An Efficient Reduced Basis Solver for Stochastic Galerkin Matrix Equations

Abstract

Abstract. Stochastic Galerkin finite element approximation of PDEs with random inputs leads to linear systems of equations with coefficient matrices that have a characteristic Kronecker product structure. By reformulating the systems as multi-term linear matrix equations, we develop an efficient solution algorithm which generalizes ideas from rational Krylov subspace approximation. The new approach determines a low-rank approximation to the solution matrix by performing a projection onto a low-dimensional space and provides an efficient solution strategy whose convergence rate is independent of the spatial approximation. Moreover, it requires far less memory than the standard preconditioned conjugate gradient method applied to the Kronecker formulation of the linear systems.Key words. generalized matrix equations, PDEs with random data, stochastic finite elements, iterative solvers, rational Krylov subspace methods.AMS subject classifications. 35R60 , 60H35, 65N30, 65F101. Introduction. This paper is concerned with the design and implementation of efficient iterative solution algorithms for high-dimensional linear algebra systems that arise from Galerkin approximation of elliptic PDE problems with correlated random inputs. The tensor product structure of the Galerkin approximation space and the low-rank structure that is inherent in the discrete systems is exploited in our innovative solution strategy. This strategy builds on recent progress in rational Krylov subspace approximation (see, for example, Simoncini [21]) and generates an accurate reduced basis approximation of the spatial component of the Galerkin solution.We focus on stochastic steady-state diffusion equations with homogeneous Dirichlet boundary conditions. To this end, let D ⊂ R 2 be a sufficiently regular spatial domain and let Ω be a sample space associated with a probability space (Ω, F , P). Our goal is to approximate u : D × Ω → R such that P-a.s.,

Optimality Properties of Galerkin and Petrov–Galerkin Methods for Linear Matrix Equations

“…Sometimes this is the only option as effective algorithms to solve (1.2) in its natural matrix equation form are still lacking in the literature in the most general case. The methods developed so far require some additional assumptions on the coefficient matrices A j , B j ; see, e.g., [7,17,20,30,34]. In this section we show that exploiting the matrix structure of equation (1.2) not only leads to numerical algorithms with lower computational costs per iteration and modest storage demands, but they also avoid some spectral redundancy encoded in the problem formulation (4.1).…”

“…that have arisen as a natural algebraic model for discretized partial differential equations, possibly including stochastic terms or parameter dependent coefficient matrices [4,8,28,30], for PDEconstrained optimization problems [39], data assimilation [13], and many other applied contexts, including building blocks of other numerical procedures [23]; see also [35] for further references. The general matrix equation (1.2) covers two well known cases, the (generalized) Sylvester equation (for = 2), and the Lyapunov equation…”

Krylov methods for low‐rank commuting generalized Sylvester equations

“…For example, the generalized Lyapunov equation , which corresponds to the special case where B = A , M i = N i , and C 1 = C 2 , arises in model order reduction of bilinear and stochastic systems (e.g., see Benner et al, Damm and references therein). Many problems arising from the discretization of PDEs can be formulated as generalized Sylvester equations . Low‐rank approximability for matrix equations has been investigated in different settings: for Sylvester equations, generalized Lyapunov equations with low‐rank correction and more generally for linear systems with tensor product structure …”

A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations

“…In recent years, many authors started to explore the Kronecker-product structure of such problems and developed iterative algorithms that exploit the structure to reduce computational efforts [3,12,13,15,18,22]. In addition, it has been shown that the solution of (1.1) in the stochastic Galerkin setting can be approximated by a tensor of low rank, which further reduces computational effort [4].…”