Tensor-Galerkin POD for Efficient Uncertainty Quantification in PDEs with Multivariate Random Parameters

Abstract

The statistically sound treatment of modelled uncertainties in simulations comes with significant additional computational costs. Since a deterministic model can already be arbitrarily complex, running statistics for general problems may soon become infeasible unless some kind of model reduction is involved.

In this talk we present a multidimensional Galerkin Proper Orthogonal Decomposition (Galerkin POD) that simultaneously reduces the physical dimensions of the model and the dimensions related to the uncertainties.

Using basic tensor calculus we extend our recent work of space-time Galerkin POD [1] to arbitrary dimensions and apply it to PDEs with multivariate uncertain. By means of a numerical example we illustrate the procedure and how it outperforms POD based on random snapshots.

References:

[1] Baumann, M.; Benner, P. & Heiland, J. Space-Time Galerkin POD with Application in Optimal Control of Semi-linear Parabolic Partial Differential Equations SIAM J. Sci. Comput., 2018, 40, A1611-A1641