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Séminaire du LJLL - 10 05 2019 14h00 : D. Peterseim

Daniel Peterseim (Université d’Augsburg)
Sparse compression of expected solution operators

We show that the expected solution operator of a prototypical linear elliptic partial differential operator with random diffusion coefficient is well approximated by a computable sparse matrix. This result holds true without structural assumptions on the random coefficient such as stationarity, ergodicity or any characteristic length of correlation. The constructive proof is based on localized orthogonal multiresolution decompositions of the solution space for each realization of the random coefficient. The decompositions lead to a block-diagonal representation of the random operator with well-conditioned sparse blocks. Hence, an approximate inversion is achieved by a few steps of some standard iterative solver. The resulting approximate solution operator can be reinterpreted in terms of classical Haar wavelets without loss of sparsity. The expectation of the Haar representation can be computed without difficulty using appropriate sampling techniques. The overall construction leads to a computationally efficient method for the direct approximation of the expected solution operator which is relevant for stochastic homogenization and uncertainty quantification.
This is joint work with Michael Feischl (Technische Universität Wien).
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