Power supply network parallel solving method based on nested region decomposition

Abstract

The invention discloses a power supply network parallel solving method based on nested region decomposition, which is used for power supply network analysis and comprises the following steps: step 1, dividing an original Laplacian matrix graph into a nested region decomposition form to form an outer layer boundary node, an inner layer boundary node and an inner layer internal node; step 2, eliminating internal nodes of the inner layer to form a Scherr complement matrix of the middle layer; step 3, eliminating inner layer boundary nodes to form a global Schel complement matrix; and step 4, solving through a top-down process. According to the method, multiple parallelization strategies are integrated, the challenges in the solution of the large-scale sparse linear system are dealt with, the calculation process of the Schel complement matrix is optimized, and a comprehensive solution is provided to remarkably improve the calculation efficiency.

Description

Technical Field

[0001] The invention belongs to the technical field of power distribution network analysis, and in particular relates to a power network parallel solution method based on nested domain decomposition. Background Art

[0002] As Moore's Law pushes transistor scaling to its limit, power distribution networks (PDNs) in integrated circuits are becoming increasingly complex, especially in technology nodes below 5nm, where the number of nodes involved has reached billions or even trillions. At the same time, due to the reduction in supply voltage, the noise margin has dropped to tens of millivolts, while power density continues to rise, making accurate and efficient power network analysis more critical than ever. However, the computational requirements for analyzing large-scale PDNs have greatly hindered design efficiency and iteration speed. Therefore, improving the efficiency of power network analysis methods has become a key research direction.

[0003] Power network analysis usually involves solving large-scale symmetrical positive definite (SPD) linear systems, which are so large that single-core processing often cannot complete them efficiently. With the development of hardware and the rise of parallel computing technology, the domain decomposition method (DDM) has become a powerful tool for large-scale power network analysis, such as Figure 1 These methods rely on graph partitioning technology to divide the PDN into smaller subdomains and significantly improve computational efficiency through parallel processing, thereby effectively solving the computational bottleneck in solving large-scale systems.

[0004] DDM relies on the Shure complement matrix, and constructing the Shure complement matrix is ​​closely related to Gaussian elimination. This process simplifies the network model by eliminating certain nodes and simplifying the relevant conductance values ​​in the admittance matrix to the equivalent conductance of the port nodes. During Gaussian elimination, the electrical connections between nodes are equivalently converted into effective conductances at the port nodes, forming a simplified system while maintaining its main electrical properties. However, constructing and solving a dense global Shure complement matrix is ​​even more difficult than directly solving the original system. This challenge mainly involves the following two aspects: (1) Calculating the local Schur complement matrix of each subdomain is inherently complex and resource-intensive because of the complex interdependencies within the subdomains.

[0005] (2) Although current computing resources are capable of processing multiple subdomains, over-partitioning will increase the size of the global Schur complement matrix, making it difficult to parallelize the matrix, which ultimately reduces computational efficiency.

[0006] Traditional DDM parallel solvers face scalability challenges. When the number of subdomain partitions exceeds a certain critical value, it is difficult to achieve effective acceleration. However, modern hardware supports a higher number of processes and threads, which provides more opportunities for enhancing parallelism. Therefore, an efficient parallel solver based on nested domain decomposition method is proposed, which is specifically used for power grid analysis, aiming to improve the scalability of traditional DDM parallel solvers. Summary of the invention

[0007] In order to make up for the shortcomings of the prior art, the present invention provides a parallel solution method for power networks based on nested domain decomposition, which is specifically used for power network analysis and aims to improve the scalability and resource utilization efficiency of the solver by introducing a parallel Schur complement calculation strategy and an intermediate layer Schur complement matrix. To achieve this purpose, the specific technical solution of the present invention is as follows: A first aspect of the present invention provides a method for parallel solution of a power network based on nested domain decomposition, comprising the following steps: Step 1, dividing the original Laplace matrix graph into a nested region decomposition form to form outer boundary nodes, inner boundary nodes and inner internal nodes; Step 2, eliminate the inner nodes of the inner layer to form the middle layer Schur complement matrix; Step 3, eliminate the inner boundary nodes to form a global Schur complement matrix; Step 4: Solve through a top-down process.

