Abstract: Max Gunzburger

Department of Scientific Computing
Florida State University

Applications of Karhunen-Loève Expansions in Numerical Methods for PDEs With Random Inputs

Truncated Karhunen-Loève expansions (KLEs) provide the underpinning for many reduced-order modeling methods, including proper orthogonal decomposition (POD). In this talk, we discuss two applications of KLEs in the design of numerical methods for PDEs with random inputs.

We first treat a class of PDEs with white noise forcing. Instead of directly discretizing the white noise, we introduce an auxiliary Ornstein-Uhlenbeck process to transform the given system into one driven by correlated noise. As a result, we can use accurate approximations of colored noise fields, such as truncated KLEs, to indirectly solve problems driven by white noise, achieving, for the same number of realizations, much higher accuracy.

We also consider the effectiveness of POD reduced-order modeling for the solution of optimization and control problems, with the Navier-Stokes setting being of particular interest. Snapshots are taken not only with respect to time, but also with respect to realizations corresponding to independent choices for the set of random parameters. Comparisons with high-dimensional discretization results shows the effectiveness of the use of the stochastic ROM for obtaining accurate optimal solutions of the stochastic control problem at low cost.


 

Aachen Institute for Advanced Study in Computational Engineering Science (AICES)
at RWTH Aachen University, Germany. Email: acces11@aices.rwth-aachen.de
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