A STOPPING CRITERION FOR ITERATIVE SOLUTION OF STOCHASTIC GALERKIN MATRIX EQUATIONS

In this paper we consider generalized polynomial chaos (gPC) based stochastic Galerkin approximations of linear random algebraic equations where the coefficient matrix and the right-hand side are parametrized in terms of a finite number of i.i.d random variables. We show that the standard stopping criterion used in Krylov methods for solving the stochastic Galerkin matrix equations resulting from gPC projection schemes leads to a substantial number of unnecessary and computationally expensive iterations which do not improve the solution accuracy. This trend is demonstrated by means of detailed numerical studies on symmetric and nonsymmetric linear random algebraic equations. We present some theoretical analysis for the special case of linear random algebraic equations with a symmetric positive definite coefficient matrix to gain more detailed insight into this behavior. Finally, we propose a new stopping criterion for iterative Krylov solvers to avoid unnecessary iterations while solving stochastic Galerkin matrix equations. Our numerical studies suggest that the proposed stopping criterion can provide up to a threefold reduction in the computational cost.