Effective resistance calculation method based on block Cholesky decomposition

Abstract

The invention discloses an effective resistance calculation method based on block Cholesky decomposition, and relates to the field of circuit simulation and graph computation.The effective resistance calculation method comprises the steps that firstly, a simulation sparse matrix is sequenced and blocked through a nested segmentation method, and a single-layer diagonal edging matrix is constructed; gradually converting the matrix structure into a multi-layer block mode through iterative optimization, reversely integrating coupling edges, and finally outputting a replacement vector and block position information; secondly, constructing a parallel processing framework based on block information, implementing parallel Cholesky decomposition on a plurality of initial matrixes with consistent structures, filling diagonal blocks of a lower triangular matrix through decomposed numerical values, and solving coupling edge numerical values; and block sorting and parallel triangular back substitution are carried out on the right end item of the linear equation set, an intermediate vector is generated, a norm and a vector product of the intermediate vector are calculated, and finally efficient solving of the effective resistance is realized. According to the method, through matrix block optimization and parallel computing strategies, the time complexity of large-scale circuit simulation is remarkably reduced while the computing precision is kept.

Description

Technical Field

[0001] The present invention relates to the technical field of circuit simulation and graph computing, and particularly to an effective resistance calculation method based on block Cholesky decomposition. Background Art

[0002] In the field of large-scale graph computing, traditional effective resistance calculation methods have significant limitations. A typical method is to calculate the effective resistance by directly solving a system of linear equations. This method relies on the precise processing of sparse matrices, and the computational complexity is relatively high. Especially in large-scale graphs, the time and resource costs required for direct solution are huge, making it difficult to execute efficiently in practical applications. Another method is to use graph sparsification techniques to simplify the problem. Although such methods can reduce the amount of calculation to a certain extent, they often sacrifice calculation accuracy and cannot meet application scenarios with high requirements for calculation accuracy.

[0003] In order to improve the calculation efficiency and maintain the calculation accuracy, approximate calculation methods based on Cholesky decomposition have been proposed in recent years. These methods perform Cholesky decomposition on the Laplacian matrix and use the sparse approximate inverse of the decomposition factors to quickly calculate the effective resistance. However, there are still certain limitations in the Cholesky decomposition and approximate inverse calculation in the prior art, making it difficult to cope with the calculation challenges in ultra-large-scale circuits and complex networks. Therefore, exploring more efficient and accurate Cholesky decomposition and its approximate inverse calculation methods is an important research direction in the current technical field. Summary of the Invention

[0004] The purpose of the present invention is to provide an effective resistance calculation method based on block Cholesky decomposition, which has higher calculation efficiency and calculation accuracy, can greatly shorten the calculation time, and at the same time ensure the accuracy of the results.

[0005] To achieve the above purpose, the present invention provides an effective resistance calculation method based on block Cholesky decomposition, and the steps are as follows: S1. Use the nested dissection method to perform a sorting operation on the simulation sparse matrix A, perform block processing according to the principle of diagonal border addition structure, and construct a single-layer diagonal border addition matrix; S2. Perform nested dissection on each sub-matrix on the diagonal of the single-layer diagonal border addition matrix, and combine the preset dissection parameters to convert the single-layer diagonal border addition matrix into a double-layer diagonal border addition matrix, and at the same time perform a synchronous sorting operation on the coupling edges that matches the sub-matrices; S3. Integrate the coupling edges corresponding to the diagonal sub-matrices of the double-layer diagonal border addition matrix in S2 into the coupling edges of the initial single-layer diagonal border addition matrix, and restore to obtain the single-layer diagonal border addition matrix; S4. Execute S2 and S3 in a cyclic iteration manner until the number of matrix diagonal blocks reaches a pre-set target number, and output the permutation vector and the position information of each block, where the diagonal blocks that do not conform to the diagonal border-added structure are processed by the approximate minimum degree sorting method; S5. Based on the permutation vector and block information obtained in S4, construct a parallel processing framework for the initial matrices with the same positions of multiple non-zero elements, and generate a multi-diagonal block single-layer diagonal border-added matrix; S6. Perform Cholesky decomposition operations on the diagonal sub-matrices of the multi-diagonal block single-layer diagonal border-added matrix in parallel, fill the decomposition results into the diagonal blocks of the lower triangular matrix L, and solve the numerical values of the coupled edges of the lower triangular matrix L by combining the coupled edge numerical values of matrix A to complete the construction of the entire lower triangular matrix L; S7. After sorting the multiple right-hand sides of the linear equations according to the permutation vector, perform parallel block processing on the right-hand sides according to the block structure of the lower triangular matrix L; S8. Perform parallel triangular back substitution operations on multiple right-hand sides to generate an intermediate vector r, and complete the back substitution calculation according to the data dependence logical order of the lower triangular matrix L; S9. Call the block vector norm calculation module to solve the norm of the intermediate vector r, and perform vector multiplication operations between different intermediate vectors r according to the effective resistance calculation requirements.

