[Paper Review] An efficient solver based on low-rank approximation and Neumann matrix series for unsteady diffusion-type partial differential equations with random coefficients

Yujun Zhu, Min Li+2 more|arXiv (Cornell University)|Jan 16, 2026

Probabilistic and Robust Engineering Design0 citations

TL;DR

The paper proposes an LRNS solver that combines a generalized low-rank approximation of stochastic stiffness matrices with Neumann matrix series to efficiently solve unsteady diffusion-type SPDEs with random coefficients, reducing computation and memory while preserving accuracy.

ABSTRACT

In this paper, we develop an efficient numerical solver for unsteady diffusion-type partial differential equations with random coefficients. A major computational challenge in such problems lies in repeatedly handling large-scale linear systems arising from spatial and temporal discretizations under uncertainty. To address this issue, we propose a novel generalized low-rank matrix approximation to represent the stochastic stiffness matrices, and approximate their inverses using the Neumann matrix series expansion. This approach transforms high-dimensional matrix inversion into a sequence of low-dimensional matrix multiplications. Therefore, the solver significantly reduces the computational cost and storage requirements while maintaining high numerical accuracy. The error analysis of the proposed solver is also provided. Finally, we apply the method to two classic uncertainty quantification problems: unsteady stochastic diffusion equations and the associated distributed optimal control problems. Numerical results demonstrate the feasibility and effectiveness of the proposed solver.

Motivation & Objective

• Motivate robust and efficient uncertainty quantification for time-dependent diffusion equations with random coefficients.
• Develop a computationally light-weight solver that handles large ensembles of stochastic linear systems arising from MC-FEM discretizations.
• Provide error analysis for the proposed LRNS approach and demonstrate applicability to stochastic diffusion and stochastic optimal control problems.

Proposed method

• Represent stochastic perturbations of the stiffness matrices via a novel generalized low-rank approximation (GLRAM) using RSVD to obtain U and V_m with a prescribed compression ratio tau.
• Approximate inverses of perturbed matrices using a truncated Neumann series: (I + A^{-1}Ã_m^*)^{-1} ≈ sum_{r=0}^R (-A^{-1}Ã_m^*)^r.
• Transform large M×L linear systems into a sequence of low-dimensional matrix multiplications and low-rank corrections to a deterministic solution.
• Provide an error analysis framework based on RMSRE and spectral considerations of the perturbation matrix N = sum_m Ã_m Ã_m^T.
• Combine the RSVD-based LRA with Neumann-series-based inversion to obtain a scalable LRNS solver (Algorithm 2).
• Demonstrate applicability to unsteady stochastic diffusion equations and distributed stochastic optimal control problems.

Experimental results

Research questions

• RQ1 How can low-rank representations accelerate the solution of time-dependent SPDEs with random coefficients in Monte Carlo FEM settings?
• RQ2 What are the accuracy and computational benefits of combining RSVD-based generalized low-rank approximation with Neumann-series inverses for perturbed system matrices?
• RQ3 Can the proposed LRNS approach efficiently solve unsteady diffusion equations with random permeability and related stochastic control problems without sacrificing accuracy?

Key findings

• The LRNS solver transforms high-dimensional matrix inversions into a sequence of low-dimensional multiplications and low-rank updates, reducing cost and storage.
• A RSVD-based generalized low-rank approximation provides efficient and accurate representation of stochastic stiffness perturbations with a prescribed compression ratio tau.
• The Neumann-series inversion yields a practical finite-term approximation whose accuracy improves with the truncation index R and the perturbation size.
• Error analysis links RMSRE to the spectral properties of the perturbation matrix N and the chosen energy capture e(tau).
• The method is applied to unsteady stochastic diffusion equations and distributed stochastic optimal control problems, illustrating feasibility and efficiency.