[0008] Furthermore, the process of dividing in step 1 is as follows: Step 1.1, preliminarily divide the graph corresponding to the original Laplacian matrix to form an external subdomain, and divide the nodes into outer boundary nodes and outer internal nodes; Step 1.2, divide the outer internal nodes in each outer subdomain into inner boundary nodes and inner internal nodes; Step 1.3: Reorder the original circuit matrix and transform the matrix into a nested domain decomposition form.

[0009] Furthermore, in step 2, the sub-matrices corresponding to the inner internal nodes, inner boundary nodes and their related outer boundary nodes of each internal subdomain are extracted in parallel, and then the inner internal nodes are eliminated by Gaussian elimination to obtain the Schur complement matrix of the middle layer for subsequent calculations.

[0010] Furthermore, in step 3, the sub-matrices corresponding to the outer boundary nodes of each external subdomain and its related inner boundary nodes are extracted in parallel, and then the inner boundary nodes are eliminated by Gaussian elimination method, so as to obtain the global Schur complement matrix.

[0011] Furthermore, in step 4, the solution process is as follows: Step 4.1, obtain the values ​​of the outer boundary nodes by solving the global Schur complement matrix; Step 4.2, substitute the obtained values ​​of the outer boundary nodes into the middle layer Schur complement matrix, and solve the values ​​of the inner boundary nodes in parallel; Step 4.3, substitute the values ​​of the inner boundary nodes back into the local network, and finally solve the values ​​of the inner internal nodes in parallel.

[0012] Furthermore, the parallel processing flow of the Schur complement matrix is ​​as follows: 1) First identify the selected port and non-port nodes; 2) Use graph partitioning tools to classify nodes; 3) Non-port nodes are further divided into internal nodes and boundary nodes of each subdomain; 4) Extract the internal nodes, boundary nodes and selected port nodes related to each subdomain in parallel; 5) The internal nodes are eliminated in parallel to obtain the Schur complement associated with the boundary nodes and the selected port nodes; 6) Finally, eliminate the boundary nodes and obtain the Schur complement corresponding to the selected port node.

[0013] Furthermore, the method utilizes the METIS graph partitioning tool to partition the power network, and uses the CHOLMOD direct solver in combination with Intel MKL to perform matrix decomposition.

[0014] A second aspect of the present invention provides an electronic device, including a memory and a processor, wherein the memory stores a program running on the processor, and the processor executes the steps of the above-mentioned method for parallel solution of power supply network based on nested area decomposition when running the program.

[0015] A third aspect of the present invention provides a computer-readable storage medium having computer instructions stored thereon, which, when executed, execute the steps of the above-mentioned method for parallel solution of a power supply network based on nested domain decomposition.

[0016] Compared with the prior art, the present invention has the following advantages: The parallel solution method of the present invention divides the nodes to be eliminated into multiple subdomains and eliminates them in parallel; an intermediate layer Shur complement matrix is ​​introduced to reduce the size of the dense global Shur complement matrix. The intermediate layer matrix can be naturally solved in parallel, further optimizing the overall process, improving computing efficiency and resource utilization; and the parallel Shur complement matrix calculation strategy and the intermediate layer Shur complement matrix are combined with the traditional parallel solver to form a parallel solution method based on nested domain decomposition. This solution method effectively solves the shortcomings of the traditional DDM parallel solver in scalability, especially when the number of partitions exceeds a certain threshold, the traditional solver is difficult to achieve significant acceleration. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 This is the flow chart of the traditional domain decomposition-based parallel solution method (DDM); Figure 2 It is a flow chart of the parallel solution method (nested DDM) based on nested domain decomposition of the present invention; Figure 3 This is a flow chart of the parallel Schur complement calculation of the present invention; Figure 4 The relationship between the graph partitioning and the corresponding matrix of the present invention: the node color in Figure (a) corresponds to the color of the sub-matrix block in the matrix in Figure (b); Figure 5 The invention provides a speedup comparison between the parallel solution method based on nested domain decomposition (nested DDM) and the parallel solution method based on domain decomposition (DDM) and the serial solution method (CHOLMOD). DETAILED DESCRIPTION

[0018] In order to make the purpose, technical solution and advantages of the present invention clearer, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0019] like Figure 2 As shown, a parallel solution method for power network based on nested domain decomposition includes the following steps: Step 1: Divide the original Laplace matrix graph into a nested region decomposition form to form outer boundary nodes, inner boundary nodes and inner internal nodes.

