Abstract

We discuss some recent developments in solution algorithms for the linear algebra problems that arise from parameter-dependent partial differential equations (PDEs). In this setting, there is a need to solve large coupled algebraic systems (which come from stochastic Galerkin methods), or large numbers of standard spatially discrete systems (from Monte Carlo or stochastic collocation methods). The ultimate goal is solutions that represent surrogate approximations that can be evaluated cheaply for multiple values of the parameters, which can be used effectively for simulation or uncertainty quantification.

Our focus is on representing parameterized solutions in reduced-basis or low-rank matrix formats. We show that efficient solution algorithms can be built from multigrid methods designed for the underlying discrete PDE , in combination with methods for truncating the ranks of iterates, which reduce both cost and storage requirements of solution algorithms. These ideas can be applied to the systems arising from many ways of treating the parameter spaces, including stochastic Galerkin and collocation. In addition, we present new approaches for solving the dense systems that arise from reduced-order models by preconditioned iterative methods and we show that such approaches can also be combined with empirical interpolation methods to solve the algebraic systems that arise from nonlinear PDEs.