A block iterative matrix solution method based on matrix dimension reduction and preprocessing technique

Abstract

The present application belongs to the technical field of block iterative matrix solution in computational electromagnetics, in particular to a block iterative matrix solution method based on matrix dimension reduction and preprocessing technology. The present application uses two technologies of matrix dimension reduction and matrix preprocessing: the right end item matrix is processed by using the matrix dimension reduction, through low rank approximation, which can effectively ensure the accuracy of the solution, significantly reduce the calculation cost, and facilitate the processing of multiple right end linear systems; and the BCOCG algorithm is applied to the preprocessing technology to further improve the matrix behavior and improve the convergence and stability of the algorithm. For the system equation generated by multiple physical excitation sources or multiple scanning angles in electromagnetic problems, the memory cost is shortened from the perspective of short recursive Krylov subspace iteration method, and the block iteration BCOCG algorithm can calculate multiple column right end item vectors at the same time, avoiding the matrix ill-conditioning problem caused by the correlation between columns and columns.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of block iterative matrix solution in computational electromagnetics, and particularly relates to a block iterative matrix solution method based on matrix dimension reduction and preprocessing technology. BACKGROUND

[0002] In computational electromagnetics, such as the finite element method, the method of moments, etc., a global linear system matrix needs to be solved finally. For different problems, the properties of the matrix are different.

[0003] For example, in the calculation of the radar cross section (RCS) of a single station in computational electromagnetics, the system equation generated by multiple physical excitation sources or multiple scanning angles often has a right end term that is no longer a single column vector but a matrix composed of multiple columns. Since the right end term is a matrix, in order to conveniently solve such a matrix problem at one time, people can start from the block iteration angle and use the block Krylov subspace numerical iteration method to obtain an approximate solution of the original problem.

[0004] In previous studies, based on the classical generalized minimum residual method GMRES and the R criterion, an inexact block Krylov subspace iteration algorithm (BGMRES-DR block iteration algorithm) was proposed, which can effectively solve the right end matrix equation. However, in actual simulation, it is found that when facing complex large-scale problems, the memory consumption is large. The main reason is that GMRES is a long recursive method, which needs to store all previous basis vectors in each iteration to ensure the optimal approximation of the residual in all Krylov subspace directions at each iteration; and its computational complexity increases with the increase of the number of iteration steps. Although GMRES has a wide range of applications and is suitable for any matrix. However, for special problems of large-scale and right end matrix being a multi-column matrix, the memory will bring serious pressure with multiple iterations, thereby affecting the simulation efficiency.

[0005] Therefore, for wide matrix right end terms, matrix problems that need to be solved simultaneously, especially large-scale matrix problems, there is an urgent need for a method that takes into account memory consumption and can significantly improve computational performance. SUMMARY

[0006] To address the aforementioned problems and shortcomings, and to resolve the issues of excessive memory consumption and computational inefficiency faced by the conventional GMRES algorithm when solving multiple right-hand side matrices and multiple column vectors simultaneously, this invention provides a block iterative matrix solving method based on matrix dimensionality reduction and preprocessing techniques. This invention addresses system equations generated by multiple physical excitation sources or multiple scanning angles. Since a right-hand side matrix of a system equation is composed of multiple right-hand side vectors, if the correlation between columns and column vectors is strong or the number of right-hand side vector columns is large, applying the Krylov subspace iterative method for matrix solving will cause the dimensionality of the Krylov subspace to grow very rapidly, resulting in excessive memory requirements and potentially slowing convergence speed or even leading to ill-conditioned matrices. This invention employs matrix dimensionality reduction to process the right-hand side matrix. Through low-rank approximation, it effectively ensures the accuracy of the solution while significantly reducing computational overhead, facilitating the handling of multi-right-hand linear systems. Furthermore, it applies preprocessing techniques to the BCOCG algorithm to further improve the algorithm's convergence and stability.

[0007] A block iterative matrix solving method based on matrix dimensionality reduction and preprocessing techniques includes the following steps:

[0008] Step A: Simulate and model the target structure by combining the material, working environment, and boundary conditions; then, perform meshing on the three-dimensional target model, and the surface discretization and volume discretization must be compatible.

