Analysis Based Computation

Seminar:

Analysis Based Computation
Tuesday, September 11, 2018
3:30PM – 5PM
POB 6.304

Wolfgang Dahmen

Among the many facets of numerically solving PDEs the perhaps two core tasks are:
(i) the efficient and accurate solvability of corresponding discretized problems,
(ii) bounding the deviation of the discrete solution from the exact solution of the underlying continuous problem.

Remarkably, both tasks are traditionally treated and analyzed separately. Regarding (ii), often only a priori estimates are available that hold under sometimes unrealistic regularity assumptions. In this talk we discuss a strategy that closely intertwines (i) and (ii). In particular, it aims at producing approximate solutions that meet a given target accuracy tolerance, eventually also in scenarios where conventional schemes fail to do so. The starting point is a suitable stable variational formulation that allows one to
bound errors in the trial norm by residuals in the dual norm of an appropriate test space. The next step is to formulate an iteration in the infinite dimensional setting that
converges with a fixed error reduction per step. The numerical scheme consists then of approximately realizing this iteration within appropriately updated accuracy
tolerances so as to still guarantee convergence to the exact solution. The availability of rigorous a posteriori error estimates for the discrete sub-problems are therefore of central importance. Roughly speaking, one clings as long as possible to the infinite dimensional problem to best exploit its intrinsic metrics. We briefly sketch how this strategy can cope with the inherent obstructions encountered in two problem scenarios, namely (a) a kinetic model of radiative transfer type, and (b) the p-Poisson problem for 1<p<2. In neither scenario the discrete sub-problems can be adequately treated by classical Galerkin schemes. Instead we show that Discontinuous Petrov Galerkin concepts can meet the essential requirements. If time permits we conclude with some remarks tying these concepts into coupling PDE models with data.

Bio
Wolfgang Dahmen is currently a chaired professor in mathematics (SmartState and Williams-Hedberg-Hedberg Chair) at the University of South Carolina in Columbia, South Carolina, USA. His research interests are in Approximation Theory, Numerical, Applied and Harmonic Analysis as well as interdisciplinary applications. Together with his collaborators he has developed adaptive and nonlinear solution concepts in a variety of contexts such as image and data analysis, machine learning as well as the numerical solution of singular integral and partial differential equations, and model reduction.

Professor Dahmen received his PhD from RWTH Aachen in 1976 and his Habilitation from the University of Bonn. After an IBM Postdoctoral Fellowship at the IBM Research Center in Yorktown Heights, NY, he took (associate and full) professor positions at the University of Bielefeld and the Free University of Berlin before joining RWTH Aachen in 1992. In 2013 he was awarded a Distinguished RWTH Professorship and in 2017 he became chaired professor at the University of South Carolina. In 2002 he received the Gottfried-Wilhelm-Leibniz Award of the German Research Foundation was elected in 2009 to the German National Academy of Sciences, Leopoldina. He had visiting professor positions at the University of South Carolina and at the Universite Marie et Pierre Curie in Paris, France. He has served on the Scientific Advisory boards of the CRM in Barcelona and the Isaac Newton Institute in Cambridge, UK. From 2014 to 2017 he was the Chair of the Board of Directors of the Society Foundations of Computational Mathematics.

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