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Solving fuzzy elliptic PDEs by means of HT tensors
Tue, 11:35--12:00
  • Corveleyn, Samuel (Department of Computer Science, K.U.Leuven, Belgium)
  • Vandewalle, Stefan (K.U.Leuven, Belgium)

We consider the solution of elliptic partial differential equations (PDE) with a high-dimensional fuzzy diffusion coefficient. Uncertainties in mathematical models are often modeled by random variables or random processes. However for the epistemic type of uncertainty, i.e., uncertainty by incomplete knowledge, an alternative based on interval or fuzzy uncertainties may be more appropriate. These equations can be solved by first constructing a polynomial approximation to the solution of the parameterized equation, by a Galerkin or collocation approach. The fuzzy quantities of interest are then calculated in a post-processing step. The specific tensor structure of the linear system of equations that is obtained by discretization of the parameterized equation allows for an approximate solution in some tensor format of low rank. Following [1], we solve the system by means of tensors in the hierarchical Tucker (HT) format using a generalized low rank GMRES method. We discuss how this approach performs in terms of computational work and accuracy in the fuzzy sense.

[1] Jonas Ballani and Lars Grasedyck. A projection method to solve linear systems in tensor format. Technical Report 309, Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Germany, April 2010.