Journal of Scientific Computing

, Volume 63, Issue 1, pp 138–162 | Cite as

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

  • Lucas C. Wilcox
  • Georg Stadler
  • Tan Bui-Thanh
  • Omar Ghattas
Article

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.

Keywords

Discontinuous Galerkin PDE-constrained optimization  Discrete adjoints Elastic wave equation Maxwell’s equations 

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Copyright information

© Springer Science+Business Media New York (outside the USA) 2014

Authors and Affiliations

  • Lucas C. Wilcox
    • 1
  • Georg Stadler
    • 2
  • Tan Bui-Thanh
    • 2
    • 3
  • Omar Ghattas
    • 2
    • 4
  1. 1.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA
  2. 2.Institute for Computational Engineering and SciencesThe University of Texas at AustinAustinUSA
  3. 3.Department of Aerospace Engineering and Engineering MechanicsThe University of Texas at AustinAustinUSA
  4. 4.Departments of Mechanical Engineering and Jackson School of GeosciencesThe University of Texas at AustinAustinUSA

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