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Séminaire du LJLL - 15 12 2017 14h00 : M. Bachmayr
Markus Bachmayr
(Université de Bonn)
Fully discrete approximations and adaptive solvers for parametric and stochastic PDEs
Résumé
We consider the approximation of PDEs with parameter-dependent coefficients by sparse polynomial approximations in the parametric variables combined with suitable discretizations in the spatial domain. Here we are especially interested in problems with countably many parameters, as they arise when coefficients with uncertainties are modelled as random fields. For the resulting fully discrete approximations of the corresponding solution maps, we obtain convergence rates in terms of the total number of degrees of freedom. In particular, in the case of affine parametrizations, we find that independent adaptive spatial approximation for each term in the polynomial expansion yields improved convergence rates. Moreover, we give an overview of a construction of near-optimal adaptive solvers for finding such approximations.
This talk is based on joint works with Albert Cohen, Wolfgang Dahmen, Ron DeVore, Dinh Dung, Giovanni Migliorati, and Christoph Schwab.