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Discretely Exact Derivatives for Hyperbolic PDE-Constrained Optimization Problems Discretized by the Discontinuous Galerkin Method

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Abstract

This paper discusses the computation of derivatives for optimization problems governed by linear hyperbolic systems of partial differential equations (PDEs) that are discretized by the discontinuous Galerkin (dG) method. An efficient and accurate computation of these derivatives is important, for instance, in inverse problems and optimal control problems. This computation is usually based on an adjoint PDE system, and the question addressed in this paper is how the discretization of this adjoint system should relate to the dG discretization of the hyperbolic state equation. Adjoint-based derivatives can either be computed before or after discretization; these two options are often referred to as the optimize-then-discretize and discretize-then-optimize approaches. We discuss the relation between these two options for dG discretizations in space and Runge–Kutta time integration. The influence of different dG formulations and of numerical quadrature is discussed. Discretely exact discretizations for several hyperbolic optimization problems are derived, including the advection equation, Maxwell’s equations and the coupled elastic-acoustic wave equation. We find that the discrete adjoint equation inherits a natural dG discretization from the discretization of the state equation and that the expressions for the discretely exact gradient often have to take into account contributions from element faces. For the coupled elastic-acoustic wave equation, the correctness and accuracy of our derivative expressions are illustrated by comparisons with finite difference gradients. The results show that a straightforward discretization of the continuous gradient differs from the discretely exact gradient, and thus is not consistent with the discretized objective. This inconsistency may cause difficulties in the convergence of gradient based algorithms for solving optimization problems.

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Acknowledgments

We would like to thank Jeremy Kozdon and Gregor Gassner for fruitful discussions and helpful comments, and Carsten Burstedde for his help with the implementation of the numerical example presented in Sect. 4. Support for this work was provided through the U.S. National Science Foundation (NSF) grant CMMI-1028889, the Air Force Office of Scientific Research’s Computational Mathematics program under the grant FA9550-12-1-0484, and through the Mathematical Multifaceted Integrated Capability Centers (MMICCs) effort within the Applied Mathematics activity of the U.S. Department of Energy’s Advanced Scientific Computing Research program, under Award Number DE-SC0009286. The views expressed in this document are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.

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Authors and Affiliations

  1. Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, USA

    Lucas C. Wilcox

  2. Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, USA

    Georg Stadler, Tan Bui-Thanh & Omar Ghattas

  3. Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX, USA

    Tan Bui-Thanh

  4. Departments of Mechanical Engineering and Jackson School of Geosciences, The University of Texas at Austin, Austin, TX, USA

    Omar Ghattas

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  1. Lucas C. Wilcox
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Correspondence to Lucas C. Wilcox.

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Wilcox, L.C., Stadler, G., Bui-Thanh, T. et al. Discretely Exact Derivatives for Hyperbolic PDE-Constrained Optimization Problems Discretized by the Discontinuous Galerkin Method. J Sci Comput 63, 138–162 (2015). https://doi.org/10.1007/s10915-014-9890-5

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