Factorizing the Stochastic Galerkin System
For researchers solving parameterized PDEs with spectral methods, this work offers a theoretical factorization that improves understanding and preconditioning of the Galerkin system.
The paper derives a factorization of the linear system arising from a stochastic Galerkin approximation of parameterized PDEs, providing eigenvalue bounds, preconditioning insights, and a flexible implementation. Numerical studies on an elliptic PDE and a CFD application demonstrate the approach.
Recent work has explored solver strategies for the linear system of equations arising from a spectral Galerkin approximation of the solution of PDEs with parameterized (or stochastic) inputs. We consider the related problem of a matrix equation whose matrix and right hand side depend on a set of parameters (e.g. a PDE with stochastic inputs semidiscretized in space) and examine the linear system arising from a similar Galerkin approximation of the solution. We derive a useful factorization of this system of equations, which yields bounds on the eigenvalues, clues to preconditioning, and a flexible implementation method for a wide array of problems. We complement this analysis with (i) a numerical study of preconditioners on a standard elliptic PDE test problem and (ii) a fluids application using existing CFD codes; the MATLAB codes used in the numerical studies are available online.