[0006] Preferably, in S1, it is default set that the proportion of the coupling block in the entire matrix is 1%; it is default set that the proportion of each diagonal block in the entire matrix is 1%; and the specific position information of the output blocks is set.

[0007] Preferably, in S2, the preset segmentation parameter is dynamically set according to the sparsity of the sub-matrix.

[0008] Preferably, in S4, the target value of the number of matrix diagonal blocks is default set to 20.

[0009] Preferably, in S6, a multi-thread parallel computing architecture is adopted to implement the Cholesky decomposition operation.

[0010] Preferably, in S7, a distributed memory parallel computing model is adopted to implement parallel block processing.

[0011] Preferably, in S8, the data dependence logic is visually verified through a directed acyclic graph model.

[0012] Preferably, in S9, the block vector norm calculation module supports L1, L2, and infinity norm calculations.

[0013] According to the specific embodiments provided by the present invention, the present invention discloses the following technical effects: (1) Significantly improve the calculation efficiency The large-scale matrix is divided into a multi-layered bordered diagonal structure through nested partitioning, and the Cholesky decomposition is performed on each sub-matrix in combination with a parallel computing strategy. This block processing transforms the traditional global matrix operation into the solution of local sub-problems, improving the utilization rate of computing resources. A block strategy with iterative loops is adopted to continuously optimize the matrix structure, and the potential characteristics of the matrix are deeply explored through reverse transformation and multi-layer nested partitioning. This effectively reduces redundant calculations and enables the algorithm to adapt to the complexity of circuit networks of different scales.

[0014] (2)Double guarantee of calculation accuracy During the partitioning process, the coupled-edge sorting and sub-matrix matching strategies are implemented synchronously, ensuring the numerical stability of the decomposition result through accurate permutation vectors and block position information. The approximate minimum degree sorting method is used to process the diagonal blocks that do not conform to the bordered diagonal structure to maintain the mathematical properties of the matrix, avoiding the problem of accuracy loss caused by traditional sparsification methods.

[0015] (3)Breakthrough in scalability The flexible conversion mechanism of single-layer / double-layer bordered diagonal matrices enables effective expansion from small-scale to large-scale networks. By adjusting the block ratio and loop termination conditions, different hardware resource configurations can be adapted. The block structure disperses the storage requirements of the large-scale matrix to each sub-matrix, and in combination with the block solution strategy for the intermediate vector r, the peak memory consumption of a single operation is significantly reduced.

[0016] (4)Adaptability to complex networks For scenarios such as power grid reduction, through coupled-edge integration technology and intermediate vector norm calculation, complex network topologies with time-varying characteristics can be effectively processed. Based on the parallel processing framework of the initial matrix with the same structure, multiple right-hand sides can be processed synchronously to meet the batch calculation requirements of scenarios such as circuit parameter scanning and Monte Carlo simulation.

[0017] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Description of the drawings

[0018] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings required for use in the embodiments will be briefly introduced below. Obviously, the accompanying drawings in the following description are only some embodiments of the present invention, and those of ordinary skill in the art can also obtain other accompanying drawings without creative efforts based on these drawings.

[0019] Figure 1 It is a schematic flow chart of an embodiment of an effective resistance calculation method based on block Cholesky decomposition of the present invention; Figure 2Schematic diagram of the partitioning structure of the diagonal bordered matrix according to an embodiment of the present invention; Figure 3 Schematic diagram of performing Cholesky decomposition and solving the coupled edges for a single-layer diagonal bordered matrix according to an embodiment of the present invention; Figure 4 Schematic diagram of solving the intermediate vector r based on the lower triangular matrix L of a single-sided diagonal block according to an embodiment of the present invention; Figure 5 Schematic diagram of the solution based on the Cholesky decomposition method according to an embodiment of the present invention; Figure 6 Schematic diagram of the solution based on the Cholesky decomposition method with sorting according to an embodiment of the present invention; Figure 7 Schematic diagram of solving different components of the solution in S9 according to an embodiment of the present invention; Figure 8 Schematic diagram of the solution time comparison between Cholesky and block Cholesky according to an embodiment of the present invention. Detailed implementation manners