[0009] Step B: Based on the solution process of the target's electromagnetic problem with multiple right-hand sides, a global linear system containing multiple right-hand sides is obtained; the corresponding matrix equation with multiple right-hand sides is:

[0010] AX = B (1)

[0011] Where A is the global system matrix with dimension m×m; the right-hand side matrix B is a multi-right-hand side matrix obtained from p incident wave excitations with dimension m×p, i.e., it has p columns of right-hand side vectors; the unknown matrix to be solved is X with dimension m×p.

[0012] Step C: Perform matrix dimensionality reduction on the matrix B with multiple right-hand side terms to obtain the dimensionality-reduced matrix newB.

[0013] From the perspective of low-rank approximation, matrix dimensionality reduction is performed on the right-hand side matrix B to reduce the number of columns p of the right-hand side while preserving the dominant direction.

[0014] The dimensionality reduction method employs selective QR decomposition, which selects the most relevant columns based on the 2-norm of the matrix columns, constructs an orthogonal basis matrix Q that satisfies orthogonality, and generates a corresponding matrix R such that B≈QR. The selection of columns is carried out iteratively, and the algorithm is terminated based on whether the tolerance condition is met.

[0015] Dimension reduction makes the original multi-right end item matrix B into a new matrix newB, and the corresponding matrix dimension changes from m x p to m x np, where np≤p, and np is the column number of the matrix after dimension reduction.

[0016] Step D, through the matrix equation newB after dimension reduction in step C, combined with the R matrix in step C, the solution (all column vectors of unknowns) of the original equation (1) is obtained by the block iterative BCOCG algorithm.

[0017] According to the dimension reduction processing of step C, the matrix equation (1) of the original target multi-right end item electromagnetic problem is converted into the equation (3) after dimension reduction, as follows:

[0018] AX * = newB (3)

[0019] Through the block iterative BCOCG algorithm, the solution X of the dimension-reduced matrix is obtained * , whose dimension is m x np; then combined with the R matrix in step C, the solution of the original equation (1) is:

[0020] X = X * x R (4)

[0021] Thus, the back substitution is completed for the matrix equation after dimension reduction.

[0022] Further, step A adopts tetrahedral elements for subdivision of the three-dimensional target model.

[0023] Further, the multi-right end item electromagnetic problem of the target is the electromagnetic problem of single-station RCS.

[0024] Further, the process of matrix dimension reduction in step C is as follows:

[0025] 1) Initialization:

[0026] ① Get the dimension size of the right end item matrix B, that is, m rows and p columns; before the loop, record the initial value of the right end item matrix B as BB = B.

[0027] ② Define the candidate column index set S = {1, 2, …, p}, which initially contains all columns; calculate the 2-norm of each column to obtain the column norm array colNorm(j) = ||B(:, j)||2, j∈S of the right end item matrix B, and select the column index ind of the maximum value in the column norm of the matrix B as the main element.

[0028] ③ np = 0, record the number of columns that have been selected, that is, the column number of the orthogonal basis matrix Q after dimension reduction.

[0029] 2) Main loop, select the leading column one by one and update the matrix.

[0030] 1. Select the column with the largest 2-norm in the current candidate set, B(:,ind), as the new dominant direction, and update np = np + 1. Add the new column to the orthogonal matrix Q, i.e., Q(:,np) = B(:,ind).

[0031] 2. Apply the Gram-Schmidt orthogonalization process to the new column Q(:,np).

[0032] 3. Normalize the new column Q(:,np) = B(:,ind) / ||B(:,ind)||2 to have unit length.

[0033] 4. Update the index set S of the candidate columns: remove the current dominant column ind from the index set S and set the norm of the column to 0.

[0034] 5. Update all the candidate columns by removing the influence of the new orthogonal basis Q(:,np) from the columns of B(:,j) for j ∈ S:

[0035] B(:,j) = B(:,j) - (Q(:,np) H ·B(:,j))·Q(:,np) (2)

[0036] Then, recalculate the 2-norm of the candidate columns, colNorm(j) = ||B(:,j)||2, j ∈ S, for the selection of the next dominant direction.

