Seminar in Numerical Analysis: Sebastian Ullmann (TU Darmstadt)
Surrogate models can be used to decrease the computational cost for uncertainty quantification in the context of parabolic PDEs with stochastic data. Projection based reduced-order modeling provides surrogates which inherit the spatial structure of the solution as well as the underlying physics. In my talk I focus on the type of models that is derived by a Galerkin projection onto a proper orthogonal decomposition (POD) of snapshots of the solution.
Standard techniques assume that all snapshots use one and the same spatial mesh. I present a generalization for unsteady adaptive finite elements, where the mesh can change from time step to time step and, in the case of stochastic sampling, from realization to realization. I will answer the following questions: How can the coding effort for creating such a reduced-order model be minimized? How can the union of all snapshot meshes be avoided? What is the main difference between static and adaptive snapshots in the error analysis of Galerkin reduced-order models?
As a numerical test case I consider a two-dimensional viscous Burgers equation with smooth initial data multiplied by a normally distributed random variable. The results illustrate the convergence properties with respect to the number of POD basis functions and indicate possible savings of computation time.
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