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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.
Monday, Sep 24, 2018 from 2PM to 3PM | POB 6.304
Hosted by Ron Elber
Sponsor: ICES Seminar-Molecular Biophysics Series
Historically, bioinformatics and computational biology are recognized as distinct endeavors. The underlying theories, experiments, software and computing resources differ significantly. We demonstrate that these differences can be overcome by exploiting existing data standards, algorithms, and web based tools to study the structure of DNA, nucleosomes and chromatin in atomic and coarse grain detail from single base pairs to megabase regions of chromatin and beyond.
In this presentation I will define the “genomics dashboard” concept, explain the mathematics and algorithms that drive G-Dash , our prototype genome dashboard, then demonstrate how experimentally determined maps of nucleosome positions for Saccharomyces cerevisiae can be used to assemble a computational karyotype. Comparative all atom molecular dynamics simulations of nucleosomes and coarse-grained models of the MMTV, CHA1, HIS3 and PHO5 promoters highlight important observations. DNA kinking in the nucleosome depends on both sequence and position. Experimentally determined nucleosome positions are insufficient to achieve tight packing of chromatin. Sequence specific material properties of DNA (conformation & flexibility) can affect chromatin bending and looping.
Thursday, Sep 27, 2018 from 3:30PM to 5PM | POB 6.304
Hosted by Tan Bui-Thanh
Sponsor: ICES Seminar
Monge-Ampère equations occur areas of Science and Engineering: Differential Geometry, Fluid Mechanics, Elasticity, Cosmology, Antenna and Windshield Design, Mesh Generation, Optimal Transport, Finance, Image Processing, etc.
Our goal in this presentation is discuss the numerical solution of the canonical Monge-Ampère equation in dimensions 2 and 3. These last two-decades, the numerical solution of Monge-Ampere equation (MA-D) has motivated a relatively large number of publications, however most of the solution methods we know use either high order finite element approximations or wide-stencil finite difference ones. The method we going to present relies on: (i) piecewise affine approximations of the unknown function u and of its second order derivatives, (ii) a time discretization by operator-splitting of an initial value problem associated with an equivalent divergence formulation of problem (MA-D), (iii) a projection operator on the cone of the symmetric positive semi-definite matrices, (iv) a Tychonoff regularization method to approximate the second order derivatives. The resulting methodology is modular and can handle easily domains with curved boundaries. It is also robust in the sense that it can also handle efficiently non- smooth situations, the non-smoothness coming from the data (if, for example, forcing is a the positive multiple of Dirac measure), or from data incompatibility. The results of numerical experiments will be presented, including those associated with the solution of the following (nonlinear) eigenvalue problem for the Monge-Ampère operator.
Friday, Sep 28, 2018 from 10AM to 11AM | POB 6.304
Hosted by Tom O'Leary-Roseberry and Kendrick Shepherd
Sponsor: ICES Seminar-Babuska Forum Series
Analysis of the physical governing equations of a system can reveal variable transformations that transform a general nonlinear model into a model with more structure. In particular, the introduction of auxiliary variables can convert a general nonlinear model to a model with polynomial nonlinearities, a so-called "lifted" system. The lifted model is equivalent to the original model; it uses a change of variables, but introduces no approximations. We present an approach that combines lifting with proper orthogonal decomposition model reduction. The approach uses a data-driven formulation to learn the low-dimensional model from high-fidelity simulation data, but a key aspect of the approach is that the state-space in which the learning is achieved is derived using the problem physics. A key benefit of the approach is that there is no need for additional approximations of the nonlinear terms, in contrast with existing nonlinear model reduction methods that require sparse sampling or hyper-reduction. A second benefit is that the lifted problem structure opens new pathways for rigorous analysis and input-independent model reduction. The method is demonstrated for nonlinear systems of partial differential equations arising in rocket combustion applications.
** Note that Professor Willcox has an open GRA position for research related to this topic.
Bio
Karen E. Willcox is Director of the Institute for Computational Engineering and Sciences (ICES) and a Professor of Aerospace Engineering and Engineering Mechanics at the University of Texas at Austin. She holds the W. A. “Tex” Moncrief, Jr. Chair in Simulation-Based Engineering and Sciences and the Peter O'Donnell, Jr. Centennial Chair in Computing Systems. Prior to joining ICES in 2018, she spent 17 years as a professor at the Massachusetts Institute of Technology, where she served as the founding Co-Director of the MIT Center for Computational Engineering and the Associate Head of the MIT Department of Aeronautics and Astronautics. Prior to joining the MIT faculty, she worked at Boeing Phantom Works with the Blended-Wing-Body aircraft design group. Her research at MIT has produced scalable computational methods for design of next-generation engineered systems, with a particular focus on model reduction as a way to learn principled approximations from data and on multi-fidelity formulations to leverage multiple sources of uncertain information. She is a Fellow of SIAM and Associate Fellow of AIAA.
Tuesday, Oct 23, 2018 from 3:30PM to 5PM | POB 6.304
Hosted by Karen Willcox
Sponsor: ICES Seminar
Numerical simulation of large-scale dynamical systems plays a crucial role in studying a great variety of complex physical phenomena. However, simulations in these large-scale settings present significant computational difficulties. Model reduction aims to resolve this computational burden by constructing simpler (reduced order) models, which are much easier and faster to simulate and yet accurately represent the original system. These simpler reduced order models can then serve as efficient surrogates for the original, replacing them, for example, in optimal control and design. In this talk, we will focus on systems theoretical methods for model reduction, with a special emphasis on interpolatory methods based on rational approximation. After reviewing the concept of interpolation in the setting of dynamical systems, we will discuss how to construct optimal interpolants. If time allows, we will also describe recent extensions to nonlinear dynamics. We will use various examples to illustrate the theoretical discussion.
Bio
Serkan Gugercin is a professor of Mathematics at Virginia Tech. He holds the the A.V. Morris Professorship and is a core faculty member in the Division of Computational Modeling and Data Analytics. In 1992, he received his B.S. degree in Electrical and Electronics Engineering from Middle East Technical University, Ankara, Turkey; and his M.S. and Ph.D. degrees in Electrical Engineering from Rice University, in 1999 and 2003, respectively. His primary research interests are model reduction, data-driven modeling, numerical linear algebra, approximation theory, and systems and control theory.
Dr. Gugercin received the Ralph Budd Award for Research in Engineering from Rice University in 2003 for the best doctoral thesis in the School of Engineering; Teaching Award from Jacobs University Bremen, in 2003; the National Science Foundation Early CAREER Award in Computational and Applied Mathematics in 2007; and the Alexander von Humboldt Research Fellowship in 2016. He is currently serving as an Associate Editor for SIAM Journal on Scientific Computing, and Systems and Control Letters.
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.