Domain Decomposition Eigenvalue Solver for Large Sparse Matrices
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Solution Overview
Problem
Current methods for computing eigenvalues and eigenvectors of large sparse symmetric matrix pencils are computationally expensive and resource-intensive, particularly in applications like spectral clustering and frequency response analysis, leading to bottlenecks in processing time and resource demand.
Innovation Solution
The proposed method employs a domain decomposition technique using a Rayleigh-Ritz projection scheme, dividing the projection subspace into interface and interior variable subspaces, and utilizing a zeroth-order truncation of the spectral Schur complement to compute eigenvalues, reducing orthogonalization costs and enhancing parallelism without requiring multiple shifts.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If conventional methods (shift-and-invert Lanczos, Automated MultiLevel Substructuring) are used to compute eigenpairs of large sparse symmetric matrix pencils, then computational resources and processing time are consumed, but the accuracy and efficiency are limited by high orthogonalization costs and lack of parallelism
Solution Approach 1:
The projection subspace is divided into two separate subspaces: interface variables and interior variables. The interface variable subspace is built by computing a few eigenvectors and derivatives of the zeroth-order approximation of the non-linear matrix-valued interface operator, while the interior variable subspace is built independently in each subdomain by exploiting local eigenmodes and matrix resolvent approximations. This segmentation enables reduced orthogonalization costs and enhanced parallelism.
Solution Approach 2:
The method pre-computes a zeroth-order approximation of the non-linear matrix-valued interface operator and its eigenvectors and derivatives before performing the final Rayleigh-Ritz projection. This preliminary action avoids the need for multiple shifts and reduces the computational burden during the main eigenvalue computation phase.
2Measurement precision
If more accurate eigenpair computations are performed using conventional methods, then measurement precision improves, but device complexity and computational demand increase
Solution Approach 1:
By segmenting the projection subspace into interface and interior variable subspaces, the method achieves high accuracy eigenpair computations without requiring complex full-domain orthogonalization. The interface variable subspace captures the essential coupling between subdomains, while the interior variable subspace can be computed independently in each subdomain using simpler local operations.
Solution Approach 2:
The zeroth-order approximation of the non-linear matrix-valued interface operator serves as an intermediary that simplifies the complex eigenvalue problem. By using this approximation to build the interface variable subspace, the method reduces the computational complexity while maintaining accuracy, as the interface operator mediates the interaction between different subdomains.
Data Source
AI summary
A device solves for eigenvalues of a matrix system. The device performs a domain decomposition of a matrix system into non-overlapping subdomains and a reordering of matrices of the matrix system. An interface variable projection subspace associated with interface variables of an adjacency graph of the matrix system is created. The interface variables are related to nodes of the adjacency graph which are connected with nodes located in neighboring partitions. An internal variable projection subspace is created that is associated with internal variables of the adjacency graph of the matrix system, wherein the internal variables are related to nodes of the adjacency graph which are connected only to nodes located in the same partition. A projection matrix is built based on the interface variable projection subspace and the internal variable projection subspace. The device determines eigenvalues that solve a Raleigh-Ritz eigenvalue problem utilizing the projection matrix.