Adjoint-state method for Hybridizable Discontinuous Galerkin discretization, application to the inverse acoustic wave problem

Abstract

In this paper, we perform non-linear minimization using the Hybridizable Discontinuous Galerkin method (HDG) for the discretization of the forward problem, and implement the adjoint-state method for the efficient computation of the functional derivatives. Compared to continuous and discontinuous Galerkin discretizations, HDG reduces the computational cost by using the numerical traces for the global linear system, hence removing the degrees of freedom that are inside the cells. It is particularly attractive for large-scale time-harmonic quantitative inverse problems which make repeated use of the forward discretization as they rely on an iterative minimization procedure. HDG is based upon two levels of linear problems: a global system to find the solution on the boundaries of the cells, followed by local systems to construct the solution inside. This technicality requires a careful derivation of the adjoint-state method, that we address in this paper. We work with the acoustic wave equations in the frequency domain and illustrate with a three-dimensional experiment using partial reflection-data, where we further employ the features of DG-like methods to efficiently handle the topography with p-adaptivity.