A bluff-and-fix algorithm for polynomial chaos methods

Abstract

Stochastic Galerkin methods can be used to approximate the solution to a differential equation in the presence of uncertainties represented as stochastic inputs or parameters. The strategy is to express the resulting stochastic solution using $$M + 1$$ terms of a polynomial chaos expansion and then derive and solve a deterministic, coupled system of PDEs with standard numerical techniques. One of the critical advantages of this approach is its provable convergence as M increases. The challenge is that the solution to the M system cannot easily reuse an already-existing computer solution to the $$M-1$$ system. We present a promising iterative strategy to address this issue. Numerical estimates of the accuracy and efficiency of the proposed algorithm (bluff-and-fix) demonstrate that it can be more effective than using monolithic methods to solve the whole M + 1 system directly.