[0020] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0021] To make the above objects, features, and advantages of the present invention more obvious and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific implementation manners. Embodiment

[0022] As Figure 1 shown, an effective resistance calculation method based on block Cholesky decomposition includes the following steps: S1. Use the nested dissection method to perform a sorting operation on the simulation sparse matrix A, perform block processing according to the principle of the diagonal bordered structure, and construct a single-layer diagonal bordered matrix; by default, set the proportion of the coupling block in the entire matrix to 1%; by default, set the proportion of each diagonal block in the entire matrix to 1%; set the specific position information of the output blocks.

[0023] S2. Perform nested dissection on each sub-matrix on the diagonal of the single-layer diagonal bordered matrix, convert the single-layer diagonal bordered matrix into a double-layer diagonal bordered matrix in combination with the preset dissection parameters, and at the same time perform a synchronous sorting operation on the coupled edges that matches the sub-matrix; the preset dissection parameters are dynamically set according to the sparsity of the sub-matrix.

[0024] S3. Integrate the coupling edges corresponding to the diagonal sub-matrices of the double-layer diagonal border-added matrix in S2 into the coupling edges of the initial single-layer diagonal border-added matrix, and restore to obtain the single-layer diagonal border-added matrix; S4. Execute S2 and S3 in a cyclic iteration manner until the number of diagonal blocks of the matrix reaches a preset target number, output the permutation vector and the position information of each block, where the diagonal blocks that do not conform to the diagonal border-added structure are processed by the approximate minimum degree sorting method; the default target value of the number of diagonal blocks of the matrix is set to 20.

[0025] S5. Based on the permutation vector and block information obtained in S4, construct a parallel processing framework for multiple initial matrices with the same non-zero element positions, and generate a multi-diagonal block single-layer diagonal border-added matrix, as Figure 2 shown; S6. As Figure 3 shown, perform Cholesky decomposition operations on the diagonal sub-matrices of the multi-diagonal block single-layer diagonal border-added matrix in parallel, fill the decomposition results into the diagonal blocks of the lower triangular matrix L, and solve the numerical values of the coupling edges of the lower triangular matrix L in combination with the coupling edge numerical values of matrix A to complete the complete construction of the lower triangular matrix L; adopt a multi-threaded parallel computing architecture to implement the Cholesky decomposition operation.

[0026] S7. After performing sorting operations on multiple right-hand sides of the linear equations according to the permutation vector, perform parallel block processing on the right-hand sides according to the block structure of the lower triangular matrix L, as Figure 4 shown, and adopt a distributed memory parallel computing model to implement parallel block processing.

[0027] S8. Perform parallel triangular back substitution operations on multiple right-hand sides to generate an intermediate vector r, as Figure 5 shown, and complete the back substitution calculation according to the data dependence logic order of the lower triangular matrix L; perform visual verification on the data dependence logic through a directed acyclic graph model.

[0028] S9. Call the block vector norm calculation module to solve the norm of the intermediate vector r, as Figure 6 shown, and perform vector multiplication operations between different intermediate vectors r according to the effective resistance calculation requirements, as Figure 7 shown. The block vector norm calculation module supports L1, L2, and infinity norm calculations.

[0029] The technical solution of this application introduces the block Cholesky decomposition technology, analyzes complex matrix operations into sub-problems that are easier to handle, thereby reducing the overall computational complexity. At the same time, the present invention also focuses on the precise control of diagonal block division and calculation errors. While achieving a 9-fold increase in computational efficiency, as Figure 8As shown, the calculation error of the effective resistance is strictly controlled within 1E-8. Compared with the prior art, the present invention not only has higher calculation efficiency and accuracy in theory, but also shows significant advantages in practical applications, such as power grid reduction, being able to greatly shorten the calculation time while ensuring the accuracy of the results, providing new ideas and methods for the research and application in related fields.

[0030] For the rest of the technical features in the above embodiments, those skilled in the art can flexibly select them according to the actual situation to meet different specific actual needs. However, it is obvious to those of ordinary skill in the art that these specific details do not have to be adopted to implement the present invention. In other instances, in order to avoid obscuring the present invention, well-known components, structures or parts have not been specifically described, and all are within the scope of the technical solutions claimed in the claims of the present invention.

