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Multi-Index Stochastic Collocation (MISC) for Elliptic PDEs with random data

Presented by: 
Lorenzo Tamellini
Tuesday 6th February 2018 - 11:30 to 12:30
INI Seminar Room 1
Co-authors: Joakim Beck (KAUST), Abdul-Lateef Haji-Ali (Oxford University), Fabio Nobile (EPFL), Raul Tempone (KAUST)

In this talk we describe the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of an elliptic PDE with random data. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. We propose an optimization procedure to select the most effective mixed differences to include in the MISC estimator: such optimization is a crucial step and allows us to build a method that, provided with sufficient solution regularity, is potentially more effective than other multi-level collocation methods already available in literature. We provide a complexity analysis both in the case of a finite and an infinite number of random variables, showing that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one dimensional problem. We show the effectiveness of MISC with some computational tests, and in particular we discuss how MISC can be efficiently combined with an isogeometric solver for PDE.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons