An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
We formulate a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with long-term integration and discontinuities in stochastic differential equations. We first present this method for Legendre-chaos corresponding to uniform random inputs, and subsequently we generalize it to other random inputs. The main idea of ME-gPC is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value. In each subdomain or random element, we then employ a generalized polynomial chaos expansion. We develop a criterion to perform such a decomposition adaptively, and demonstrate its effectiveness for ODEs, including the KraichnanOrszag three-mode problem, as well as advectiondiffusion problems. The new method is similar to spectral element method for deterministic problems but with hp discretization of the random space.
polynomial chaos discontinuities uncertainty