[0031] Modifications and changes made by those skilled in the art without departing from the spirit and scope of the present invention shall fall within the protection scope of the appended claims of the present invention. In the above description, in order to provide a thorough understanding of the present invention, a large number of specific details have been set forth. However, it is obvious to those of ordinary skill in the art that these specific details do not have to be adopted to implement the present invention. In other instances, in order to avoid obscuring the present invention, well-known technologies have not been specifically described, such as specific construction details, working conditions and other technical conditions.

[0032] Specific examples are used in this article to illustrate the principles and implementation manners of the present invention. The description of the above embodiments is only used to help understand the method and its core idea of the present invention; at the same time, for those of ordinary skill in the art, according to the idea of the present invention, there will be changes in the specific implementation manners and application scopes. In summary, the content of this specification should not be construed as a limitation to the present invention.

Claims

1. An effective resistance calculation method based on block Cholesky decomposition, characterized in that The steps are as follows: S1. Perform a sorting operation on the simulation sparse matrix A using the nested partitioning method, perform block processing according to the principle of diagonal bordering structure, and construct a single-layer diagonal bordered matrix; S2. Perform nested partitioning on each sub-matrix on the diagonal of the single-layer diagonal bordered matrix, convert the single-layer diagonal bordered matrix into a double-layer diagonal bordered matrix in combination with preset partitioning parameters, and simultaneously perform a synchronous sorting operation on the coupling edges that matches the sub-matrices; S3. Integrate the coupling edges corresponding to the diagonal sub-matrices of the double-layer diagonal bordered matrix in S2 into the coupling edges of the initial single-layer diagonal bordered matrix to restore the single-layer diagonal bordered matrix; S4. Sequentially execute S2 and S3 in a cyclic iteration manner until the number of diagonal blocks of the matrix reaches a preset target number, output the permutation vector and the position information of each block, and use the approximate minimum degree sorting method to process the diagonal blocks that do not conform to the diagonal bordering structure; S5. Based on the permutation vector and block information obtained in S4, construct a parallel processing framework for the initial matrices with the same non-zero element positions to generate a multi-diagonal block single-layer diagonal bordered matrix; S6. Parallelly perform Cholesky decomposition operations on the diagonal sub-matrices of the multi-diagonal block single-layer diagonal bordered matrix, fill the decomposition results into the diagonal blocks of the lower triangular matrix L, and numerically solve the values of the coupling edges of the lower triangular matrix L in combination with the coupling edge values of matrix A to complete the construction of the entire lower triangular matrix L; S7. After performing a sorting operation on multiple right-hand sides of the linear equations according to the permutation vector, perform parallel block processing on the right-hand sides according to the block structure of the lower triangular matrix L; S8. Parallelly perform triangular back substitution operations on multiple right-hand sides to generate an intermediate vector r, and complete the back substitution calculation according to the data dependence logical order of the lower triangular matrix L; S9. Call the block vector norm calculation module to solve the norm of the intermediate vector r, and perform vector multiplication operations between different intermediate vectors r according to the effective resistance calculation requirements.

2. The effective resistance calculation method based on block Cholesky decomposition according to claim 1, wherein: In S1, it is default set that the proportion of the coupling block in the entire matrix is 1%; it is default set that the proportion of each diagonal block in the entire matrix is 1%; set the specific position information of the output blocks.

3. An effective resistance calculation method based on block Cholesky decomposition according to claim 1, characterized in that: In S2, the preset partitioning parameters are dynamically set according to the sparsity of the sub-matrices.

4. An effective resistance calculation method based on block Cholesky decomposition according to claim 1, characterized in that: In S4, the default target value of the number of matrix diagonal blocks is set to 20.

5. The effective resistance calculation method based on block Cholesky decomposition according to claim 1, wherein: In S6, a multi-thread parallel computing architecture is adopted to implement the Cholesky decomposition operation.

6. The effective resistance calculation method based on block Cholesky decomposition according to claim 1, characterized in that: In S7, a distributed memory parallel computing model is adopted to implement parallel block processing.

7. An effective resistance calculation method based on block Cholesky decomposition according to claim 1, characterized in that: In S8, the data dependence logic is visually verified through a directed acyclic graph model.

8. An effective resistance calculation method based on block Cholesky decomposition according to claim 1, characterized in that: In S9, the block vector norm calculation module supports L1, L2, and infinity norm calculations.

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