CN114692562A - High-precision hybrid dynamic priority multi-objective optimization method - Google Patents

High-precision hybrid dynamic priority multi-objective optimization method Download PDF

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CN114692562A
CN114692562A CN202210256682.7A CN202210256682A CN114692562A CN 114692562 A CN114692562 A CN 114692562A CN 202210256682 A CN202210256682 A CN 202210256682A CN 114692562 A CN114692562 A CN 114692562A
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李艳秋
韦鹏志
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Abstract

The invention provides a high-precision hybrid dynamic priority multi-objective optimization method, which comprises the following specific processes: establishing an objective optimization function, and calculating gradient values of optimization variables relative to optimization targets based on the objective optimization function; calculating a loss function value of each optimization target of the iteration, and calculating an overall rate factor reflecting the change rate of the target loss function, a gradient factor reflecting the update of target parameters and an overall scale factor reflecting the optimization degree of the target based on the loss function value and the gradient value; calculating a function value of each factor by using the focus loss function, and calculating a total gradient value of each optimization target based on the function values of the three factors; updating the optimization variables based on the total gradient values. Because the algorithm adopts a mixed priority strategy weighted by three factors, the problem of difference increase between targets caused by the existing multi-target algorithm based on a single priority rule can be avoided.

Description

High-precision hybrid dynamic priority multi-objective optimization method
Technical Field
The invention relates to a high-precision hybrid priority multi-objective optimization method, and belongs to the technical field of multi-objective optimization.
Background
For the single-target optimization, the quality of any two solutions can be compared according to a single target, an undisputed optimal solution can be obtained, and the multi-target optimization is opposite to the traditional single-target optimization. The concept of multi-objective optimization is that when multiple objectives are needed to be achieved in a certain scenario, because intrinsic conflicts between the objectives are easy to exist, the optimization of one objective is at the cost of the degradation of other objectives, so that a unique optimal solution is difficult to occur, and instead, coordination and compromise are made among the objectives to make the overall objective as optimal as possible.
With the rapid development of the fields of deep learning and multi-task learning, the multi-target algorithm based on gradient gets further attention, and the method is mainly characterized in that the priorities of targets are distributed according to different priority distribution rules, and the priority distribution rules mainly comprise three types: the first class homogenizes convergence rate, the second class improves difficult targets, and the third class searches for pareto improvements.
In the aspect of a homogenization convergence rate rule, a related document (Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks, ICML, 2018) provides Gradnorm, and the problem of multi-target weight distribution is converted into a target optimization problem based on learning, so that the Gradient size of each task is balanced. Related documents (End-to-End Multi-Task Learning with Attention, CVPR, 2018) propose a lightweight Multi-objective algorithm of a Dynamic Weight averaging method (DWA for short), and the method does not need to establish an optimization objective, distributes the optimization Weight through the Loss change rate in the iteration process, and homogenizes the update speed of each Task.
In the aspect of improving the difficult objective law, related documents (Dynamic Task Prioritization for multi Task Learning, ECCV,2018) propose a Dynamic Task Prioritization (DTP for short) multi-objective method, which designs a specific function to strengthen the Learning of the 'difficult' Task and endows the 'difficult' Task with higher optimization weight. Related literature (Multi-object textual source mask optimization to reduce the uniform impact of polarization interaction at full exposure field, OE,2019) proposes an adaptive weight Multi-objective gradient descent algorithm, which assigns weight factors by using the specific gravity of each objective loss function in the previous round.
In terms of searching for the pareto improvement law, a Multi-gradient descent algorithm (Multi-Task Learning as Multi-Objective Optimization, neuroips, 2018) in the related art approaches the non-dominant solution by searching for the pareto improvement direction of the Multi-Objective problem.
However, each of the above algorithms is based on a "single" priority assignment rule, and the inherent contradictory relationship between the priority rules is ignored. In fact, there is a contradictory relationship between the "homogenization convergence rate" and the "improvement of the difficult target rule" priority rule. When the optimization problem is concerned with the 'homogenization convergence rate' rule in a transition way, the order of magnitude of each target is unified, and the difficult target cannot obtain a larger order of magnitude, so that the optimization process cannot be improved well; on the contrary, when the optimization problem is focused on the 'difficult target improvement' rule, the gradient magnitude of the difficult task is continuously strengthened, so that the magnitude difference between targets is enlarged, and the optimization precision is influenced.
