A Factorization of the Spectral Galerkin System for Parameterized Matrix Equations: Derivation and Applications

Authors:
Paul G. Constantine;David F. Gleich;Gianluca Iaccarino
Affiliations:
pconsta@sandia.gov;dgleich@purdue.edu;jops@stanford.edu
Venue:
SIAM Journal on Scientific Computing
Year:
2011

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Abstract

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.