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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 PDF

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CN109165416A
CN109165416A CN201810853780.2A CN201810853780A CN109165416A CN 109165416 A CN109165416 A CN 109165416A CN 201810853780 A CN201810853780 A CN 201810853780A CN 109165416 A CN109165416 A CN 109165416A
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radiation energy
iteration
backtracking
sparse
conjugate gradient
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CN109165416B (en
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张雁峰
黄运保
李海艳
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Guangdong University of Technology
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Abstract

本发明属于惯性约束聚变试验中靶丸靶腔的初步设计领域，以及大规模和超大规模非线性辐射能量平衡方程组的求解，涉及一种基于回溯的共轭梯度迭代硬阈值辐射能量求解方法。本发明主要由两部分组成，第一部分是设计不同拓扑结构的靶腔和靶丸辐射能量稀疏表示方法，并构建关于稀疏系数的非线性辐射能量平衡方程模型，第二部分是提出一种基于回溯的共轭梯度迭代硬阈值求解该能量平衡方程。本发明大规模减少所求能量平衡方程的个数，减少模型的求解时间，从而极大地提高了辐射对称性评估效率；充分利用了前次迭代中有用的信息，优化了支撑集的选择策略，确保支撑集被快速准确找到；适合于非线性稀疏模型的求解，且求解效率和精度都能得到保障。

The invention belongs to the field of preliminary design of target cavities in inertial confinement fusion experiments, and the solution of large-scale and ultra-large-scale nonlinear radiation energy balance equations, and relates to a backtracking-based conjugate gradient iteration hard threshold radiation energy solution method. The invention is mainly composed of two parts, the first part is to design the sparse representation method of the radiation energy of the target cavity and the target pellet with different topological structures, and build the nonlinear radiation energy balance equation model about the sparse coefficient; the second part is to propose a backtracking-based The conjugate gradient of the iterative hard thresholding solves this energy balance equation. The invention greatly reduces the number of energy balance equations to be sought, and reduces the solution time of the model, thereby greatly improving the radiation symmetry evaluation efficiency; fully utilizing the useful information in the previous iteration, optimizing the selection strategy of the support set, Ensure that the support set is found quickly and accurately; it is suitable for solving nonlinear sparse models, and the solution efficiency and accuracy can be guaranteed.

Description

Conjugate gradient iteration hard -threshold radiation energy method for solving based on backtracking
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)