Disclosure of Invention
In view of this, the present invention provides a high-precision hybrid dynamic priority multi-target method, which can avoid the problem of increasing the inter-target gap caused by a single priority rule.
The technical solution for realizing the invention is as follows:
a high-precision hybrid dynamic priority multi-objective optimization method comprises the following specific processes:
establishing an objective optimization function, and calculating gradient values of optimization variables relative to optimization targets based on the objective optimization function;
calculating a loss function value of each optimization target of the iteration, and calculating an overall rate factor reflecting the change rate of the target loss function, a gradient factor reflecting the update of target parameters and an overall scale factor reflecting the optimization degree of the target based on the loss function value and the gradient value;
calculating a function value of each factor by using a focus loss function, and calculating a total gradient value of each optimization target based on the function values of the three factors;
updating the optimization variables based on the total gradient values.
Further, the present invention calculates a function value of each factor based on the segmented focus loss function; the function values of the overall rate factor and the gradient factor calculated using the focus loss function are inversely proportional to themselves, and the function values of the overall scale factor calculated using the focus loss function are proportional to themselves.
Further, the focal loss function of the present invention is:
Figure BDA0003548850130000031
wherein r isiAs a whole rate factor, giIs a gradient factor, siFor the global scale factor, γ is the focusing parameter, and is usually equal to 1.
Further, the invention starts the next iteration when the loss function of each optimization target reaches the preset convergence condition.
Further, the overall rate factor is determined by performing exponential moving average on the change rate of the overall rate factor in the DWA algorithm in each turn.
Further, the overall rate factor of the present invention is:
Figure BDA0003548850130000032
EMAt(x)＝[βEMAt-1(x)+(1-β)x(t-1)]/(1-βt)
wherein, beta represents a moving average weight factor, t represents an iteration turn number, and RiRepresenting the instantaneous rate factor in the DWA algorithm.
Further, the gradient factor of the invention is L of each optimized target gradient matrix2And (4) norm.
Further, the overall scale factor is determined by performing moving average on each turn of instantaneous scale factor.
Furthermore, the invention also comprises the step of assigning coefficients to the function values calculated by the overall rate factor, the gradient factor and the overall scale factor respectively, wherein the coefficients of the overall rate factor and the gradient factor are smaller than the coefficient of the overall scale factor.
Advantageous effects
Firstly, calculating the total gradient value of each optimization target based on an overall rate factor, a gradient factor and an overall scale factor, wherein the overall rate factor reflects the change rate of a target loss function, the gradient factor reflects the update of target parameters, and the overall scale factor reflects the optimization degree of the target; the problem of the increase of the difference between targets caused by a single priority rule is avoided.
Secondly, the invention utilizes the focus loss function to distribute the weight, and distributes and focuses the priority of the target on the sudden change factor, thereby realizing the reasonable priority ordering among the optimization targets.
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FIG. 1 is a general algorithm flow diagram of the present invention.
FIG. 2 is a schematic diagram of an EUV lithography arc field of view.
FIG. 3 is a distribution diagram of an EUV lithography arc field and typical field points.
FIG. 4 is a diagram of an error distribution diagram of the dot pattern of each field of view after the optimization of the multi-objective algorithm provided by the present invention.
FIG. 5 shows the source pattern, mask pattern and photoresist image corresponding to the F2 and F5 fields of view after the algorithm of the present invention has been optimized.
Detailed Description
The present invention will be further described in detail with reference to the accompanying drawings.
The embodiment of the application discloses a high-precision hybrid dynamic priority multi-objective optimization method, as shown in fig. 1, the specific process is as follows:
establishing an objective optimization function, and calculating gradient values of optimization variables relative to optimization targets based on the objective optimization function; calculating a loss function value of each optimization target of the iteration, and calculating an overall rate factor reflecting the change rate of the target loss function, a gradient factor reflecting the update of target parameters and an overall scale factor reflecting the optimization degree of the target based on the loss function value and the gradient value; respectively endowing the overall rate factor, the gradient factor and the overall scale factor with weight values, and calculating the total gradient value of each optimization target based on the weight values; updating the optimization variables based on the total gradient values.
