Space and chaos‐expansion Galerkin proper orthogonal decomposition low‐order discretization of partial differential equations for uncertainty quantification

Abstract

The quantification of multivariate uncertainties in partial differential equations can easily exceed any computing capacity unless proper measures are taken to reduce the complexity of the model. In this work, we propose a multidimensional Galerkin proper orthogonal decomposition that optimally reduces each dimension of a tensorized product space. We provide the analytical framework and results that define and quantify the low-dimensional approximation.We illustrate its application for uncertainty modeling with polynomial chaos expansions and show its efficiency in a numerical example.