Cost reduction of stochastic Galerkin method by adaptive identification of significant polynomial chaos bases for elliptic equations

One widely used and computationally efficient method for uncertainty quantification using spectral stochastic finite element is the stochastic Galerkin method. Here the solution is represented in polynomial chaos expansion, and the residual of the discretized governing equation is projected on the polynomial chaos bases. This results in a system of deterministic algebraic equations with the polynomials chaos coefficients as unknown. However, one impediment for its large scale applications is the curse of dimensionality, that is, the exponential growth of the number of polynomial chaos bases with the stochastic dimensionality and degree of expansion. Here, for a stochastic elliptic problem, an adaptive selection of polynomial chaos bases is proposed. Accordingly, during the first few iterations in the preconditioned conjugate gradient method for solving the system of linear algebraic equations, the chaos bases with maximal contribution -in an appropriately defined metric - to the solution are first identified. Subsequently, only these bases are retained for further iterations until convergence is achieved. Using numerical studies a three times cost saving over the existing method is observed. Furthermore, for enhancing the computational cost gain, the stochastic Galerkin method is reformulated as a generalized Sylvester equation. This step allowed efficient usage of the sparsity of moments of product of polynomial chaos bases. Through numerical studies on problems with large stochastic dimensionality, an additional cost saving of up to one order of magnitude -twenty times -is observed. This amounts to sixty times speedup over the existing method, when adaptive selection and generalized Sylvester equation formulation are used together. The proposed methodology can be easily incorporated in an existing standard stochastic Galerkin method solver for elliptic problems.