In the embodiment of the application, the function value of each factor is calculated based on the sectional type focus loss function; the function values of the overall rate factor and the gradient factor calculated using the focus loss function are inversely proportional to themselves, and the function values of the overall scale factor calculated using the focus loss function are proportional to themselves.
In the Hybrid Dynamic Priority (HDP for short) multi-objective optimization method of the embodiment of the present application, a total gradient value of each optimization target is calculated based on an overall rate factor, a gradient factor and an overall scale factor, the three factors respectively reflect a target loss function change rate, a target update rate and a target optimization degree, and a final Priority level is configured to be a weighted form in which three evaluation factors pass through a focus loss function, so that a purpose of integrating a "homogenization convergence rate" and a "difficult-to-improve target" dual-Priority rule is achieved, and precision reduction caused by excessive attention to a certain Priority rule is avoided. The multiple factors are brought into a specific focusing function, the priority of the target is distributed and focused on the sudden-change factors, more reasonable priority sequencing of the targets is realized, and the calculation precision is improved.
Example 1:
the method can be applied to Extreme Ultraviolet Lithography (EUVL) multi-target Resolution Enhancement Technology (RET), and belongs to the technical field of integrated circuit design and Lithography Resolution Enhancement.
In the embodiment of the present application, a hybrid dynamic priority multi-objective optimization method is used, an optimization target of the embodiment is a full-chip light source, as shown in fig. 2, one chip is composed of multiple clips, and a multi-objective light source optimization problem needs to perform synchronous optimization and performance balance for the multiple clips. The specific optimization process comprises the following steps:
step one, establishing a multi-objective optimization function of a full-chip light source optimization problem:
Figure BDA0003548850130000061
wherein Z isiFor the actual photoresist pattern of the ith clip,
Figure BDA0003548850130000062
target resist pattern for ith clip, ωiTo weight the priority factor, J represents the parameter matrix of the light source. Photo resist image Z of each clipiThe calculation is as follows:
Figure BDA0003548850130000063
wherein Z is a binary matrix for describing a photoresist pattern, 0 represents an area which is not developed by exposure, 1 represents an area which is developed by exposure, binarization processing is carried out by a threshold function gamma { }, and t isrAs the photoresist threshold, if the corresponding pixel point value is greater than tr，Z(Ai)m,n1, whereas Z (A)i)m,n＝0。PSFrThe point spread function of a photoresist is generally expressed by actual measurements
Fitting the photoresist image function to obtain the photoresist image function; i is the aerial image of the lithography, which in the absence of stray light can be expressed as:
Figure BDA0003548850130000064
wherein the operator [ ] is multiplication of corresponding elements of the matrix
Figure BDA0003548850130000065
Is a convolution of (x)s,ys) For normalized light source pupil space coordinates,
Figure BDA0003548850130000066
is RN×NThe scattering matrix of (a) is,
Figure BDA0003548850130000067
is a point spread function, and M is a mask structure matrix.
Figure BDA0003548850130000068
Representing the light intensity J (x) of each light source points,ys) The sum of (a) and (b).
Step two, calculating the gradient of the light source parameter relative to each target clip
Figure BDA0003548850130000069
Figure BDA0003548850130000071
Wherein the source variable Ω is unconstrainedJ＝arg cos(2J+1)。
Step three, calculating Loss function values Loss of each optimized target clip in the current iteration processi；
Figure BDA0003548850130000072
Step four, according to
Figure BDA0003548850130000073
And LossiAnd calculating three evaluation factors of an integral rate factor, a gradient factor and an integral scale factor, wherein the integral rate factor reflects the change rate of the loss function of the target, the gradient factor reflects the magnitude of the updating order of target parameters, and the integral scale factor reflects the optimization degree of the target.
Integral rate factor riThe calculation is as follows:
Figure BDA0003548850130000074
therein, it is noted that the instantaneous rate factor R is compared to the DWA algorithmi(t)＝Lossi(t-1)/Lossi(t) of (d). The application adopts an integral rate factor r for carrying out exponential moving average on each round change rateiTherefore, the problem of unreasonable priority caused by the rate mutation of a certain round can be effectively solved. The function EMA (x) is an exponential moving average with bias correction, whose development recursion is as follows:
EMAt(x)＝[βEMAt-1(x)+(1-β)x(t-1)]/(1-βt)
where β denotes a moving average weight factor, and is usually equal to 0.9, and t denotes an iteration number.