[0037] 6. Check the termination condition: find the index of the column with the largest 2-norm, ind = max(colNorm), and the value of the largest 2-norm, val. If the largest norm is less than the tolerance threshold tol, i.e., val < tol, stop the iteration; otherwise, continue the selection and update of the next round.

[0038] 3) Finally, output the orthogonal basis matrix Q = Q(:,1:np) with the first np columns, ensuring that the reduced matrix Q has m rows and np columns. Then, calculate the matrix R = Q H ·BB, which has dimensions np × p.

[0039] Thus, the selective QR decomposition algorithm selects the most contributing direction vectors from the original right-hand side matrix B to generate the orthogonal basis matrix Q, while removing redundant directions in the candidate columns, achieving the dimension reduction of the matrix B.

[0040] The right-hand side matrix Q obtained after dimension reduction is set as newB = Q to distinguish the original multi-right-hand side matrix B before dimension reduction from the right-hand side matrix newB after dimension reduction.

[0041] Further, the step B multiple right end item matrix equation is preprocessed by multiple wavefront block incomplete Cholesky decomposition and then subjected to matrix dimension reduction processing.

[0042] Further, the specific process of the multiple wavefront block incomplete Cholesky decomposition preprocessing is as follows:

[0043] Before solving equation (1), another non-singular matrix is used to preprocess the coefficient matrix to reduce the power of the minimum eigenvalue corresponding to the original coefficient matrix, so that the preprocessed matrix equation has a good form, thereby facilitating the subsequent solution of the BCOCG iterative algorithm.

[0044] First, set M as a left preprocessing sub, and process AX=B by using the left preprocessing sub M, so that M -1 AX=M -1 B; the preprocessed matrix is regarded as a whole B←M -1 B, and the dimension reduction processing of step C is adopted.

[0045] In summary, the present application uses two technologies of matrix dimension reduction and matrix preprocessing: the right end item matrix is processed by using matrix dimension reduction, and through low rank approximation, the accuracy of the solution can be effectively guaranteed, the calculation cost is significantly reduced, and the multiple right end linear system is facilitated to be processed; and the BCOCG algorithm is applied to the preprocessing technology to further improve the behavior of the matrix and the convergence and stability of the algorithm. For the complex symmetric linear system generated in the electromagnetic problem, the memory cost is shortened from the perspective of the short recursive Krylov subspace iteration method, the block iteration BCOCG algorithm can calculate multiple column right end item vectors at the same time, and the matrix ill-conditioned problem caused by the correlation between columns is avoided. BRIEF DESCRIPTION OF DRAWINGS

[0046] Figure 1 is a flowchart of the present application.

[0047] Figure 2 is a structural schematic diagram of an embodiment-combination model of a ball and a square.

[0048] Figure 3 is a single station RCS result graph of four methods of an embodiment and comparative examples. DETAILED DESCRIPTION

[0049] The present application will be further described in detail below in combination with the drawings and embodiments.

[0050] The solving problem of the embodiment is a single station RCS problem, that is, the transmitting and receiving radars belong to the same position. In order to analyze the single station RCS value in a certain direction range and help the user to evaluate the detection and concealment performance of the target at different angles, a series of different incident plane wave excitations in the direction range are first defined. The different incident plane wave excitations constitute a multi-right end item vector, which needs to be solved by combining the HDGTD numerical algorithm. In order to further efficiently solve the integrated global linear system in the HDGTD algorithm, the application proposes a multi-right end item technology containing matrix preconditioning. The application will be further described in detail in combination with the drawings and embodiments.

[0051] A block iteration matrix solving method based on matrix dimension reduction and preconditioning technology, with reference to Figure 1 , comprising the following steps:

[0052] Step A, the target structure is simulated and modeled in combination with the material, the combination working environment and the boundary conditions. Then, the three-dimensional target model is divided, and the surface discretization and the volume discretization must be compatible.

