Discretely Exact Derivatives for Hyperbolic PDE-Constrained Optimization Problems Discretized by the Discontinuous Galerkin Method
Discretely Exact Derivatives for Hyperbolic PDE-Constrained Optimization Problems Discretized by the Discontinuous Galerkin Method
复制标题
用间断伽辽金法离散的双曲偏微分方程约束优化问题的离散精确导数
DOI：
10.1007/s10915-014-9890-5
复制
发表时间：
2013
影响因子：
2.5
通讯作者：
O. Ghattas
中科院分区：
文献类型：
--
作者：
L. Wilcox;G. Stadler;T. Bui;O. Ghattas
本文讨论了用间断Galerkin(DG)方法离散的线性双曲型偏微分方程组(PDE)优化问题的导数计算。例如，在反问题和最优控制问题中，有效和准确地计算这些导数是很重要的。这种计算通常基于伴随偏微分方程组，而本文所讨论的问题是这种伴随偏微分方程组的离散化如何与双曲型状态方程的DG离散化相联系。基于伴随的导数既可以在离散化之前计算，也可以在离散化之后计算；这两种选择通常称为先优化后离散化和先离散化后优化的方法。我们讨论了DG空间离散和Runge-Kutta时间积分的两种选择之间的关系。讨论了不同的DG公式和数值积分的影响。推导了几个双曲型优化问题的离散精确离散格式，包括平流方程、麦克斯韦方程和弹性-声波耦合方程。我们发现，离散的伴随方程从状态方程的离散继承了自然的DG离散化，并且离散精确梯度的表达式通常必须考虑单元表面的贡献。对于弹性-声波耦合方程，通过与有限差分梯度的比较，说明了导数表达式的正确性和准确性。结果表明，连续梯度的直接离散化不同于离散的精确梯度，因此与离散化的目标不一致。这种不一致性可能会导致基于梯度的算法在求解优化问题时的收敛困难。
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.