Gradient factor giDefined as L of each target gradient matrix2Norm, calculated as follows:
Figure BDA0003548850130000075
gradient factor is determined by taking the gradient of each target as L2The norm reflects the order of magnitude information of each target gradient, so that the targets with large order of magnitude and the targets with small order of magnitude can be distinguished conveniently, and the correct priority sequencing is realized.
Integral scale factor siThe calculation is as follows:
Figure BDA0003548850130000081
wherein the instantaneous scale factor
Figure BDA0003548850130000082
Similarly, the instantaneous scale factors of each round are subjected to moving average, so that unreasonable weight distribution caused by the mutation of the scale factors in certain round optimization is eliminated.
And the magnitude of each target and the optimization degree are quantitatively reflected through the calculation of the three factors, so that the optimization level is conveniently distributed in the subsequent step according to the magnitude of each target and the optimization degree.
Step five, establishing a priority distribution model of the focus function:
after each evaluation factor is substituted into the focus loss function, the target containing abnormal factors is focused, and the condition that each factor is substituted into the focus loss function
Figure BDA0003548850130000083
The above formula shows that the focus loss function is in a piecewise function form, and the integral rate factor and the gradient factor are taken as independent variables to be brought into the upper half of the focus loss function for calculation. For the target with the larger integral rate factor and gradient factor, the upper half of the focus loss function will return a smaller weight, and for the target with the smaller integral rate factor and gradient factor, the upper half of the focus loss function will return a larger weight, so that the priority assignment tends to "focus on" the target with the smaller update rate.
The 'integral scale factor' is taken as an independent variable and is brought into the area of the lower half section of the focus loss function, for a difficult target with a large integral scale factor, the lower half section of the focus loss function returns a large weight, and for an easy target with a small integral scale factor, the lower half section of the focus loss function returns a small weight, so that priority assignment tends to pay attention to improvement of the 'difficult' target.
Compared with the method that the weights are directly distributed through the evaluation factors, the weights are distributed through the focus loss function, so that the weight difference between the focused target and the non-focused target can be further enlarged, and the optimization focus is improved on the difficult target and the target with larger magnitude.
Step six, calculating the weighted priority:
since the return value magnitudes of the upper half and the lower half of the focus loss function are not uniform, the coefficient term C is set1,C2,C3The orders of magnitude are unified. Since the lower half of the focus loss function is significantly smaller in magnitude than the upper half, C3Should be taken to be equivalent to C1,C2Larger, typical reference value C1＝0.5,C2＝0.5,C3＝20。
ωi(t)＝C1FL(ri)+C2FL(gi)+C3FL(si)
Therefore, the comprehensive priority distribution rule of 'order of magnitude balance among targets' and 'difficult improvement target' is realized by substituting each evaluation factor into a focus function weighting mode, so that the precision reduction caused by a single priority distribution method of transitional attention is avoided.
Step five, weighting gradient according to priority:
under the condition of reasonably distributing the priority of each target, the total gradient descending direction of the light source variable is a weighted form of each target gradient and a weight factor:
Figure BDA0003548850130000091
step six, updating parameters by a gradient descent method:
returning to the main optimization flow, updating light source parameter optimization parameters according to the total gradient descending direction:
Figure BDA0003548850130000092
where k is the step size.
After updating the optimization parameters, judging whether the convergence conditions of the multi-objective optimization problem are met, and if so, ending the multi-objective optimization process; otherwise, returning to the step two for continuous optimization.
In the embodiment of the application, when the difference value of the adjacent iteration loss functions of each optimization target is smaller than the set threshold value, convergence is considered, a better optimization effect is achieved at the moment, and updating of the optimization variables can be stopped.
Example 2:
the problem of full-field light source mask collaborative optimization is solved by using a hybrid dynamic priority multi-objective algorithm, as shown in fig. 3, imaging performance in a full-field range is attenuated due to aberration or polarization aberration. Compared with the single-target light source mask collaborative optimization problem, the multi-target light source mask collaborative optimization problem considering the full field range has better full field imaging uniformity, and the specific optimization process is as follows:
step one, establishing a multi-objective optimization function of a full-field light source mask collaborative optimization problem:
Figure BDA0003548850130000101
wherein, Z (A)i) For imaging at the i-th typical field of view point wave aberration or polarization aberration, AiIs the phase factor of the ith field of view.
