A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem

Abstract

We study a multigrid method for solving large linear systems of equations with tensor product structure. Such systems are obtained from stochastic finite element discretization of stochastic partial differential equations such as the steady-state diffusion problem with random coefficients. When the variance in the problem is not too large, the solution can be well approximated by a low-rank object. In the proposed multigrid algorithm, the matrix iterates are truncated to low rank to reduce memory requirements and computational effort. The method is proved convergent with an analytic error bound. Numerical experiments show its effectiveness in solving the Galerkin systems compared to the original multigrid solver, especially when the number of degrees of freedom associated with the spatial discretization is large.

On the convergence of Krylov methods with low-rank truncations

“…We would like to mention that also different solution procedures based on, e.g., a low-rank multigrid method tailored to (7.2) which exploits the possible data-sparse format of the involved coefficient matrices can be employed as well. See, e.g., [64,65].…”

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

“…Low-rank multigrid. We developed a low-rank geometric multigrid method in [9] for solving linear systems with the same structure as (4.8). The complete algorithm for solving (4.9)…”

“…We also use the idea of inexact inverse iteration methods [14,22] so that in the first few steps of subspace iteration, the systems (2.6) are solved with milder error tolerances than in later steps. Specifically, we set the multigrid tolerance as mg , rel = 10 −2 [9]. This is shown to be useful in reducing the computational costs while not affecting the convergence of the subspace iteration algorithm (see Figure 4.1b).…”

“…Combined with rank compression techniques, the iterates are forced to stay in low-rank format. This idea has been used with Krylov subspace methods [2,5,20,23] and multigrid methods [9,15]. The low-rank solution methods solve the linear systems to a certain accuracy with much less computational effort and facilitate the treatment of larger problem scales.…”

“…Low-rank approximation and truncation have been used for Krylov subspace methods [5,20,23] and multigrid methods [9]. More details can be found in these references.…”