1.一种基于回溯的共轭梯度迭代硬阈值辐射能量求解方法，其特征在于，包括以下步骤：1. a conjugate gradient iteration hard threshold radiation energy solution method based on backtracking, is characterized in that, comprises the following steps: S1.建立基于离散视角因子的辐射能量平衡方程模型；S1. Establish a radiation energy balance equation model based on discrete viewing angle factors; S2.设计合适的稀疏基对靶丸、靶腔进行稀疏表示，并构建关于稀疏系数的辐射能量平衡方程；S2. Design a suitable sparse basis to sparsely represent the target pellet and target cavity, and construct the radiation energy balance equation about the sparse coefficient; S3.对关于稀疏系数的辐射能量平衡方程进行压缩采样；S3. Compressed sampling of the radiation energy balance equation with respect to the sparse coefficient; S4.基于回溯的共轭梯度迭代硬阈值求解关于稀疏系数的辐射能量平衡方程；S4. Solving the radiation energy balance equation with respect to the sparse coefficients based on the backtracking conjugate gradient iterative hard threshold; S5.根据所求的稀疏系数反求辐射能量。S5. Reverse the radiation energy according to the required sparse coefficient. 2.根据权利要求1所述的基于回溯的共轭梯度迭代硬阈值辐射能量求解方法，其特征在于，所述的S1步骤具体包括：2. the conjugate gradient iteration hard threshold radiation energy solution method based on backtracking according to claim 1, is characterized in that, described S1 step specifically comprises: S11.建立ICF基于离散视角因子非线性能量平衡方程：S11. Establish ICF nonlinear energy balance equation based on discrete viewing angle factor: 式中，Bi表示待求的辐射能量，Ei表示第i个面元初级入射光源转化为软x光后对其他面元的直接能量，Aj表示第j个面元的面积，Fji表示视角因子矩阵，是一个对称矩阵，λi是第i个面元的反照率；In the formula, B i represents the radiation energy to be determined, E i represents the direct energy of the i-th surface element after the primary incident light source is converted into soft x-rays to other surface elements, A j represents the area of the j-th surface element, F ji Represents the viewing angle factor matrix, which is a symmetric matrix, and λ i is the albedo of the i-th surface element; 其中，Ei和λi用如下计算公式求得：Among them, E i and λ i are obtained by the following formulas: 式中，表示初级入射光源的能量，C、α和β是常数；In the formula, represents the energy of the primary incident light source, C, α and β are constants; S12.将ICF基于离散视角因子非线性能量平衡方程写成矩阵形式：S12. Write the ICF nonlinear energy balance equation based on discrete viewing angle factors in matrix form: 令y＝E，则：Let y=E, then: 式中，I表示单位矩阵；V表示由Fij组成的视角因子矩阵，即V＝{Fij}；E表示由Ei构成的向量，即E＝[E1,E2,…En]T；表示向量a中每个元素都以b做指数幂。In the formula, I represents the unit matrix; V represents the viewing angle factor matrix composed of F ij , that is, V={F ij }; E represents the vector composed of E i , that is, E=[E 1 ,E 2 ,...E n ] T ; Indicates that each element in the vector a is raised to the power of b. 3.根据权利要求2所述的基于回溯的共轭梯度迭代硬阈值辐射能量求解方法，其特征在于，所述的S2步骤具体包括：3. the conjugate gradient iteration hard threshold radiation energy solution method based on backtracking according to claim 2, is characterized in that, described S2 step specifically comprises: S21.将辐射能量用稀疏基表示为：S21. Represent the radiation energy with a sparse basis as: B＝ψxB=ψx S22.将S12步骤的能量平衡方程改写为关于稀疏系数的方程 S22. Rewrite the energy balance equation of step S12 as an equation about sparse coefficients 式中，其中ψ是稀疏基，上式是一个非线性方程组，它的一阶泰勒展开可表示成：where ψ is a sparse basis, the above formula is a nonlinear equation system, and its first-order Taylor expansion can be expressed as: f(x)＝f(xn)+f'(xn)(xn+1-xn)f(x)=f(x n )+f'(x n )(x n+1 -x n ) 其中，f'(xn)＝J×Basis，J为S22步骤中能量平衡方程的雅可比矩阵，Basis为稀疏基；Among them, f'(x n )=J×Basis, J is the energy balance equation in step S22 The Jacobian matrix of , Basis is a sparse basis; 令f'(xn)＝Φ、f(x)-f(xn)+f'(xn)xn＝y，则，一阶泰勒展开式可表示为：Let f'(x n )=Φ, f(x)-f(x n )+f'(x n )x n =y, then, the first-order Taylor expansion can be expressed as: y＝Φx。y=Φx. 4.根据权利要求3所述的基于回溯的共轭梯度迭代硬阈值辐射能量求解方法，其特征在于，所述的S3步骤具体包括：4. the conjugate gradient iteration hard threshold radiation energy solution method based on backtracking according to claim 3, is characterized in that, described S3 step specifically comprises: S31.