Convergence acceleration of polynomial chaos solutions via sequence transformation

We investigate convergence acceleration of the solution of stochastic differential equations characterized by their polynomial chaos expansions. Specifically, nonlinear sequence transformations are adapted to these expansions, viewed as a one-parameter family of functions with the parameter being the polynomial degree of the expansion. These transformations can be generally viewed as nonlinear generalizations of Richardson Extrapolation and permit the estimation of coefficients in higher order expansions having knowledge of the coefficients in lower order ones. Stochastic Galerkin closure that typically characterizes the solution of such equations yields polynomial chaos representations that have the requisite analytical properties to ensure suitable convergence of these nonlinear sequence transformations. We investigate specifically Shanks and Levin transformations, and explore their properties in the context of a stochastic initial value problem and a stochastic elliptic problem.