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MICDE Seminar: Pavel Bochev, Center for Computing Research, Sandia National Laboratories

October 5 @ 3:00 pm - 4:00 pm

1084 East Hall

BioPavel Bochev is a Distinguished Member of the Technical Staff at Sandia National Laboratories in Albuquerque where he works in the Center for Computing Research. He joined Sandia in 2000 after six years of teaching and research at the University of Texas at Arlington.

Pavel’s research interests include compatible discretizations for partial differential equations, optimization and control problems, and the development of new, property preserving heterogeneous numerical methods for complex applications relevant to the mission of the US Department of Energy and the National Nuclear Security Administration.

Pavel’s thesis was awarded the SIAM Student paper prize in 1994. In 2012 he was elected a Fellow of the Society for Industrial and Applied Mathematics. Pavel is a recipient of 2014 US Department of Energy’s E. O. Lawrence Medal in the category of “Computer, information and knowledge sciences”. This award honors U.S. scientists and engineers, at mid-career, for exceptional contributions in research and development supporting the Department of Energy and its mission to advance the national, economic and energy security of the United States. In 2017 Pavel was awarded the Thomas J.R. Hughes Medal by the U.S. Association for Computational Mechanics for his contributions to the field of numerical partial differential equations.

Pavel has authored and co-authored over 100 research papers, two books and several book chapters, and has given numerous plenary and invited lectures in the US and abroad. He served two terms as Editor-in-Chief of the SIAM Journal on Numerical Analysis and is currently member of the editorial board of SINUM.

 Optimization-based property preserving methods, or going boldly beyond compatible discretizations

Homological approaches such as finite element exterior calculus, mimetic finite differences and Discrete Exterior Calculus [1, 2] have been a game changer in the quest for structure preserving discretization of PDEs. However, these techniques face difficulties in at least two contexts. First, many practically important models don’t fit neatly in an exterior calculus structure. Examples include multiphysics problems in which different constituent model components may place conflicting requirements on the representation of the variables and heterogeneous problems which combine fundamentally different mathematical models. Preservation of relevant physical properties such as maximum principles, local solution bounds, symmetries, and Geometric Conservation laws provide another context in which homological techniques don’t fare well. Indeed, while they ensure stability and accuracy of the discretization, stable and accurate does not imply property preserving.

In this talk we examine the application of optimization and control ideas to the formulation of feature-preserving heterogeneous numerical methods (HNM). An HNM is a collection of dissimilar numerical “subparts” functioning together as a unified simulation tool. We present a general optimization framework, which couches the assembly of these subparts into an HNM and the preservation of the relevant physical properties into a constrained optimization problem [7] with virtual controls. The misfit between the states of the subparts and suitable target solutions define the optimization objective, while the relevant physical properties provide the optimization constraints. Three complementary case studies illustrate the scope of the optimization approach. In the first study we apply the framework to formulate an optimization-based heterogeneous numerical method, which couples local and nonlocal material models [3, 4]. The second study focusses on optimization based property-preserving methods for passive tracer transport [5], and the third study presents an optimization approach for enforcing a Geometric Conservation Law in Lagrangian methods [6]. The talk is based on joint work with M. D’Elia, S. Moe, K. Peterson, D. Ridzal, M. Shashkov, M.Perego and D. Littlewood. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research (ASCR).

This is a joint seminar with the Applied and Interdisciplinary Mathematics program. Dr. Bochev is being hosted by Prof. Robert Krasny (Mathematics). If you would like to meet with him, please send an email to micde-events@umich.edu

[1] D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications. Acta Numerica, 15:1–155, 2006.
[2] K. Lipnikov, G. Manzini, and M. Shashkov. Mimetic finite difference method. Journal of Computational Physics, 254:1163–1227, 2014.
[3] M. D’Elia, M. Perego, P. Bochev, and D. Littlewood. A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions. Computers & Mathematics with Applications, 71(11):2218 – 2230, 2016.
[4] M. D’Elia, P. Bochev, M. Perego, and D. Littlewood. An optimization-based coupling of local and nonlocal models with applications to peridynamics. In George Z. Voyiadjis, editor, Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Springer Verlag, Berlin, Heidelberg, 2017.
[5] P. Bochev, S. Moe, K. Peterson, and D. Ridzal. A conservative, optimization-based semi-lagrangian spectral element method for passive tracer transport. In B. Schrefler, E. Onate, and M. Papadrakakis, editors, COUPLED PROBLEMS 2015, VI International Conference on Computational Methods for Coupled Problems in Science and Engineering, pages 23–34, Barcelona, Spain, April 2015. International Center for Numerical Methods in Engineering (CIMNE).
[6] M. D’Elia, D. Ridzal, K. Peterson, P. Bochev, and M. Shashkov. Optimization-based mesh correction with volume and convexity constraints. Journal of Computational Physics, 313:455–477, 2016.
[7] P. Bochev, D. Ridzal, and K. Peterson. Optimization-based remap and transport: A divide and conquer strategy for feature-preserving discretizations. Journal of Computational Physics, 257, Part B(0):1113 – 1139, 2014. Physics-compatible numerical methods.


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