A parallel solution method for power networks based on nested domain decomposition

Abstract

The present invention discloses a parallel solution method for power networks based on nested domain decomposition, which is used for power network analysis. The method comprises the following steps: Step 1: dividing the original Laplace matrix graph into a nested domain decomposition form to form outer boundary nodes, inner boundary nodes, and inner internal nodes; Step 2: eliminating the inner internal nodes to form an intermediate layer Schur complement matrix; Step 3: eliminating the inner boundary nodes to form a global Schur complement matrix; Step 4: solving the problem through a top-down process. The method integrates multiple parallelization strategies to address the challenges of solving large-scale sparse linear systems, optimize the calculation process of the Schur complement matrix, and provide a comprehensive solution to significantly improve computational efficiency.

Description

Technical Field

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

[0002] As Moore's Law pushes transistor scaling to its limits, power distribution networks (PDNs) in integrated circuits (ICs) are becoming increasingly complex, especially at technology nodes below 5nm, where the number of nodes involved has reached billions or even trillions. Simultaneously, noise margins have fallen to tens of millivolts due to decreasing supply voltages, while power density continues to rise. This makes accurate and efficient power network analysis more critical than ever. However, the computational demands of analyzing large-scale PDNs significantly hinder 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. However, these linear systems 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 Schur complement matrix, the construction of which is closely related to Gaussian elimination. This process simplifies the network model by eliminating certain nodes and reducing 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 to the effective conductance at the port nodes, forming a simplified system while maintaining its key electrical properties. However, constructing and solving the dense global Schur complement matrix is even more difficult than directly solving the original system. This challenge mainly involves the following two aspects:

[0005] (1) Calculating the local Schur complement matrix of each subdomain is inherently complex and resource-intensive because of the complex interdependencies within the subdomains.

[0006] (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.

[0007] Traditional DDM parallel solvers face scalability challenges, becoming ineffective when the number of subdomain partitions exceeds a critical value. However, modern hardware supports higher numbers of processes and threads, providing more opportunities for increased parallelism. Therefore, an efficient parallel solver based on the nested domain decomposition method (NDM) is proposed for power grid analysis, aiming to improve the scalability of traditional DDM parallel solvers. Summary of the Invention

[0008] To address the shortcomings of existing technologies, this invention provides a parallel solution method for power networks based on nested domain decomposition, specifically for power network analysis. By introducing a parallel Schur complement calculation strategy and an intermediate-layer Schur complement matrix, this method aims to improve the solver's scalability and resource utilization efficiency. To achieve this goal, the specific technical solutions of this invention are as follows:

[0009] A first aspect of the present invention provides a parallel solution method for a power network based on nested domain decomposition, comprising the following steps:

[0010] Step 1: Divide the original Laplace matrix graph into a nested region decomposition form, forming outer boundary nodes, inner boundary nodes, and inner internal nodes;

[0011] Step 2: Eliminate the inner nodes of the inner layer to form the middle layer Schur complement matrix;

[0012] Step 3: Eliminate the inner boundary nodes to form a global Schur complement matrix;

[0013] Step 4: Solve through a top-down process.

[0014] Furthermore, the process in step 1 is divided into:

[0015] Step 1.1: Perform a preliminary partition on the graph corresponding to the original Laplacian matrix to form an outer subdomain, and divide the nodes into outer boundary nodes and outer internal nodes;

[0016] Step 1.2: Divide the outer internal nodes in each outer subdomain into inner boundary nodes and inner internal nodes;

[0017] Step 1.3: Reorder the original circuit matrix and convert it into a nested domain decomposition form.

[0018] 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.

[0019] Furthermore, in step 3, the submatrices 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 to obtain the global Schur complement matrix.

[0020] Furthermore, in step 4, the solution process is as follows:

[0021] Step 4.1: Obtain the values of the outer boundary nodes by solving the global Schur complement matrix;

[0022] 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;

[0023] 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.

[0024] Furthermore, the parallel processing flow of the Schur complement matrix is as follows:

[0025] 1) First, identify the selected port and non-port nodes;

[0026] 2) Use graph partitioning tools to classify nodes;

[0027] 3) Non-port nodes are further divided into internal nodes and boundary nodes of each subdomain;

[0028] 4) Extract the internal nodes, boundary nodes and selected port nodes related to each subdomain in parallel;

[0029] 5) The internal nodes are eliminated in parallel to obtain the Schur complement associated with the boundary nodes and the selected port nodes;

[0030] 6) Finally, eliminate the boundary nodes and obtain the Schur complement corresponding to the selected port node.

