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

TL;DR

This work tackles the computational challenge of solving unsteady diffusion-type SPDEs with random coefficients by introducing a generalized low-rank approximation combined with Neumann-series based inverse corrections (LRNS). The method replaces costly full-matrix inversions with sequences of low-dimensional matrix multiplications, enabling efficient MC-FEM for time-dependent uncertainty quantification. The authors provide an RMSRE-based error analysis and demonstrate the approach on unsteady stochastic diffusion equations and distributed stochastic diffusion optimal control, achieving significant reductions in memory and compute with controlled accuracy. The LRNS framework offers a scalable path for UQ in time-dependent PDEs and can be integrated with various sampling and optimization schemes for complex stochastic systems.

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.

Figures (9)

• Figure 1.1: The flowchart of the efficient solver.
• Figure 3.1: Raw image from LFW dataset (left), the images compressed by conventional GLRAM (middle) and Algorithm \ref{['alg1']} (right) using the data compression ratio $\tau=10\%$.
• Figure 4.1: Four randomly selected MC realizations of the permeability $a(\bm{x},\omega)$.
• Figure 4.2: Four randomly selected samples of reference solution.
• Figure 4.3: Comparison among the reference solution, the numerical solutions computed by the LRNS solver in Algorithm \ref{['alg3']} under data compression ratios $\tau = 88\%$ and $10\%$.

Theorems & Definitions (12)

• Theorem 3.1: eckart1936approximation
• Theorem 3.2: zhu2024low
• Theorem 3.3: zhu2024low
• Theorem 3.4: halko2011finding
• Lemma 3.5
• Theorem 3.6: zhu2024low
• Lemma 3.7: stewart1998matrix
• Lemma 3.8: wedin1973perturbation
• Theorem 3.9
• proof