Sparse Matrix Graph Decomposition for Faster Finite Element Solving
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Solution Overview
Problem
Current methods for solving large sparse matrices in EDA simulation tools are inefficient and lack precision, particularly in analyzing non-linear circuits and 3D electromagnetic fields, leading to prolonged computation times and reduced accuracy.
Innovation Solution
The method involves restoring connection relationships between port nodes, converting the matrix to an undirected graph, decomposing it optimally, renumbering nodes, and resequencing to generate a final finite element matrix, using techniques like Metis program and evaluation functions to improve efficiency and accuracy.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional matrix solvers are used to solve large sparse matrices in EDA simulations, then computation accuracy is maintained, but computation speed becomes extremely slow when the number of unknown quantities reaches millions
Solution Approach 1:
The patent applies segmentation by dividing the large sparse matrix into multiple smaller sub-matrices through graph decomposition. The undirected graph representing the matrix is decomposed into connected components, allowing the large matrix inversion problem to be broken down into smaller, more manageable sub-problems that can be solved more efficiently while maintaining numerical accuracy.
Solution Approach 2:
The patent transforms the matrix problem into a graph problem by converting the sparse matrix into an undirected graph where nodes represent matrix rows/columns and edges represent non-zero elements. This dimensionality change allows the application of graph decomposition techniques (connected component analysis) that are more efficient for large-scale problems than traditional direct matrix solvers.
2Measurement precision
If the number of unknown quantities in 3D electromagnetic field analysis is increased to model finer grids and more complex structures, then analysis precision is improved, but computation time increases from hours to days or weeks
Solution Approach 1:
The patent segments the large system matrix arising from fine-grid 3D electromagnetic field analysis into smaller sub-matrices through graph decomposition. By identifying connected components in the associated undirected graph, the method divides the computation into smaller tasks that can be solved faster, reducing computation time from days/weeks to more practical durations while preserving the fine grid resolution needed for high precision.
Solution Approach 2:
The patent performs preliminary graph decomposition and connected component analysis before the actual matrix solving process. This preliminary action identifies the optimal decomposition structure in advance, allowing the subsequent matrix operations to proceed more efficiently. The undirected graph is constructed and decomposed beforehand, creating a roadmap for efficient computation that prevents time-consuming operations during the main simulation phase.
3Productivity
If non-zero detection and memory storage optimization are implemented to reduce reading amount during computation, then computation speed is accelerated, but treatment precision cannot be promised
Solution Approach 1:
The patent segments the matrix computation into disconnected sub-problems through graph decomposition. Each connected component forms an independent sub-matrix that can be solved separately. This segmentation maintains numerical precision because each sub-problem is solved independently with appropriate numerical methods, while the overall computation speed benefits from the reduced problem size and potential parallelization of independent sub-problems.
Data Source
AI summary
The present invention discloses a method for accelerated solving of large sparse matrix, system and storage medium thereof, which falls into the technical field of electromagnetic field computation. In view of the problem that solving of the current large sparse matrix is slow and not accurate enough, the present invention provides a method for accelerated solving of large sparse matrix, comprising: restoring connection relationships between port nodes of an initial finite element matrix, and obtaining restored second-order finite element matrix; converting the second-order finite element matrix to an undirected graph; decomposing the undirected graph, and selecting optimal decomposition via an evaluation function; renumbering nodes of the second-order finite element matrix according to the renumbered nodes and generating a new final finite element matrix; and solving the final finite element matrix. In the present invention, by restoring the connection relationships of port nodes the subsequent solving accuracy is promised, by converting the matrix to the undirected graph and decomposing, the optimal decomposition is obtained and by going on with the subsequent operations as per the optimal decomposition, the solving of the matrix is accelerated.