Discretely exact derivatives for hyperbolic PDE-constrained optimization problems discretized by the discontinuous Galerkin method, Draft
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Authors
Wilcox, Lucas C.
Stadler, Georg
Bui-Thanh, Tan
Ghattas, Omar
Subjects
Discontinuous Galerkin
PDE-constrained optimization
Discrete adjoints
Elastic wave equation
Maxwell’s equations
PDE-constrained optimization
Discrete adjoints
Elastic wave equation
Maxwell’s equations
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Date of Issue
2013
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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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Applied Mathematics
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This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.