[0053] The embodiment takes the model shown in Figure 2 as an example. Since the application mainly solves the solving problem of the multi-right end item matrix, the solving of the single station RCS is taken as an example in the embodiment. The three-dimensional target model is divided by using the tetrahedral element, and the surface discretization and the volume discretization must be compatible. The division technology is a known process, and therefore the step will not be described in detail.

[0054] Step B, according to the solving process of the single station RCS problem of the model shown in Figure 2 , a global linear system containing a multi-right end item is obtained.

[0055] According to the electromagnetic problem of the single station RCS, a multi-right end item matrix equation as shown below is finally obtained:

[0056] AX=B (1)

[0057] Wherein A is the global system matrix, which is often a complex symmetric matrix, and the dimension is m x m. The right end item matrix B is a multi-right end item matrix obtained by p incident wave excitations, and the dimension is m x p, that is, there are p columns of right end item vectors. The unknown matrix to be solved is X, and the dimension is m x p.

[0058] By solving equation (1), once X is obtained, the final single station RCS response can be obtained according to the post-processing of the field value and the RCS.

[0059] Since the traditional short recursion COCG algorithm can only solve a single right end item vector at a time, if the COCG algorithm is used to solve formula (1), it needs to be looped p times, and each loop can only solve the unknown quantity of one column vector of X. The present application is to expand the COCG algorithm into the BCOCG algorithm of block iteration, so that all column vector unknown quantities can be solved at the same time. Since the BCOCG block iteration algorithm is a well-known theory, it will not be described in detail here.

[0060] Preferably, the multifrontal block incomplete Cholesky decomposition preprocessing technique is integrated into the BCOCG block iteration algorithm for further acceleration of the convergence of the algorithm.

[0061] When solving the general sparse matrix linear system AX=B, directly using the Krylov subspace method (such as GMRES) may lead to very slow convergence, especially when the condition number of A is poor (ill-conditioned matrix). In order to improve the computational performance and numerical stability of matrix solving, the present embodiment also uses the preprocessing technique of multifrontal block incomplete Cholesky (MFBIC), combined with the BCOCG block iteration algorithm, to further reduce the storage space and computational amount.

[0062] The specific implementation process is as follows: before solving formula (1), another non-singular matrix is used to preprocess the coefficient matrix to reduce the power of the minimum eigenvalue corresponding to the original coefficient matrix, so that the preprocessed matrix equation has good form, thereby facilitating the subsequent solution of the BCOCG iteration algorithm.

[0063] First, set M as a left preprocessing sub, and process AX=B with the left preprocessing sub M, then M -1 AX=M -1 B. The preprocessed matrix is regarded as a whole B←M -1 B is processed again by the dimension reduction of step C. The present embodiment uses the MFBIC preprocessing method, which is especially suitable for solving large sparse matrices, and mainly includes two stages of analysis and decomposition (see Table 1). Since the implementation of MFBIC preprocessing is a well-known process, the formula derivation will not be described in detail here.

[0064] Table 1 MFBIC preprocessing construction flow:

[0065]

[0066]

[0067] Step C, with the number of right end item vectors increasing, in order to avoid the matrix singular value appearing in the direct application of BCOCG, the matrix dimension reduction processing is carried out on the multiple right end item matrix B, and the right end item matrix newB after dimension reduction is obtained.

[0068] Although the block iterative BCOCG algorithm is a very efficient Lanczos type Krylov subspace algorithm for solving complex symmetric linear systems, however, with the increase of the number of right end items, the algorithm has the following shortcomings:

[0069] (1) When p increases, all columns of the right end item matrix B are operated each time, and multiple iteration directions are maintained, which greatly increases the storage requirements of matrix and vector.

[0070] (2) In the process of BCOCG algorithm, the search direction matrix P m is used in the Krylov subspace m When p is large, although U m is a small matrix, its column vectors may be highly correlated, and with the increase of the iteration step m, the condition number of the matrix U m may increase rapidly, and even may become close to singular, so that the residual matrix R m+1 and the direction matrix P m+1 are invalid, and the algorithm cannot continue.

