Adaptive method for solving large sparse ill-conditioned linear systems

By adaptively adjusting the subspace dimension and introducing a random sketch matrix to reduce computational complexity, the problems of high computational cost and unstable convergence performance of GMRES in large-scale sparse ill-conditioned linear systems are solved, and efficient solutions for sparse ill-conditioned linear systems are achieved.

CN122196311APending Publication Date: 2026-06-12SOUTHWEST PETROLEUM UNIV
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Patent Information

Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
SOUTHWEST PETROLEUM UNIV
Filing Date
2026-03-06
Publication Date
2026-06-12

AI Technical Summary

⚠Technical Problem

Existing generalized minimum residual method (GMRES) and its variants suffer from problems such as high computational cost, high storage requirements, high communication overhead in the orthogonalization process, lack of adaptability in the restart strategy, and insufficient preservation of spectral information when solving large-scale sparse ill-conditioned linear systems, resulting in low computational efficiency and unstable convergence performance.

⚗Method used

An adaptive approach is adopted to reduce the computational complexity of the orthogonalization process by constructing a random sketch matrix. The subspace dimension is dynamically adjusted by utilizing the norm of the coefficient vector, and the harmonic Ritz vector is extracted as key spectral information to construct an enhanced Krylov subspace, thereby achieving the solution of the low-dimensional least squares problem.

🎯Benefits of technology

It significantly reduces computational complexity and memory usage, improves the algorithm's adaptability and convergence speed to different ill-conditioned problems, and enhances the overall efficiency of solving large-scale sparse ill-conditioned linear systems.

✦ Generated by Eureka AI based on patent content.

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Abstract

The present application relates to computer numerical calculation and high performance computing technical field, disclose a kind of adaptive method for solving large-scale sparse ill-conditioned linear system, the method is first constructed random sketch matrix satisfying the subspace embedding property;In iterative loop, according to the norm of coefficient vector, the dimension of subspace is adaptively determined to cope with convergence stagnation;Random sketch matrix is used to project the dimension reduction of coefficient matrix, combined with random Arnoldi process and the harmonic Ritz vector reserved in the last cycle to construct enhanced Krylov subspace;Then the S-orthogonality of extended basis matrix is used to transform the original problem into a low-dimensional least squares problem to solve, update the approximate solution;If not converged, the eigenvector with the smallest modulus is extracted as the key spectral information and passed to the next cycle.The present application effectively solves the convergence stagnation problem, greatly reduces the orthogonalization calculation complexity, significantly improves the convergence speed and stability of the algorithm.
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