Explicit cost bounds of stochastic Galerkin approximations for parameterized PDEs with random coefficients

This work analyzes the overall computational complexity of the stochastic Galerkin finite element method (SGFEM) for approximating the solution of parameterized elliptic partial differential equations with both affine and non-affine random coefficients. To compute the fully discrete solution, such approaches employ a Galerkin projection in both the deterministic and stochastic domains, produced here by a combination of finite elements and a global orthogonal basis, defined on an isotopic total degree index set, respectively. To account for the sparsity of the resulting system, we present a rigorous cost analysis that considers the total number of coupled finite element systems that must be simultaneously solved in the SGFEM. However, to maintain sparsity as the coefficient becomes increasingly nonlinear in the parameterization, it is necessary to also approximate the coefficient by an additional orthogonal expansion. In this case we prove a rigorous complexity estimate for the number of floating point operations (FLOPs) required per matrix-vector multiplication of the coupled system. Based on such complexity estimates we also develop explicit cost bounds in terms of FLOPs to solve the stochastic Galerkin (SG) systems to a prescribed tolerance, which are used to compare with the minimal complexity estimates of a stochastic collocation finite element method (SCFEM), shown in our previous work [16]. Finally, computational evidence complements the theoretical estimates and supports our conclusion that, in the case that the coefficient is affine, the coupled SG system can be solved more efficiently than the decoupled SC systems. However, as the coefficient becomes more nonlinear, it becomes prohibitively expensive to obtain an approximation with the SGFEM.