[0071] (3) When p increases, the column vectors of the right end item matrix B in the actual problem may not be completely independent. If the columns of B are in a low-rank subspace, the redundant information will cause the algorithm to try to generate meaningless search directions in the redundant direction, and at the same time, these redundant information will reduce the robustness of the numerical solution, thereby slowing down the convergence speed of the algorithm.

[0072] Therefore, in order to alleviate the above shortcomings, the present application reduces the dimension of the right end item matrix B from the low-rank approximation point of view, so as to reduce the column number p of the right end item and thus retain the dominant direction. The dimension reduction method of the present application is to use selective QR decomposition, which is an algorithm for extracting dominant directions (dimension reduction) from the original matrix. The goal is to select the most relevant columns according to the 2-norm of the matrix columns, construct an orthogonal basis matrix Q (satisfying orthogonality), and generate a corresponding matrix R, so that B≈QR. The columns are selected by iterative steps, and whether to terminate the algorithm is judged according to whether the tolerance condition is met.

[0073] Through the dimension reduction technology of the present application, the original right end item matrix B becomes a new matrix Q, and the corresponding matrix dimension changes from m×p to m×np, wherein np≤p.

[0074] The specific dimension reduction steps are as follows:

[0075] 1) Initialization:

[0076] 1) Get the size of the right-hand side matrix B, i.e. m rows and p columns. Since the value of B will change in the later loop, here we first record the initial value of the right-hand side matrix B as BB = B before the loop.

[0077] 2) Define the candidate column index set S = {1, 2, …, p}, which initially contains all columns. Calculate the 2-norm of each column to obtain the column norm array of the right-hand side matrix B, colNorm(j) = ||B(:, j)||2, j ∈ S, and select the column with the maximum column norm of B as the leading element by finding the column index ind.

[0078] 3) np = 0, record the number of columns that have been selected so far, i.e. the number of columns of the orthogonal basis matrix Q after dimension reduction.

[0079] 2) Main loop, select the leading column and update the matrix in turn.

[0080] 1) Select the column with the maximum 2-norm in the current candidate column B(:, ind) as the new leading direction column, at this time np = np + 1, add the current column (the column with the maximum norm B(:, ind)) as the new orthogonal basis to Q, i.e. Q(:, np) = B(:, ind).

[0081] 2) Use the Gram-Schmidt orthogonalization process for the new column Q(:, np).

[0082] 3) Normalize the new column Q(:, np) = B(:, ind) / ||B(:, ind)||2 to make it have unit length.

[0083] 4) Update the index set S of the candidate columns: remove the current leading column ind from the candidate column index set S, and set the column norm to 0 to prevent it from being selected repeatedly.

[0084] 5) Update all candidate columns, remove the influence of the new orthogonal basis Q(:, np) on all columns j ∈ S of the matrix B(:, j), and perform orthogonalization:

[0085] B(:, j) = B(:, j) - (Q(:, np) H ·B(:, j)) · Q(:, np) (2)

[0086] Then, recalculate the 2-norm of the candidate columns, i.e. colNorm(j) = ||B(:, j)||2, j ∈ S, for selecting the next leading direction column.

[0087] ⑥ Check termination condition: find the column index of the largest 2-norm as ind = max (colNorm), and the value of the largest 2-norm under the corresponding index as val; if the largest norm is less than the tolerance threshold tol, that is, val < tol, stop iteration; otherwise, continue to the next round of selection and update.

[0088] 3) The final output is the solved orthogonal basis matrix Q = Q(:,1: np), where only the first np columns are retained, ensuring that the reduced matrix Q is m rows and np columns. Then, calculate the matrix R = Q H · BB, which has dimensions np x p. Here, the superscript H represents the transpose conjugate.

[0089] At this point, the selective QR decomposition algorithm generates an orthogonal basis matrix Q (also the reduced right-hand side matrix) by selecting the most contributing direction vector in the original multi-right-hand side matrix B column by column, while eliminating redundant directions in the candidate column, achieving matrix dimension reduction. The final generated Q contains the main direction, reducing the computational complexity, while the number of columns np of the reduced right-hand side matrix is controlled by the tolerance tol, ensuring the accuracy and efficiency of the dimension reduction. In order to facilitate the description and differentiation below, the invention will set the right-hand side matrix Q obtained by the matrix dimension reduction method as newB = Q; to facilitate the differentiation between the original right-hand side matrix B before dimension reduction and the right-hand side matrix newB after dimension reduction.

