Space and Chaos-Expansion Galerkin POD Low-order Discretization of PDEs for Uncertainty Quantification

TL;DR

This paper introduces a multidimensional Galerkin Proper Orthogonal Decomposition method that efficiently reduces the complexity of PDE models for uncertainty quantification, demonstrated through Polynomial Chaos Expansions.

Contribution

It presents a novel tensorized Galerkin POD approach that optimally reduces each dimension of the model space, with analytical framework and practical application for uncertainty modeling.

Findings

Efficient low-dimensional approximation of PDEs for uncertainty quantification.

Demonstrated effectiveness with Polynomial Chaos Expansions.

Numerical example confirms computational efficiency.

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.