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# ACMD Seminar: Multiscale stochastic domain decomposition methods for multiphysics problems

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Ivan Yotov*Chair, Department of Mathematics, University of Pittsburgh*

Tuesday, May 16, 2017, 11:00 - 12:00Building 101, Lecture Room FGaithersburg

Tuesday, May 16, 2017, 09:00 - 10:00Room 1-4058Boulder

**Abstract:** We discuss a multiscale framework for multiphysics problems based on domain decomposition methods. The domain is decomposed into a series of subdomains (coarse grid) with different physical processes, mathematical models, and numerical methods. The equations are discretized locally on a fine scale, while interface conditions are imposed weakly on a coarse scale using mortar finite elements. By eliminating the subdomain unknowns, the global problem is reduced to a coarse scale interface problem that is solved efficiently using a multiscale flux basis. The framework is extended to include uncertainty quantification for stochastic problems. The stochastic parameters are represented with different Karhunen-Loeve (KL) expansions in different subdomains. Stochastic collocation is utilized to sample the stochastic space. We precompute a multiscale basis, which involves solving subdomain problems for each realization of the local KL expansion.  The basis is then used to solve the coarse scale mortar interface problem for each global KL realization. The resulting algorithm is orders of magnitude faster than a global stochastic collocation approach. Applications to flow in porous media, as well as coupled Stokes-Darcy flows are presented.

**Bio:** Ivan Yotov is Professor and Chair of the Department of Mathematics at University of Pittsburgh. His research interests are in numerical analysis of partial differential equations and large scale scientific computing with applications to flow in porous media, computational fluid dynamics, and biomedical problems. His recent work spans multiscale modeling of multiphysics systems of coupled flow and mechanics, advanced finite element and finite volume discretizations, scalable parallel solvers and preconditioners, stochastic modeling, uncertainty quantification, and parameter estimation.Professor Yotov obtained his Ph.D. in Computational and Applied Mathematics from Rice University in 1996. He held a postdoctoral position at the Institute for Computational Engineering and Sciences at The University of Texas at Austin before joining University of Pittsburgh in 1998. He is an author of more than 80 scientific papers. He is Editor in Chief of the journal Computational Geosciences and has served as Associate Editor of SIAM Journal on Numerical Analysis and Numerical Linear Algebra with Applications.

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