[0031] Furthermore, the method utilizes the METIS graph partitioning tool to partition the power network, and uses the CHOLMOD direct solver combined with Intel MKL to perform matrix decomposition.

[0032] A second aspect of the present invention provides an electronic device comprising a memory and a processor, wherein the memory stores a program running on the processor, and when the processor runs the program, the processor executes the steps of the above-mentioned method for parallel solution of a power supply network based on nested region decomposition.

[0033] 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 network based on nested domain decomposition.

[0034] Compared with the prior art, the present invention has the following advantages:

[0035] 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 Schur complement matrix is introduced to reduce the size of the dense global Schur complement matrix. The intermediate-layer matrix can be naturally solved in parallel, further optimizing the overall process and improving computational efficiency and resource utilization. The parallel Schur complement matrix calculation strategy and the intermediate-layer Schur complement matrix are combined with traditional parallel solvers to form a parallel solution method based on nested domain decomposition. This solution method effectively addresses the scalability shortcomings of traditional DDM parallel solvers, especially the difficulty of achieving significant acceleration when the number of partitions exceeds a certain threshold. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 This is the flow chart of the traditional domain decomposition-based parallel solution method (DDM);

[0037] Figure 2 Flowchart of the nested domain decomposition-based parallel solution method (nested DDM) of the present invention;

[0038] Figure 3 This is a flow chart of the parallel Schur complement calculation of the present invention;

[0039] 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);

[0040] Figure 5 This is a speedup comparison between the parallel solution method based on nested domain decomposition (nested DDM) of the present invention and the parallel solution method based on domain decomposition (DDM) and the serial solution method (CHOLMOD). DETAILED DESCRIPTION

[0041] In order to make the purpose, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to 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 intended to limit the present invention.

[0042] like Figure 2 As shown, a parallel solution method for power network based on nested domain decomposition includes the following steps:

[0043] Step 1: Divide the original Laplace matrix graph into a nested region decomposition form, forming outer boundary nodes, inner boundary nodes, and inner internal nodes.

[0044] 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 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 brown nodes in the middle) and inner nodes ( Figure 4 ).

[0045] Reorder the original circuit matrix and transform it into the form of nested domain decomposition.

[0046] Graph partitioning involves minimizing the number of boundary nodes in each block (and maximizing the number of internal nodes) after specifying the number of blocks. This reduces coupling between blocks. This application's two-part graph partitioning involves calling the graph partitioning toolkit twice. The first time, all nodes are partitioned into "outer boundary nodes" (few nodes) and "outer internal nodes" (many nodes). The second time, after removing the "outer boundary nodes," the "outer internal nodes" are partitioned into "inner boundary nodes" (few nodes) and "inner internal nodes" (many nodes).

[0047] Step 2: Eliminate the inner nodes of the inner layer to form the middle layer Schur complement matrix.

[0048] The submatrices 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 through Gaussian elimination to obtain the middle-layer Schur complement matrix for subsequent calculations.

[0049] Step 3: Eliminate the inner boundary nodes to form the global Schur complement matrix.

[0050] The submatrices corresponding to the outer boundary nodes and associated inner boundary nodes of each external subdomain are extracted in parallel. The inner boundary nodes are then eliminated through Gaussian elimination to obtain the global Schur complement matrix. This process of eliminating the inner internal nodes first and then the inner boundary nodes utilizes our proposed parallel Schur complement calculation method, which transfers the conductance to the outer boundary nodes (selected ports).

[0051] The parallel processing flow of the Schur complement matrix is:

[0052] 1) First, identify the selected port and non-port nodes;

[0053] 2) Use graph partitioning tools to classify nodes;

[0054] 3) Non-port nodes are further divided into internal nodes and boundary nodes of each subdomain;

[0055] 4) Extract the internal nodes, boundary nodes and selected port nodes related to each subdomain in parallel;

[0056] 5) The internal nodes are eliminated in parallel to obtain the Schur complement associated with the boundary nodes and the selected port nodes;

[0057] 6) Finally, eliminate the boundary nodes and obtain the Schur complement corresponding to the selected port node.

[0058] Step 4: Solve through a top-down process.

[0059] The solution process uses a top-down approach. First, the global Schur complement matrix is solved to obtain the values of the outer boundary nodes. 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 boundary node values are substituted back into the local network and the values of the inner internal nodes are solved in parallel.

