Year: 2026
Type: article
Abstract: . This paper proposes a new framework for computing low-rank solutions to nonlinear matrix equations arising from spatial discretization of nonlinear partial differential equations: low-rank Anderson acceleration (lrAA). lrAA is an adaptation of Anderson acceleration, a well-known approach for solving nonlinear fixed point problems, to the low-rank format. In particular, lrAA carries out all linear and nonlinear operations in low-rank form with rank truncation using an adaptive truncation tolerance. We propose a simple scheduling strategy to update the truncation tolerance throughout the iteration according to a residual indicator. This controls the intermediate rank and iteration number effectively. To perform rank truncation for nonlinear functions, we propose a new cross approximation, which we call Cross-DEIM, with adaptive error control that is based on the discrete empirical interpolation method (DEIM). Cross-DEIM employs an iterative update between the approximate singular value decomposition and cross approximation. It naturally incorporates a warm-start strategy for each lrAA iterate. We demonstrate the superior performance of lrAA applied to a range of linear and nonlinear problems, including those arising from finite difference discretizations of Laplace’s equation, the Bratu problem, the elliptic Monge–Ampére equation, and the Allen–Cahn equation.
DOI: 10.1137/25m173942x
Authors: Daniel Appelö, Yingda Cheng
Institution: Virginia Tech
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416111970
FWCI:
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