A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem
It addresses the computational challenge of solving stochastic PDEs with random coefficients for researchers in uncertainty quantification.
This paper proposes a low-rank multigrid method for solving large linear systems with tensor product structure arising from stochastic finite element discretization. The method is proven convergent with an analytic error bound and shown to be more effective than the original multigrid solver, especially for large spatial discretizations.
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.