Low-rank solutions to a class of parametrized systems using Riemannian optimization

TL;DR

This paper introduces a Riemannian optimization framework for efficiently computing low-rank solutions to parametrized systems, applicable to both linear and nonlinear cases, with theoretical guarantees and numerical validation.

Contribution

It develops a novel approach that formulates the problem as a Riemannian optimization task over fixed-rank matrices, extending low-rank approximation techniques to nonlinear parametrized systems.

Findings

Achieves significant computational savings over independent solves.

Provides theoretical conditions for accurate low-rank approximations.

Demonstrates effectiveness on large-scale nonlinear problems.

Abstract

We propose a computational framework for computing low-rank approximations to the ensemble of solutions of a parametrized system of the form $A(\xi)x(\xi)+g(x(\xi))=b(\xi)$ for multiple parameter values. The central idea is to reinterpret the parametrized system as the first-order optimality condition of an optimization problem set over the space of real matrices, which is then minimized over the manifold of fixed-rank matrices. This formulation enables the use of Riemannian optimization techniques, including conjugate gradient and trust-region methods, and covers both linear and nonlinear instances under mild assumptions on the structure of the parametrized system. We further provide a theoretical analysis establishing conditions under which the solution matrix admits accurate low-rank approximations, extending existing results from linear to nonlinear problems. To enhance computational efficiency and robustness, we discuss tailored preconditioning strategies and a rank-compression mechanism to control the rank growth induced by nonlinearities. Numerical experiments demonstrate that the proposed approach achieves significant computational savings compared to solving each system independently, as well as highlight the potential of Riemannian optimization methods for low-rank approximations in large-scale parametrized nonlinear problems.