[0060] This can be understood as follows: there are three unknown vectors: x (the value of the outer boundary node), y (the value of the inner boundary node), and z (the value of the inner internal node). The first system of equations is only about x, the second system is about x and y, and the third system is about x / y / z. First, solve equation 1 for x; then, use the x value from equation 1 to solve equation 2 for y; finally, use the x value from equation 1 and the y value from equation 2 to solve equation 3 for z.

[0061] To evaluate the performance of a proposed parallel solver based on the nested domain decomposition method, this application was developed in C++. The power network was partitioned using the METIS graph partitioning tool, and matrix decomposition was performed using the CHOLMOD direct solver in conjunction with Intel MKL. Experiments were conducted on the IBMPG (IBM's power grid) and THUPG (Tsinghua University's power grid) benchmarks. The proposed solution was compared with two state-of-the-art methods: a CHOLMOD-based serial solver and a traditional domain decomposition method (DDM) parallel solver. All experiments were conducted on a Linux server equipped with two Intel Xeon Silver 4210R processors and 128GB of memory.

[0062] 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 serial solution method (CHOLMOD) are demonstrated. It can be found that with the increase in the number of threads, the speed of the DDM parallel solver is significantly improved compared with the serial solver. However, with the further increase in 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 Schur complement matrix to increase sharply, and the global Schur 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 Schur 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 Schur complement matrix. On the other hand, the middle-layer Schur complement matrix can be processed in parallel, thereby effectively improving the computing efficiency. Therefore, as Figure 5 As shown in FIG, 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.

[0063] In summary, the nested DDM parallel solver proposed in this paper offers improved scalability compared to traditional DDM parallel solvers. It not only accelerates the solution process but also optimizes resource utilization, making it highly adaptable to the growing computational demands of modern integrated circuit design.

[0064] In a specific embodiment, a method for parallel solving of a power network based on nested domain decomposition includes the following steps:

[0065] 1) First, the power distribution network (PDN) is partitioned into p outer subdomains. Each outer subdomain is then further partitioned into q inner subdomains, forming local networks numbered 11 to pq. This hierarchical strategy significantly reduces the size of the top-level problem by introducing an intermediate-level Schur complement matrix. Although this intermediate-level Schur complement matrix remains dense, it is more amenable to parallel computation.

[0066] 2) Next, for each local network i1 to iq within the external subdomain i, an intermediate network i is constructed by computing the Schur complement matrix, which concentrates the admittance of most nodes onto a specific node. Then, through further node elimination, the p intermediate networks are merged into a top-level network.

[0067] 3) The solution process starts with solving the top-level 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 2shown.

[0068] Among them, step 2) uses a method for parallel calculation of Schul complements as one of the core technologies of the nested domain decomposition method. The 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 Schul complements corresponding to the boundary nodes and selected port nodes. Finally, the boundary nodes are eliminated to obtain the Schul complements corresponding to the selected port nodes. The specific process is as follows Figure 3 shown.

[0069] 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 when the processor runs the program, the steps of the above-mentioned method for parallel solution of power supply network based on nested domain decomposition are executed.

[0070] The present invention also provides a computer-readable storage medium having computer instructions stored thereon. When the computer instructions are executed, the steps of the above-mentioned method for parallel solution of a power network based on nested domain decomposition are executed.

[0071] 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 above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above 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: Divide the original Laplace matrix graph into a nested region decomposition form, forming 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 the problem through a top-down process; 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.

2. The method for parallel solving of power network based on nested domain decomposition according to claim 1, characterized in that: The process of step 1 is divided into: Step 1.1: Perform a preliminary partition on the graph corresponding to the original Laplacian matrix to form an outer 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 convert it 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 inner 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 calculation.

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

5. A parallel solution method for power network based on nested domain decomposition according to any one of claims 1 to 4, 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) extracting the internal nodes, boundary nodes and selected port nodes associated with each subdomain in parallel; 5) The internal nodes are eliminated in parallel to obtain the Schur complements 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.

6. A parallel solution method for power network based on nested domain decomposition according to any one of claims 1 to 4, characterized in that: The method uses the METIS graph partitioning tool to partition the power network and uses the CHOLMOD direct solver combined with IntelMKL to perform matrix decomposition.

7. 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 parallel solution method for a power supply network based on nested domain decomposition according to any one of claims 1 to 4 when running the program.

8. A computer-readable storage medium having computer instructions stored thereon, characterized in that: When the computer instructions are executed, the steps of the method for parallel solving a power network based on nested domain decomposition according to any one of claims 1 to 4 are executed.

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