A PDE-constrained optimization approach to the discontinuous Petrov–Galerkin method with a trust region inexact Newton-CG solver

Abstract

We propose a partial-differential-equation-constrained (PDE-constrained) approach to the discontinuous Petrov–Galerkin (DPG) of Demkowicz and Gopalakrishnan (2010, 2011). This view opens the door to invite all the state-of-the-art PDE-constrained techniques to be part of the DPG framework, and hence enabling one to solve large-scale and difficult (nonlinear) problems efficiently. That is, our proposed method preserves all the attractive features of the DPG framework while enjoying all advances from the PDE-constrained optimization community. The proposed approach can be considered as a Rayleigh–Ritz method for the DPG minimum residual statement. It is equipped with a trust region inexact Newton conjugate gradient (TRINCG) method which prevents over-solving when optimization iterates are far away from the optimal solution but converges quadratically otherwise for sufficiently smooth residual. The PDE-constrained approach together with the TRINCG solver is therefore a robust iterative method for the DPG framework. It is robust in the sense that the approximate solution is improved after each optimization iterate until the algorithm converges to a local minimum of the residual. As numerically shown, it is also scalable in the sense that the number of Newton iterations remains constant as either the mesh or the solution order is refined. Moreover, the proposed approach solves neither the optimal test functions nor the original PDE directly, though the discretization of the latter is still necessary. The gradient and Hessian-vector product, which are required by the TRINCG solver, are explicitly derived for both abstract variational problems and viscous Burger equation. This reveals a fact that the complexity of each Newton iteration scales like O(N×nCG), where N and nCG are the number of unknowns and the number of conjugate iterations, respectively. Optimal h- and p- convergences of the proposed approach are demonstrated for Laplace and Helmholtz equations. Numerical results for the viscous Burger equation and the Euler equation of gas dynamics are very promising.