We propose a computational framework for computing low-rank approximations to the ensemble of solutions of a parametrized system of the form $A(ξ)x(ξ)+g(x(ξ))=b(ξ)$ 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.
Automatic differentiation for Riemannian optimization on low-rank matrix and tensor-train manifolds
Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences（俄罗斯科学院马尔丘克计算数学研究所）；HSE University（高等经济学院）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）
Low-rank Riemannian eigensolver for high-dimensional Hamiltonians
ETH Zurich（苏黎世联邦理工学院）；Seminar for Applied Mathematics, ETH Zurich（苏黎世联邦理工学院应用数学研讨会）；Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences（俄罗斯科学院马丘克数值数学研究所）；National Research University Higher School of Economics（国立高等经济大学）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）
Fast Optimization Algorithm on Riemannian Manifolds and Its Application in Low-Rank Representation
Beijing Key Laboratory of Multimedia and Intelligent Technology（北京多媒体与智能技术重点实验室）；College of Metropolitan Transportation, Beijing University of Technology（北京理工大学城市交通学院）；School of Computing and Mathematics, Charles Sturt University（查尔斯·斯特劳特大学计算机与数学学院）