Figure BDA0003548850130000102
Target resist pattern, ω, for the ith typical field of view pointiIs a weight priority factor. J and M represent parameter matrixes of the light source and the mask. Photoresist image Z (A) at each typical field of view pointi) Is defined as:
Figure BDA0003548850130000103
wherein Z is a binary matrix for describing a photoresist pattern, 0 represents an area which is not developed by exposure, 1 represents an area which is developed by exposure, binarization processing is carried out by a threshold function gamma { }, and t isrIf the corresponding pixel point value is greater than t, the value is the photoresist threshold valuer，Z(Ai)m,n1, whereas Z (A)i)m,n0. In the absence of stray light, it can be stated that the aerial image of the ith typical field-of-view point is defined as:
Figure BDA0003548850130000104
wherein the operator is a multiplication of corresponding elements of the matrix
Figure BDA0003548850130000105
Is a convolution of (x)s,ys) For normalized light source pupil space coordinates,
Figure BDA0003548850130000106
is RN×NThe scattering matrix of (a) is,
Figure BDA0003548850130000107
is a point spread function containing aberration or polarization aberration, and M is a mask structure matrix.
Step two, calculating gradients of light source and mask parameters relative to each target clip
Figure BDA0003548850130000108
Figure BDA0003548850130000111
Wherein the unconstrained light source variable and mask variable are respectively defined as omegaJ＝arg cos(2J+1)，ΩM＝arg cos(2M+1)。
Step three, calculating Loss function values Loss of various clips in the current iteration processi；
Figure BDA0003548850130000112
Step four, according to
Figure BDA0003548850130000113
And LossiAnd calculating three evaluation factors of an integral rate factor, a gradient factor and an integral scale factor:
integral rate factor riThe calculation is as follows:
Figure BDA0003548850130000114
gradient factor giIs defined as L of each target gradient matrix2The norm, the gradient factors for the source and mask are calculated as follows:
Figure BDA0003548850130000115
integral scale factorSon siThe calculation is as follows:
Figure BDA0003548850130000116
wherein the instantaneous scale factor
Figure BDA0003548850130000117
Step five, establishing a priority distribution model of the focus function:
after each evaluation factor is substituted into the focus loss function, the target containing abnormal factors is focused, and the condition that each factor is substituted into the focus loss function
Figure BDA0003548850130000118
Step four, calculating the weighted priority (including the light source parameter and the mask parameter):
similar to the case of one variable, for ri,gi,siWeighting is performed by a Focus Loss function. Since the gradient factors of the light source variable and the mask variable are not consistent, the priorities of the light source variable and the mask variable need to be calculated respectively.
Figure BDA0003548850130000121
Figure BDA0003548850130000122
Step five, weighting gradient according to priority:
under the condition of reasonably distributing the priority of each target, respectively calculating the total gradient descending direction of the light source variable and the mask variable, and taking the total gradient descending direction as a weighting form of each target gradient and a weighting factor:
Figure BDA0003548850130000123
step six, updating parameters by a gradient descent method:
returning to the main optimization flow, updating light source parameter optimization parameters according to the total gradient descending direction:
Figure BDA0003548850130000124
Figure BDA0003548850130000125
where k is the step size.
After updating the optimization parameters, judging whether the convergence conditions of the multi-objective optimization problem are met, and if so, ending the multi-objective optimization process; otherwise, returning to the step two for continuous optimization.
Implementation examples of the invention:
as fig. 2 is a schematic diagram of an EUVL arc Field of View, since wavefront errors in EUVL depend on the position of the Field of View (FOV), it is difficult for the single target RET technique considering only a single Field point to meet the resolution requirement in the full Field range. Therefore, the optimization objective function of the multi-objective RET considering the low sensitivity of the wavefront error in the full field of view is established as follows:
Figure BDA0003548850130000126
wherein, Z (A)i) Is the photoresist pattern under the ith sampling field-of-view point wave aberration, and Z is the target photoresist pattern. Therefore, the method takes the figure fidelity under each view field point as a starting point, and converts the optimization problem of the imaging consistency under each view field point into a multi-target optimization problem containing the figure fidelity of each typical view field point.
The multi-target problem is taken as an implementation example, and the calculation accuracy of the gradient multi-target algorithm (GradNarm, DWA, DTP, Ada-Weight and MGDA) which is mainstream at present and the HDP algorithm provided by the invention is compared.
