A reduced basis method for matrix equations associated with stochastic Galerkin approximation of parameter-dependent saddle point problems

We are interested in the efficient numerical solution of the structured linear systems that arise when we apply stochastic Galerkin mixed finite element methods (SGMFEMs) to systems of partial differential equations (PDEs) with inputs that depend on a possibly large number of uncertain parameters. In particular, we are interested in PDEs arising in engineering applications for which SGMFEMs give rise to saddle point systems, such as linear elasticity and fluid flow problems. Despite being highly structured, saddle point systems associated with SGMFEMs are challenging to solve due to their extremely large size. The number of equations is the product of (i) the number of degrees of freedom associated with the chosen mixed finite element method on the spatial domain and (ii) the dimension of the polynomial space associated with the parameter domain. When we refine the finite element mesh and/or increase the degree of the parametric polynomial approximation to improve accuracy, the dimension of the associated linear system increases. When working on standard desktop computers, one cannot use conventional Krylov subspace methods for very fine SGMFEM discretisations because storing the required matrices and vectors quickly exhausts available memory. One potential remedy is to recast the discrete problem as a linear multi-term matrix equation (LMTME) and use reduced basis methods. Such methods construct low rank factored approximations to the solution matrix by projecting the problem onto a lower-dimensional space. Our main aim is to develop a memory efficient solver for the discrete problems that arise when we apply SGMFEMs to a three-field linear elasticity model with parameter-dependent Young modulus. The starting point is a reduced basis method known as Multi-RB that was recently proposed for LMTMEs associated with symmetric and positive definite problems. After reformulating the matrix equation, the scheme iteratively constructs a reduced basis using a strategy inspired by rational Krylov subspace approximation, then applies Galerkin projection and solves a reduced problem. When the problem is not positive definite, the best choice of preconditioner and projection technique to apply is not clear. For large-scale problems with solutions that cannot be approximated well by very low rank matrices, selecting a computationally feasible stopping condition is also problematic. We modify the Multi-RB method to solve LMTMEs associated with symmetric and indefinite problems. For the linear elasticity problem of main interest, we critically assess the performance of the method using two distinct preconditioning strategies and two projection techniques. We also provide new eigenvalue analysis for the preconditioned systems. For smaller problems, we examine the convergence of the method by tracking the preconditioned and unpreconditioned residuals. Finally, to compare how the solver behaves on a different LMTME with the same structure, but whose solution matrix has different rank characteristics, we also consider a two-field groundwater flow model with parameter-dependent permeability coefficient.