  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.