In fig. 3, histograms of different patterns represent the imaging conditions of the system at different view points after different algorithms are optimized. The graph Error (PAE) of the ordinate represents the pixel Error between the actual exposure graph and the target exposure graph, and the smaller the PAE, the higher the graph fidelity. By integrating the imaging conditions of each typical field-of-view point, the HDP algorithm provided by the invention has optimal optimization performance and minimum average value PAE in the full field-of-view range. The HDP algorithm thus further improves EUVL full field imaging uniformity over other mainstream multi-objective gradient algorithms.
As shown in fig. 4, the system and the partial field of view imaging situation after the optimization by the HDP algorithm are shown, 401 is a light source graph obtained after the optimization by the HDP algorithm; 402 is an optimized mask pattern; 403 is the imaging of the optimized system at F2 field of view point, with PAE of 348; 404 for the optimized system imaging at F5 field of view point, its PAE is 268.
From the above analysis, the HDP multi-target algorithm provided by the research effectively improves the uniformity of full-field imaging in the EUVL multi-target RET technology. Compared with other mainstream gradient-based multi-target algorithms, the method has higher full-field imaging uniformity. Because the HDP multi-objective algorithm has universality, the application range is not limited to the HDP multi-objective algorithm, and the HDP multi-objective algorithm is theoretically suitable for any multi-objective optimization problem based on gradients.
Although the embodiment of the present invention has been described with reference to the drawings, it should be understood that the present invention is not limited to the specific embodiments, but may be embodied in various other forms without departing from the spirit or scope of the present invention.

Claims (9)

1. A high-precision hybrid dynamic priority multi-objective optimization method is characterized by comprising the following specific processes:
establishing an objective optimization function, and calculating gradient values of optimization variables relative to optimization targets based on the objective optimization function;
calculating a loss function value of each optimization target of the iteration, and calculating an overall rate factor reflecting the change rate of the target loss function, a gradient factor reflecting the update of target parameters and an overall scale factor reflecting the optimization degree of the target based on the loss function value and the gradient value;
calculating a function value of each factor by using the focus loss function, and calculating a total gradient value of each optimization target based on the function values of the three factors;
updating the optimization variables based on the total gradient values.
2. The high-precision hybrid dynamic priority multi-objective optimization method according to claim 1, wherein the function value of each factor is calculated based on a segmented focus loss function; the function values of the overall rate factor and the gradient factor calculated using the focus loss function are inversely proportional to themselves, and the function values of the overall scale factor calculated using the focus loss function are proportional to themselves.
3. The high-precision hybrid dynamic priority multi-objective optimization method according to claim 2, wherein the focus loss function is:
Figure FDA0003548850120000011
wherein r isiAs a whole rate factor, giIs a gradient factor, siGamma is the set focus parameter, which is the global scale factor.
4. The high-precision hybrid dynamic priority multi-objective optimization method according to claim 1, wherein when the loss function of each optimization objective reaches a preset convergence condition, a next iteration is started.
5. The high-precision hybrid dynamic priority multi-objective optimization method according to claim 4, wherein the overall rate factor is an exponential moving average of the rate of change of the overall rate factor in the DWA algorithm in each round.
6. The high-precision hybrid dynamic priority multi-objective optimization method according to claim 5, wherein the overall rate factor is:
Figure FDA0003548850120000021
EMAt(x)＝[βEMAt-1(x)+(1-β)x(t-1)]/(1-βt)
wherein beta represents a moving average weight factor, t represents an iteration turn number, and RiRepresenting the instantaneous rate factor in the DWA algorithm.
7. The high-precision hybrid dynamic priority multi-objective optimization method according to claim 1, wherein the gradient factor is L of each optimization target gradient matrix2And (4) norm.
8. The high-precision hybrid dynamic priority multi-objective optimization method according to claim 1, wherein the overall scale factor is a moving average of instantaneous scale factors for each round.
9. The high-precision hybrid dynamic priority multi-objective optimization method according to claim 1, further comprising assigning coefficients to the function values calculated by the overall rate factor, the gradient factor and the overall scale factor, respectively, wherein the coefficients of the overall rate factor and the gradient factor are smaller than the coefficients of the overall scale factor.
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