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

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.

Automated summary

In this work, a multidimensional Galerkin proper orthogonal decomposition method is proposed to reduce the complexity of quantifying multivariate uncertainties in partial differential equations. The analytical framework and results are provided to define and quantify the low-dimensional approximation. An application for uncertainty modeling using polynomial chaos expansions is illustrated, showing the efficiency of the proposed method.