利用拉丁超立方采样发对y＝Φx进行压缩采样，采样个数根据下式计算：S31. Use Latin hypercube sampling to compress and sample y=Φx, and the number of samples is calculated according to the following formula: m≥k log(N)m≥k log(N) 式中，m为采样数，k为辐射能量在稀疏基上的稀疏度，N为总的面元数量；In the formula, m is the number of samples, k is the sparsity of radiation energy on the sparse basis, and N is the total number of bins; S32.采样后的y＝Φx是一个以l0范数为约束的优化问题，用下式表示：S32. The sampled y=Φx is an optimization problem constrained by the l 0 norm, which is expressed by the following formula: 式中，A为对Φ进行采样后的矩阵。In the formula, A is the matrix after sampling Φ. 5.根据权利要求4所述的基于回溯的共轭梯度迭代硬阈值辐射能量求解方法，其特征在于，所述的S4步骤在每步迭代中首先利用支撑集回溯的方法求得近似解，然后使用改进的共轭梯度法更新近似解。5. the conjugate gradient iteration hard threshold radiation energy solution method based on backtracking according to claim 4, is characterized in that, described S4 step first utilizes the method of support set backtracking to obtain approximate solution in each step iteration, then The approximate solution is updated using the modified conjugate gradient method. 6.根据权利要求5所述的基于回溯的共轭梯度迭代硬阈值辐射能量求解方法，其特征在于，由于s.t.||x||0≤k是一个欠定方程组，求解它是一个NP难问题，但是如果矩阵A满足一定的有限等距条件，则可通过以下算法精确重构x；其算法的具体步骤包括：6. The conjugate gradient iterative hard threshold radiation energy solution method based on backtracking according to claim 5, is characterized in that, due to st||x|| 0 ≤k is an underdetermined system of equations, and it is an NP-hard problem to solve, but if the matrix A satisfies certain finite equidistant conditions, x can be accurately reconstructed by the following algorithm; the specific algorithm Steps include: 以下步骤中，表示空集，Hs(α)是一个非线性算子，表示取α中绝对值最大的s个元素，μ为迭代步长，Γn表示第n次迭代的索引集合，∪表示并集，＜·,·＞表示求内积，supp(·)表示向量非零元素索引，ε表示迭代停止误差阈值；In the following steps, represents the empty set, H s (α) is a nonlinear operator, which means to take the s elements with the largest absolute value in α, μ is the iteration step size, Γ n represents the index set of the nth iteration, ∪ represents the union, ＜·,·>represents the inner product, supp(·) represents the non-zero element index of the vector, and ε represents the iteration stop error threshold; S41.初始化x0＝0，残差r0＝y，初始支撑集初始梯度：g0＝r0，初始寻优方向d0＝g0，步长x1＝Hs(x0+μ0d0)，Γ1＝supp(x1)；S41. Initialize x 0 =0, residual r 0 =y, initial support set Initial gradient: g 0 =r 0 , initial optimization direction d 0 =g0, step size x 1 =H s (x 0 +μ 0 d 0 ), Γ 1 =supp(x 1 ); S42.迭代过程：S42. Iterative process: S421.更新梯度方向gn＝AT(y-Ax)；S421. Update gradient direction g n = AT (y-Ax); S422.判断是否Γn＝Γn-1，如果是，转步骤S424，如果否，转步骤S423；S422. Determine whether Γ n = Γ n-1 , if yes, go to step S424, if not, go to step S423; S423.梯度下降法：S423. Gradient descent method: S4231.计算步长 S4231. Calculation step size S4232.迭代an＝Hs(xn-1+μngn)；S4232. Iteration an = H s (x n -1 +μ n g n ); S424.共轭梯度法：S424. Conjugate gradient method: S4241.计算方向因子 S4241. Calculate direction factor S4242.更新寻优方向dn＝gn+βndn-1；S4242. Update the optimization direction d n =g n +β n d n-1 ; S4243.更新步长 S4243. Update step size S4244.迭代an＝Hs(xn-1+μndn)；S4244. Iteration an = H s (x n -1 + μ n d n ); S425.回溯：S425. Backtracking: S4251.合并支撑集Γn＝supp(an+1)∪supp(xn)；S4251. Merge support set Γ n =supp(a n+1 )∪supp(x n ); S4252. S4252. S426.更新残差rn+1＝y-Axn+1；S426. Update residual r n+1 =y-Ax n+1 ; S43.判断是否||rn+1||2＜ε，若是，迭代停止，否则返回S422步骤继续迭代。S43. Determine whether ||r n+1 || 2 <ε, if yes, stop the iteration, otherwise return to step S422 to continue the iteration. 7.根据权利要求6所述的基于回溯的共轭梯度迭代硬阈值辐射能量求解方法，其特征在于，所述的S4步骤为：当辐射能量稀疏表示模型求解完成后，得到x，然后即可通过B＝ψx得到所求靶丸和靶腔上的辐射能量。7. The conjugate gradient iterative hard threshold radiation energy solution method based on backtracking according to claim 6, wherein the step S4 is: when the radiation energy sparse representation model is solved, obtain x, and then The radiation energy on the desired target ball and target cavity is obtained by B=ψx.
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