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Optimization-based computational physics and high-order methods: from optimized analysis to design and data assimilation

  • April 02, 2018
  • 4:00 p.m.
  • 260 New Classroom Building
  • Matthew J. Zahr, Alvarez Postdoctoral Fellow
  • Lawrence Berkeley National Laboratory
  • Faculty Host: Dr. Kevin G. Wang

Optimization problems governed by physical processes are ubiquitous in modern science and engineering. They play a central role in optimal design and control of multiphysics systems, data assimilation and inverse problems. When the governing physics exhibit complex, time-dependent features such as those seen in compressible fluid flows, and the number of optimization parameters exceeds a few, gradient-based optimization methods become rather attractive.

In the first part of this talk, I will present a fully discrete adjoint method for a globally high-order discretization of partial differential equations and their quantities of interest. The adjoint method provides the desired gradients of optimization functionals in a gradient-based PDE-constrained optimization setting. Examples are illustrated with applications to the design of energetically optimal flapping motions, the design of energy harvesting mechanisms, and data assimilation to dramatically enhance the resolution of magnetic resonance images. This deterministic framework is extended to address the challenges posed by stochastic optimization problems characterized by input data that is not known with certainty.

In the second part of this talk, I will demonstrate that the role of optimization in computational physics extends well beyond these traditional design and control problems. I will introduce a new method for the discovery and subsequent high-order accurate resolution of shock waves in compressible flows using PDE-constrained optimization techniques. The key feature of this method is an optimization formulation that aims to align discontinuous features of the solution basis with the discontinuities in the solution. The method is demonstrated on a number of one- and two-dimensional transonic and supersonic flow problems. In all cases, the framework tracks the discontinuity closely with curved mesh elements and provides accurate solutions on extremely coarse meshes.

Bio: Matthew is the Luis W. Alvarez Postdoctoral Fellow in the Department of Mathematics at the Lawrence Berkeley National Laboratory and University of California, Berkeley. In January 2019, he will be an assistant professor in the Department of Aerospace and Mechanical Engineering at the University of Notre Dame. He received his PhD in Computational and Mathematical Engineering from Stanford University in 2016 and his BSc in Civil and Environmental Engineering at the University of California, Berkeley in 2011. His research interests include high-order methods for computational physics, PDE-constrained optimization, model reduction, computational methods for handling shocks and discontinuities, and multiscale methods.