matrix form:
Enable y=E, then:
In formula, I indicates unit matrix；V is indicated by FijThe viewing factor matrix of composition, i.e. V={ Fij}；E is indicated by EiIt constitutes
Vector, i.e. E=[E1,E2,…En]T；Each element does exponential depth with b in expression vector a.
3. the conjugate gradient iteration hard -threshold radiation energy method for solving according to claim 2 based on backtracking, feature
It is, the S2 step specifically includes:
S21. radiation energy is indicated with sparse basis are as follows:
B=ψ x
S22., the energy-balance equation of S12 step is rewritten as to the equation about sparse coefficient
In formula, wherein ψ is sparse basis, and above formula is a Nonlinear System of Equations, its first order Taylor expansion can be expressed as:
F (x)=f (xn)+f'(xn)(xn+1-xn)
Wherein, f'(xn)=J × Basis, J are energy-balance equation in S22 stepJacobian matrix, Basis be it is sparse
Base；
Enable f'(xn)=Φ, f (x)-f (xn)+f'(xn)xn=y, then, first order Taylor may be expressed as:
Y=Φ x.
4. the conjugate gradient iteration hard -threshold radiation energy method for solving according to claim 3 based on backtracking, feature
It is, the S3 step specifically includes:
S31. it is sent out using Latin Hypercube Sampling and compression sampling is carried out to y=Φ x, number of samples calculates according to the following formula:
m≥k log(N)
In formula, m is hits, and k is degree of rarefication of the radiation energy in sparse basis, and N is total face element quantity；
S32. the y=Φ x after sampling is one with l0Norm is the optimization problem of constraint, is indicated with following formula:
In formula, A is the matrix after sampling to Φ.
5. the conjugate gradient iteration hard -threshold radiation energy method for solving according to claim 4 based on backtracking, feature
It is, the S4 step acquires approximate solution first with the method that supported collection is recalled in every step iteration, then using improvement
Conjugate gradient method update approximate solution.
6. the conjugate gradient iteration hard -threshold radiation energy method for solving according to claim 5 based on backtracking, feature
It is, due tos.t.||x||0≤ k is a underdetermined system of equations, and solving it is a np hard problem, still
If matrix A meets certain limited equidistant condition, following algorithm Accurate Reconstruction x can be passed through；The specific steps packet of its algorithm
It includes:
In following steps,Indicate empty set, Hs(α) is a nonlinear operator, indicates s element for taking maximum absolute value in α, μ
For iteration step length, ΓnIndicate the index set of nth iteration, ∪ indicates union, and inner product, supp () are asked in <, > expression
Indicate that vector nonzero element index, ε indicate iteration stopping error threshold；
S41. x is initialized0=0, residual error r0=y, initial support collectionInitial Gradient: g0=r0, initial search direction d0=
G0, step-lengthx1=Hs(x0+μ0d0), Γ1=supp (x1)；
S42. iterative process:
S421. gradient direction g is updatedn=AT(y-Ax)；
S422. judge whether Γn=Γn-1, if so, S424 is gone to step, if not, going to step S423；
S423. gradient descent method:
S4231. material calculation
S4232. iteration an=Hs(xn-1+μngn)；
S424. conjugate gradient method:
S4241. direction factor is calculated
S4242. search direction d is updatedn=gn+βndn-1；
S4243. step-length is updated
S4244. iteration an=Hs(xn-1+μndn)；
S425. recall:
S4251. merge supported collection Γn=supp (an+1)∪supp(xn)；
S4252.
S426. residual error r is updatedn+1=y-Axn+1；
S43. judge whether | | rn+1||2< ε, if so, iteration stopping, otherwise returns to S422 step and continue iteration.
7. the conjugate gradient iteration hard -threshold radiation energy method for solving according to claim 6 based on backtracking, feature
It is, the S4 step are as follows: after the completion of the solution of radiation energy sparse representation model, obtain x, then can pass through B=ψ x
Obtain the radiation energy on required pellet and target chamber.
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