A CUR Krylov Solver for Large-Scale Linear Matrix Equations

TL;DR

This paper presents a novel CUR Krylov solver that efficiently tackles large-scale multi-term linear matrix equations by decomposing problems, selecting strategic subsets, and dynamically adjusting ranks, applicable to various complex matrix equations.

Contribution

The paper introduces a CUR decomposition-based Krylov method with a new iterative scheme for independent subproblem solving, scalable to extremely large matrices and adaptable in rank.

Findings

Successfully solves large-scale equations up to 10^13 unknowns.

Demonstrates effectiveness in time integration, Lyapunov, and nonlinear PDE contexts.

Provides a scalable, structure-agnostic approach for complex matrix equations.

Abstract

Developing efficient solvers for large-scale multi-term linear matrix equations remains a central challenge in numerical linear algebra and is still largely unresolved. This paper introduces a methodology leveraging CUR decomposition for solving large-scale generalized Sylvester as well as non-Sylvester multi-term equations on low-rank matrix manifolds. The approach decomposes the original equation into two smaller subproblems: one involving all columns with a small subset of rows, and the other involving all rows with a small subset of columns. The rows and columns are strategically selected using the discrete empirical interpolation method. We further utilize the CUR properties and propose a novel iterative scheme that removes the dependencies between selected and unselected rows (and likewise for columns), thereby enabling the subset problems to be solved independently. We present a Krylov-based scheme for solving the resulting subproblems, which scales effectively to large problems and does not rely on a Sylvester structure. The method incorporates rank adaptivity, dynamically adjusting computational rank to reach the desired accuracy. The methodology is demonstrated in three representative settings: (i) implicit time integration of matrix differential equations on low-rank manifolds, leading to multi-term linear matrix equations; (ii) large-scale steady-state generalized Lyapunov equations including cases of size up to $10^{13}$ unknown entries; and (iii) non-Sylvester linear matrix equations with Hadamard product terms, such as those arising in nonlinear partial differential equations.