Low-Rank Solution Methods for Discrete Parametrized Partial Differential Equations

Abstract

Stochastic partial differential equations are widely used to model physical problems with uncertainty. For numerical treatment, the stochastic Galerkin discretization in general gives rise to large, coupled algebraic systems that are computationally expensive to solve. In this thesis, we develop efficient iterative algorithms to reduce the costs, by taking advantage of the structures of the systems and computing low-rank approximations to the discrete solutions.

We demonstrate this idea by exploring three types of problems: (i) the stochastic diffusion equation, in which the diffusion coefficient is a random field; (ii) a collection of stochastic eigenvalue problems arising from models of diffusion and fluid dynamics; (iii) stochastic version of the time-dependent incompressible Navier--Stokes equations with an uncertain viscosity. These problems range from a relatively straightforward linear elliptic problem for which we are able to obtain rigorous results on convergence rates for solvers, to more complex models that include eigenvalue computations and nonlinear and time-dependent computations.

For the diffusion problem, we propose a low-rank multigrid method for solving the linear system obtained from the stochastic Galerkin discretization. In the algorithm, the iterates are represented as low-rank matrices, with which the associated computations become much cheaper. We conduct a rigorous error analysis for the convergence of the low-rank multigrid method. Numerical experiments show significant cost savings from low-rank approximation.

We design a low-rank variant of the inverse subspace iteration algorithm for stochastic eigenvalue problems. We apply low-rank iterative methods to efficiently solve the large algebraic systems required at each step of the algorithm, and show that the costs of other computations, including the Gram--Schmidt process and the Rayleigh quotient, are also greatly reduced. The accuracy of the solutions and efficiency of the algorithm are illustrated in numerical tests.

For the time-dependent Navier--Stokes problem, we consider an all-at-once formulation where the discrete solutions at all the time steps are represented in a three-dimensional tensor. In the nonlinear iteration, we compute low-rank tensor approximations to explore further reduction in memory and computation. Effective mean-based preconditioners are derived for the all-at-once systems. The low-rank algorithm is able to efficiently handle large-size problems.