[0090] Step D, through the matrix equation newB after dimension reduction in step C, combined with the R matrix in step C, the solution of the original equation (1) is obtained by the block iterative BCO CG algorithm.

[0091] According to the dimension reduction processing of step C, the matrix equation (1) of the original problem is converted into the reduced equation (3), which is:

[0092] AX * = newB (3)

[0093] Through the block iterative BCO CG algorithm, the solution X * of the reduced matrix can be obtained, which has dimensions m x np. Then, combined with the R matrix in step C, the solution of the original equation (1) is:

[0094] X = X * x R (4)

[0095] Thus, the reduced matrix equation is completed. By solving equation (3), according to the post-processing of the field value and RCS, the final single station RCS response is obtained.

[0096] The computational complexity is optimized by reducing the column number of the original right-hand side matrix. The high-dimensional problem is reduced to a low-rank approximation space, and the computational efficiency of the original BCOCG algorithm is improved. In addition, the reduced subspace is also convenient for parallel implementation, which can further reduce the storage requirement.

[0097] In the process of solving the monostatic radar cross section (RCS), in order to study the local scattering characteristics of large and complex targets, the target can be appropriately simplified in shape. For example, when analyzing the RCS of the connection area between the wing and the fuselage of an airplane, the fuselage can be simplified as a sphere and the wing can be simplified as a cube. Using such a simplified model can efficiently explore the local scattering behavior and significantly reduce the computational complexity. The embodiment uses the combined model (a combination of a sphere and a cube) shown in the figure as an approximate substitute model for the connection area between the wing and the fuselage of an airplane, thereby verifying the technology of the application. Figure 2

[0098] Figure 2 The radius of the sphere is 500 mm, and the center of the sphere is at the origin. The cube has a specified point (centered at the origin) with a side length of 500 mm, and the two are subjected to Boolean operation. The material is pec; the shell is a vacuum tetrahedron, and the surrounding is a radiation boundary. The specified point of the shell tetrahedron is (-700 mm, -700 mm, -700 mm), and the side length is 1400 mm. The calculation frequency is 300 MHz. The embodiment mainly analyzes the monostatic RCS value in the range of θ=0~180° (with an interval angle of 30), at which time the number of right-hand side vectors is 61. The number of discrete tetrahedrons of the combined model is 22338, and there are 5006 vertices. The dimension of the finite element matrix is 147612.

[0099] In order to verify the BCOCG block iteration algorithm proposed in the application, the block iteration BGMRES-DR algorithm studied in the previous stage, and the classical GCR algorithm and GMRES algorithm for sequentially solving a single right-hand side vector are used. The difference is that: the GCR and GMRES algorithms sequentially solve each individual right-hand side vector, and need to loop 61 times; while the BCOCG block iteration algorithm and the BGMRES-DR block iteration algorithm are equivalent to a matrix of 61 vectors, and only one solution is needed to solve each right-hand side vector. Finally, through electromagnetic post-processing, the RCS target value is obtained. In addition, in order to ensure the rigor of the comparison, the multi-wavefront block incomplete Cholesky decomposition MFBIC preprocessing technology is used for the four algorithms.

[0100] Comparison of the results of several iteration algorithms:

[0101] (1) Accuracy: Figure 3 ​​is calculated by using GCR algorithm, GMRES algorithm, BGMRES-DR block iteration algorithm and BCOCG algorithm. Figure 3 It can be seen that the BCOCG block iteration algorithm proposed in the application is consistent with the numerical RCS calculated by the classical GCR and GMRES algorithms and the block iteration BGMRES-DR algorithm in the early research, and has a consistent curve. The correctness of the application is further verified.

