Backward error and condition number of a generalized Sylvester equation, with application to the stochastic Galerkin method

Abstract

The governing equation of the stochastic Galerkin method can be formulated as a generalized Sylvester equation. Therefore, developing solvers for it is attracting a lot of attention from the uncertainty quantification community. In this regard Krylov subspace based iterative solvers, that are used for standard linear systems are being used for the generalized Sylvester equations as well. This is achieved by converting the generalized Sylvester equation to a standard linear system using the Kronecker product. Accordingly the residual norm is used as a stopping criterion for the iterations, and the condition number of linear systems is used for the generalized Sylvester equations as well. For a linear system a small residual norm implies a small backward error, and hence using residual norm as a stopping criterion is justified. In this work we prove that this need not be the case for the generalized Sylvester equation. We introduce two definitions for the backward error, and then derive an upper bound on each of them. We also verify the predictions of the analysis using numerical experiments. For the special case of the stochastic Galerkin method we show that the upper bound on the backward error can be computed with minimal computational overhead, and hence it can be used as a stopping criterion in the iterative solvers. For the matrices stemming from the stochastic Galerkin method we numerically demonstrate that the actual backward error can be up to 3 orders of magnitude higher than the relative residual. Finally by taking into account the structure of the equation we derive an expression for the condition number, and discuss an algorithm for its computation in the special case of the stochastic Galerkin method.