Adaptive stochastic Galerkin finite element approximation: accurate error estimation and memory-smart solvers.

Stochastic Galerkin finite element methods (SGFEMs) are commonly used to approximate solutions to PDEs with random inputs. When the inputs are represented by a countable number of random variables, one can reformulate the problem as a high-dimensional parametric one. In the SGFEM approach, an approximation is sought in a space which is the tensor product of a finite element space on the spatial domain and a space of polynomials on the parameter domain. When the number of active parameters is high, the dimension of standard approximation spaces is unmanageable. One remedy is to generate an approximation in a low-dimensional space and then use a posteriori error estimators to decide whether it is necessary to enrich the finite element space and/or the polynomial space. This allows us to build up a tailored sequence of approximation spaces, so that the dimension of the final one is balanced against an error tolerance. 

We discuss two issues related to the efficient implementation of adaptive SGFEMs for elliptic PDEs with random coefficients: error estimation and fast solvers. First, we discuss an error estimation strategy proposed in [A. Bespalov, C.E. Powell, D. Silvester, Energy norm a posteriori error estimation for parametric operator equations, SIAM Journal Sci. Comp. 36(2), A339--A363, 2014]. Next, we discuss the associated discrete linear systems. When recast as matrix equations, the solution matrix often has low rank. We describe an adaptive  reduced basis algorithm proposed in [C.E. Powell, V. Simoncini, D. Silvester, An efficient reduced basis solver for stochastic Galerkin matrix equations, SIAM J. Comp. Sci. 39(1), pp A141-A163 (2017)] that exploits this and has lower memory requirements than standard Krylov methods.