[0020] The original Laplacian matrix can be transformed into a nested domain decomposition form through a two-step graph partitioning process, as Figure 4 As shown. First, the nodes corresponding to the entire matrix are divided once to form external subdomains; then, the outer internal nodes in each external subdomain are further divided into inner boundary nodes and inner internal nodes. This process divides the nodes into three categories: outer boundary nodes ( Figure 4 Gray nodes in the middle), inner boundary nodes ( Figure 4 ) and the inner nodes ( Figure 4 ).

[0021] The original circuit matrix is ​​reordered and converted into the form of nested domain decomposition.

[0022] Graph partitioning is to make the number of boundary nodes of each block as small as possible (and the number of internal nodes as large as possible) after specifying the number of partition blocks, so that the coupling between blocks is reduced. The two graph partitionings in this application are to call the graph partitioning toolkit twice. The first time, all nodes are divided into "outer boundary nodes" (very few) and "outer internal nodes" (many). The second time, after removing the "outer boundary nodes", the "outer internal nodes" are divided into "inner boundary nodes" (very few) and "inner internal nodes" (many).

[0023] Step 2: Eliminate the inner nodes of the inner layer to form the middle layer Schur complement matrix.

[0024] The sub-matrices corresponding to the inner internal nodes, inner boundary nodes and associated outer boundary nodes of each inner subdomain are extracted in parallel; the inner internal nodes are then eliminated by Gaussian elimination to obtain the middle-layer Schur complement matrix for subsequent calculations.

[0025] Step 3: Eliminate the inner boundary nodes to form a global Schur complement matrix.

[0026] The submatrices corresponding to the outer boundary nodes and the associated inner boundary nodes of each external subdomain are extracted in parallel; the inner boundary nodes are then eliminated by Gaussian elimination to obtain the global Schur complement matrix. The process of eliminating the inner internal nodes first and then the inner boundary nodes adopts the parallel Schur complement calculation method we proposed, which equates the conductance to the outer boundary nodes (selected ports).

[0027] The parallel processing flow of the Schur complement matrix is: 1) First identify the selected port and non-port nodes; 2) Use graph partitioning tools to classify nodes; 3) Non-port nodes are further divided into internal nodes and boundary nodes of each subdomain; 4) Extract the internal nodes, boundary nodes and selected port nodes related to each subdomain in parallel; 5) The internal nodes are eliminated in parallel to obtain the Schur complement associated with the boundary nodes and the selected port nodes; 6) Finally, eliminate the boundary nodes and obtain the Schur complement corresponding to the selected port node.

[0028] Step 4: Solve through a top-down process.

[0029] The solution process adopts a top-down approach. First, the values ​​of the outer boundary nodes are obtained by solving the global Schur complement matrix; then, these values ​​are substituted into the middle layer Schur complement matrix, and the values ​​of the inner boundary nodes are solved in parallel; finally, the values ​​of the boundary nodes are substituted back into the local network, and the values ​​of the inner internal nodes are finally solved in parallel.

[0030] It can be understood as follows: there are three unknown vectors x (values ​​of outer boundary nodes), y (values ​​of inner boundary nodes), and z (values ​​of inner internal nodes). The first set of equations is only about x, the second set of equations is about x and y, and the third set of equations is about x / y / z. First, solve equation 1 for x; then, use the x obtained from equation 1 to solve equation 2 for y; finally, use the x obtained from equation 1 and the y obtained from equation 2 to solve equation 3 for z.