[0102] (2) Calculation performance: Table 2 shows the calculation performance results of the four algorithms;

[0103] Table 2:

[0104]

[0105] In Table 2, it can be seen that the block iteration algorithm has a larger peak memory than the single right end item vector iteration algorithm, although the actual solving calculation memory is not much different. This is because the single right end item vector iteration adopts the form of matrix multiplication vector, while the multiple right end item is in the form of matrix multiplication matrix, so it naturally consumes more memory than the single right end item vector iteration.

[0106] However, after applying the BCOCG algorithm proposed in the application, it can be seen that the peak memory is significantly lower than the BGMRES-DR block iteration algorithm in the early research, and is not much different from GCR and GMRES. Therefore, it can be considered that the short recursion and dimension reduction BCOCG block iteration algorithm can improve the problem of memory consumption of the block iteration algorithm. In addition, the BCOCG block iteration can reduce the original 61 right end item vectors to 16 right end item vectors through the matrix dimension reduction technology proposed in the application. By solving the dimension-reduced solution and then performing matrix back substitution processing, the solution of the original problem can be obtained. It can also be seen from Table 2 that the calculation time of the BCOCG block iteration algorithm is the shortest. If a larger matrix size is faced, the calculation performance is more advantageous.

[0107] It can be seen from the above examples that the application uses two technologies of matrix dimension reduction and matrix preprocessing: the right end item matrix is processed by using the matrix dimension reduction, and through the low rank approximation, the accuracy of the solution can be effectively ensured, the calculation cost is significantly reduced, and the multiple right end linear system is facilitated to be processed; and after the BCOCG algorithm is applied to the preprocessing technology, the matrix behavior is further improved, and the convergence and stability of the algorithm are improved. The application is aimed at the system equation generated by multiple physical excitation sources or multiple scanning angles in the electromagnetic problem, realizes the shortening of the memory cost from the perspective of the short recursive Krylov subspace iteration method, adopts the block iteration BCOCG algorithm to simultaneously calculate multiple column right end item vectors, and avoids the matrix ill-conditioned problem caused by the correlation between columns.

Claims

1. A block iterative matrix solving method based on matrix dimensionality reduction and preprocessing techniques, characterized in that, Includes the following steps: Step A: Simulate and model the target structure by combining the material, working environment, and boundary conditions; then, mesh the 3D target model, and the surface discretization and volume discretization must be compatible. Step B: Based on the solution process of the target's electromagnetic problem with multiple right-hand sides, a global linear system containing multiple right-hand sides is obtained; the corresponding matrix equation with multiple right-hand sides is: AX = B(1) Where A is the global system matrix with dimension m×m; the right-hand side matrix B is a multi-right-hand side matrix obtained from p incident wave excitations with dimension m×p, i.e., it has p columns of right-hand side vectors; the unknown matrix to be solved is X, with dimension m×p. The electromagnetic problem of the target with multiple right-hand terms is an electromagnetic problem of the single-station radar cross section (RCS). Step C: Perform matrix dimensionality reduction on the matrix B with multiple right-hand side terms to obtain the dimensionality-reduced matrix newB. From the perspective of low-rank approximation, matrix dimensionality reduction is performed on the right-hand side matrix B to reduce the number of columns p of the right-hand side while preserving the dominant direction; The dimensionality reduction method employs selective QR decomposition, which selects the most relevant columns based on the 2-norm of the matrix columns, constructs an orthogonal basis matrix Q that satisfies orthogonality, and generates a corresponding matrix R such that B≈QR. The selection of columns is carried out iteratively, and the algorithm is terminated based on whether the tolerance condition is met. Dimension reduction transforms the original matrix B with multiple right-hand terms into a new matrix newB, with the corresponding matrix dimension changing from m×p to m×np, where np≤p, and np is the number of columns in the dimension-reduced matrix. Step D: Using the matrix equation newB after dimensionality reduction in step C, and combining it with the R matrix in step C, the solution of the original equation (1) is obtained through the block iterative BCOCG algorithm. Based on the dimensionality reduction process in step C, the matrix equation (1) of the original target's multi-right-hand side electromagnetic problem is transformed into the dimensionality-reduced equation (3), as follows: AX * =newB(3) The solution X of the dimension-reduced matrix is ​​obtained by using the block iterative BCOCG algorithm. * Its dimension is m×np; then, combined with the R matrix in step C, the solution to the original equation (1) is: X=X * ×R(4) Therefore, the dimension-reduced matrix equation can be replaced back.

