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Tuesday, Sep 11, 2018 from 3:30PM to 5PM | POB 6.304
Hosted by Leszek Demkowicz
Sponsor: ICES Seminar
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.
Thursday, Sep 20, 2018 from 3:30PM to 5PM | POB 6.304
Hosted by Leszek Demkowicz
Sponsor: ICES Seminar
We present VoroCrust: a novel approach to polyhedral meshing that simultaneously generates a quality mesh of the surface of a Piecewise Linear Complex (PLC) model and decomposes the enclosed volume by unweighted Voronoi cells with good aspect ratios conforming to the surface mesh, without clipping or bad normals. VoroCrust has an embedded sizing function that capture the curvature of the model, and robustly represents sharp features and narrow regions that may be associated with the input model.
Up to our knowledge, VoroCrust is the first to solve this open problem. VoroCrust also outputs an approximation of the medial axis of the input model and provides a fast technique for in/out point classification. A variation of VoroCrust can also handle non-manifold and non-watertight inputs. We illustrate the robustness and output quality of VoroCrust through a collection of models of varying complexity. In this talk we also present the recently released VoroCrust software and discuss its performance in practice.
Bio
Mohamed Ebeida is an expert in computational geometry related to Voronoi diagrams, hyperplane sampling and sphere packing. He is the creator of several novel Voronoi-based algorithms with application to meshing, high-dimensional sampling, uncertainty quantification, and optimization. He graduated from University of California Davis in 2008 with a PhD in Mechanical and Aeronautical Engineering and a Masters in Applied Mathematics. He worked for two years as a Postdoc at Carnegie Mellon University. In 2010, he joined Sandia National Laboratories where he actively works in exploring the potential of Voronoi decompositions for a wide range on non-traditional applications. Mohamed is the inventor on three patents for novel applications of the emerging VoroCrust technology in low and high-dimensions.
Thursday, Oct 25, 2018 from 3:30PM to 5PM | POB 6.304
Hosted by Karen Willcox
Sponsor: ICES Seminar
Work with Sergey Dolgov, University of Bath, UK, Akwum Onwunta and Martin Stoll, Faculty of Mathematics, TU Chemnitz, Germany
We discuss optimization and control of unsteady partial differential equations (PDEs), where some coefficient of the PDE as well as the control may be uncertain. This may be due to the lack of knowledge about the exact physical parameters like material properties describing a real-world problem ("epistemic uncertainty") or the inability to apply a computed optimal control exactly in practice. Using a stochastic Galerkin space-time discretization of the optimality system resulting from such PDE-constrained optimization problems under uncertainty leads to large-scale linear or nonlinear systems of equations in saddle point form. Nonlinearity is treated witha Picard-type iteration in which linear saddle point systems have to be solved in each iteration step. Using data compression based on separation of variables and the tensor train (TT) format, we show how these large-scale indefinite and (non)symmetric systems that typically have $10^8$ to $10^{15}$ unknowns can be solved without the use of HPC technology. The key observation is that the unknown and the data can be well approximated in a new block TT format that reduces complexity by several orders of magnitude. As examples, we consider control and optimization problems for the linear heat equation, the unsteady Stokes and Stokes-Brinkman equations, as well as the incompressible unsteady Navier-Stokes equations. The talk reviews the results published in {BenOS16,BenDOS16} and provides new results for the Navier-Stokes case.
Bibliography
{BenOS16} P. Benner, A. Onwunta, M. Stoll, Block-diagonal preconditioning for optimal control problems constrained by {PDE}s with uncertain inputs, SIAM Journal on Matrix Analysis and Application, 37(2):491--518, 2016.
{BenDOS16} P. Benner, S. Dolgov, A. Onwunta, M. Stoll, Low-rank solvers for unsteady Stokes-Brinkman optimal control problem with random data,
Computer Methods in Applied Mechanics and Engineering, 304:26--54, 2016.
Bio
Peter Benner received the Diplom in mathematics from RWTH Aachen, Germany, in 1993. From 1993 to 1997, he worked on his Ph.D. at the University of Kansas, Lawrence, KS USA, and the TU Chemnitz-Zwickau, Germany, where he received the Ph.D. degree in February 1997. In 2001, he received the Habilitation in Mathematics from the University of Bremen, Germany, where he held an Assistant Professor position from 1997 to 2001. After spending a term as a Visiting Associate Professor at TU Hamburg-Harburg, Germany, he was a Lecturer in mathematics at TU Berlin 2001–2003. Since 2003, he has been a Professor for "Mathematics in Industry and Technology" at TU Chemnitz. In 2010, he was appointed as one of the four directors of the Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany. Since 2011, he has also been an Honorary Professor at the Otto-von-Guericke University of Magdeburg.
Benner's research interests are in the areas of scientific computing, numerical mathematics, systems theory, and optimal control. A particular emphasis has been on applying methods from numerical linear algebra and matrix theory in systems and control theory. Recent research focuses on numerical methods for optimal control of systems modeled by evolution equations (PDEs, DAEs, SPDEs), model order reduction, preconditioning in optimal control and UQ problems, and Krylov subspace methods for structured or quadratic eigenproblems. Research in all these areas is accompanied by the development of algorithms and mathematical software suitable for modern and high-performance computer architectures.
Tuesday, Nov 27, 2018 from 3:30PM to 5PM | POB 6.304
Hosted by Karen Willcox
Sponsor: ICES Seminar
This tutorial will focus on compressed sensing approaches to sparse polynomial approximation of complex functions in high dimensions. Of particular interest to the UQ community is the parameterized PDE setting, where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we will present and analyze several procedures for exactly reconstructing a set of (jointly) sparse vectors, from incomplete measurements. These include novel weighted $\ell_1$ minimization, improved iterative hard thresholding, mixed convex relaxations, as well as nonconvex penalties. Theoretical recovery guarantees will also be presented based on improved bounds for the restricted isometry property, as well as unified null space properties that encompass all currently proposed nonconvex minimizations. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the described compressed sensing methods.
Tuesday, Feb 5, 2019 from 3:30PM to 5PM | POB 6.304
Important Update: NOTE: Title and Abstract ChangeHosted by Tan Bui-Thanh
Sponsor: ICES Seminar
In the framework of steady-state diffusion problems, we show how the ideas of static condensation and hybridization lead to the introduction of the hybridizable discontinuous Galerkin methods.
Bio
Professor Cockburn received his Ph.D from University of Chicago in 1986 under the direction of Jim Douglas, Jr. He has spent all his academic career at University of Minnesota where he is now Distinguished McKnight University Professor. His research interests include the development of Discontinuous Galerkin methods for nonlinear conservation laws, second-order elliptic problems, electro-magnetism, wave propagation and elasticity.