[0031] In order to evaluate the performance of the proposed parallel solver based on the nested domain decomposition method, this application is developed in C++, the METIS graph partitioning tool is used to partition the power network, and the CHOLMOD direct solver is combined with Intel MKL for matrix decomposition. Experiments are conducted on the IBMPG (IBM's power grid) and THUPG (Tsinghua University's power grid) benchmarks, and the solution method of this application is compared with two state-of-the-art solution methods: a CHOLMOD-based serial solver and a traditional domain decomposition method (DDM) parallel solver. All experiments are performed on a Linux server equipped with two Intel Xeon Silver 4210R processors and 128GB of memory.

[0032] Figure 5The acceleration effects of the parallel solution method based on nested domain decomposition (nested DDM) proposed in this application and the traditional parallel solution method based on domain decomposition (DDM) and the serial solution method (CHOLMOD) are demonstrated. It can be found that with the increase of the number of threads, the speed of the DDM parallel solver is significantly improved compared with the serial solver. However, with the further increase of the number of threads, the acceleration effect begins to saturate, and even when the number of threads is too large, the overall performance may decline. This is because too many partition blocks will cause the scale of the global Shur complement matrix to increase sharply, and the global Shur complement matrix is ​​a dense matrix, and its solution process may occupy the main computing time and become a performance bottleneck, thereby weakening the advantages of parallel computing. Relatively speaking, in nested DDM, due to the introduction of the middle-layer Shur complement matrix, some top-level tasks are assigned to the middle layer. On the one hand, this greatly reduces the dimension of the dense global Shur complement matrix, and on the other hand, the middle-layer Shur complement matrix can realize parallel processing, thereby effectively improving the computing efficiency. Therefore, as Figure 5 As shown, when the traditional DDM parallel solver reaches a performance bottleneck, the nested DDM parallel solver proposed in this application can still maintain a good acceleration effect.

[0033] In summary, the nested DDM parallel solver proposed in the present invention has better scalability than the traditional DDM parallel solver. It not only accelerates the solution process, but also optimizes resource utilization, making it highly adaptable in dealing with the growing computing needs of modern integrated circuit design.

[0034] In a specific embodiment, the specific steps of a method for parallel solving of a power network based on nested domain decomposition are as follows: 1) First, the power distribution network (PDN) is divided into p external subdomains, and then each external subdomain is further divided into q internal subdomains to form local networks labeled 11 to pq. This hierarchical strategy significantly reduces the size of the top-level problem by introducing an intermediate-level Schur complement matrix. Although the intermediate-level Schur complement matrix is ​​still dense, it is more suitable for parallel computing.

[0035] 2) Next, for each local network i1 to iq within the external subdomain i, an intermediate network i is constructed by calculating the Schur complement matrix, which concentrates the admittance of most nodes to a specific node. Then, through further node elimination, the p intermediate networks are merged into the top-level network.

[0036] 3) The solution process starts with solving the top network, and the solution is then used to solve the p intermediate networks. Finally, the solution of the intermediate network is substituted back into each local network to finally obtain the overall solution. The specific process is as follows Figure 2 shown.

[0037] Among them, step 2) uses a method for parallel calculation of Schur's complement as one of the core technologies of the nested domain decomposition method. This method first identifies the selected port and non-port nodes, and then uses a graph partitioning tool to classify the nodes. In order to achieve load balancing, a higher weight is given to the selected ports to ensure that the nodes can be evenly distributed in each subdomain. Subsequently, the non-port nodes are further divided into internal nodes and boundary nodes of each subdomain. This classification not only helps the elimination process, but also organizes the system into a block structure format, thereby optimizing the efficiency of parallel computing. In order to improve the efficiency of parallel processing, the internal nodes, boundary nodes and selected port nodes associated with each subdomain are first extracted in parallel. Subsequently, the internal nodes are eliminated in parallel to obtain the Schur's complements corresponding to the boundary nodes and selected port nodes. Finally, the boundary nodes are eliminated to obtain the Schur's complements corresponding to the selected port nodes. The specific process is as follows Figure 3 shown.