2. The block iterative matrix solving method based on matrix dimensionality reduction and preprocessing techniques as described in claim 1, characterized in that: Step A involves partitioning the three-dimensional target model using tetrahedral elements.

3. The block iterative matrix solving method based on matrix dimensionality reduction and preprocessing techniques as described in claim 1, characterized in that, The matrix dimensionality reduction process in step C is specifically as follows: 1) Initialization: ① Obtain the dimension of the right-hand side matrix B, i.e., m rows and p columns; before the loop, record the initial value of the right-hand side matrix B as BB = B; ② Define a candidate column index set S = {1, 2, ..., p}, initially containing all columns; calculate the 2-norm of each column to obtain the column norm array colNorm(j) = ||B(:,j)||2,j∈S, and select the main element by finding the column index ind of the maximum value of the column norm of B; ③np=0, record the number of columns currently selected, that is, the number of columns of the orthogonal basis matrix Q after dimensionality reduction; 2) Main loop: Select the dominant column in sequence and update the matrix. ① Select the column B(:,ind) with the largest 2-norm among the current candidate columns and mark it as the new dominant direction column. At this time, np = np + 1. Add the current column as a new orthogonal basis to matrix Q, that is, Q(:,np) = B(:,ind); ② Apply the Gram-Schmidt orthogonalization process to the new column Q(:,np); ③ For the new column Q(:,np)=B(:,ind) / ||B(:,ind)||2, normalize it to make it have a unit length; ④ Update the candidate column index set S: Remove the current dominant column ind from the candidate column index set and set its norm to 0; ⑤ Update all candidate columns. For all columns j∈S of matrix B(:,j), remove the influence of the newly added orthogonal basis Q(:,np) and perform orthogonalization: B(:,j)=B(:,j)-(Q(:,np) H ·B(:,j))·Q(:,np) (2) Then, the 2-norm of the candidate column is recalculated, i.e., colNorm(j)=||B(:,j)||2,j∈S, to select the next dominant direction column; ⑥ Check the termination condition: Find the column index with the largest 2-norm as ind = max(colNorm), and the value of the largest 2-norm under this index as val; if the largest norm is less than the tolerance threshold tol, i.e. val < tol, then stop the iteration; otherwise, continue to the next round of selection and update; 3) Finally, output the solved orthogonal basis matrix Q = Q(:,1:np), retaining only the first np columns to ensure that the dimensionality-reduced matrix Q has only m rows and np columns; then, calculate the matrix R = Q H ·BB, whose dimension is np×p; Thus, the selective QR decomposition algorithm generates an orthogonal basis matrix Q by selecting the most contributing direction vector in the original multi-right-side term matrix B column by column, and removes redundant directions in the candidate columns, thereby achieving dimensionality reduction of matrix B. The right-hand side matrix Q obtained after dimensionality reduction is set to newB = Q to distinguish the original multi-right-hand side matrix B before dimensionality reduction from the right-hand side matrix newB after dimensionality reduction.

4. The block iterative matrix solving method based on matrix dimensionality reduction and preprocessing techniques as described in claim 1, characterized in that, The multi-right-end term matrix equation in step B is preprocessed using multi-wavefront block incomplete Chuleski decomposition before matrix dimensionality reduction.

5. The block iterative matrix solving method based on matrix dimensionality reduction and preprocessing techniques as described in claim 4, characterized in that, The specific process of the preprocessing for the incomplete Chuleski decomposition of the multi-wavefront block is as follows: Before solving equation (1), the coefficient matrix is ​​preprocessed using another non-singular matrix to reduce the power of the minimum feature polymorphism corresponding to the original coefficient matrix. First, let M be a left preprocessor. Process AX = B using the left preprocessor M, then we have M -1 AX = M -1 B; Treat the preprocessed matrix as a whole B←M -1 B, then proceed with step C for dimensionality reduction.

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