[0038] The present invention also provides an electronic device, including a memory and a processor, wherein the memory stores a program running on the processor, and the processor executes the steps of the above-mentioned method for parallel solution of power supply network based on nested area decomposition when running the program.

[0039] The present invention also provides a computer-readable storage medium on which computer instructions are stored. When the computer instructions are executed, the steps of the above-mentioned method for parallel solution of power supply network based on nested area decomposition are executed.

[0040] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A parallel solution method for power network based on nested domain decomposition, characterized in that: The following steps are involved: Step 1, dividing the original Laplace matrix graph into a nested region decomposition form to form outer boundary nodes, inner boundary nodes and inner internal nodes; Step 2, eliminate the inner nodes of the inner layer to form the middle layer Schur complement matrix; Step 3, eliminate the inner boundary nodes to form a global Schur complement matrix; Step 4: Solve through a top-down process.

2. A method for parallel solution of power network based on nested domain decomposition according to claim 1, characterized in that: The process of dividing in step 1 is as follows: Step 1.1, preliminarily divide the graph corresponding to the original Laplacian matrix to form an external subdomain, and divide the nodes into outer boundary nodes and outer internal nodes; Step 1.2, divide the outer internal nodes in each outer subdomain into inner boundary nodes and inner internal nodes; Step 1.3: Reorder the original circuit matrix and transform the matrix into a nested domain decomposition form.

3. The method for parallel solving of power network based on nested domain decomposition according to claim 1, characterized in that: In step 2, the sub-matrices corresponding to the inner internal nodes, inner boundary nodes and their related outer boundary nodes of each internal subdomain are extracted in parallel, and then the inner internal nodes are eliminated by Gaussian elimination to obtain the Schur complement matrix of the middle layer for subsequent calculations.

4. The method for parallel solving of power supply network based on nested domain decomposition according to claim 1, characterized in that: In step 3, the sub-matrices corresponding to the outer boundary nodes and the related inner boundary nodes of each external sub-domain are extracted in parallel, and then the inner boundary nodes are eliminated by Gaussian elimination method, so as to obtain the global Schur complement matrix.

5. The method for parallel solving of power network based on nested domain decomposition according to claim 1, characterized in that: In step 4, the solution process is as follows: Step 4.1, obtain the values ​​of the outer boundary nodes by solving the global Schur complement matrix; Step 4.2, substitute the obtained values ​​of the outer boundary nodes into the middle layer Schur complement matrix, and solve the values ​​of the inner boundary nodes in parallel; Step 4.3, substitute the values ​​of the inner boundary nodes back into the local network, and finally solve the values ​​of the inner internal nodes in parallel.

6. A method for parallel solution of power network based on nested domain decomposition according to any one of claims 1 to 5, characterized in that: The parallel processing flow of the Schur complement matrix is ​​as follows: 1) First identify the selected port and non-port nodes; 2) Use graph partitioning tools to classify nodes; 3) Non-port nodes are further divided into internal nodes and boundary nodes of each subdomain; 4) Extract the internal nodes, boundary nodes and selected port nodes related to each subdomain in parallel; 5) The internal nodes are eliminated in parallel to obtain the Schur complement associated with the boundary nodes and the selected port nodes; 6) Finally, eliminate the boundary nodes and obtain the Schur complement corresponding to the selected port node.

7. A method for parallel solution of power network based on nested domain decomposition according to any one of claims 1 to 5, characterized in that: The method uses the METIS graph partitioning tool to partition the power network, and uses the CHOLMOD direct solver combined with Intel MKL to perform matrix decomposition.

8. An electronic device, characterized in that: The method comprises a memory and a processor, wherein the memory stores a program running on the processor, and the processor executes the steps of a method for parallel solution of a power supply network based on nested area decomposition as described in any one of claims 1 to 5 when running the program.

9. A computer-readable storage medium having computer instructions stored thereon, characterized in that: When the computer instructions are executed, the steps of a method for parallel solving a power network based on nested domain decomposition as described in any one of claims 1 to 5 are executed.

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