Acoustic Full Waveform Inversion via Optimal Control: First- and Second-Order Analysis

This paper proposes and analyzes an optimal control approach for acoustic full waveform inversion (FWI) to reconstruct the speed wave parameter entering the hyperbolic acoustic PDE model in the coefficient of the second-order time derivative of the acoustic pressure. We develop a novel technique accounting for an auxiliary first-order system. In contrast to the original state equation, the underlying control parameter appears in the auxiliary system not only as the coefficient of the time derivative but also as the initial data under the image of the solution operator for a specific elliptic problem. On this basis, we construct an adjoint state explicitly using the corresponding dual semigroup, leading to necessary optimality conditions with a low adjoint regularity such that no Sobolev smoothing effect occurs in the optimal solution. The final part of the paper is devoted to the second-order analysis of the optimal control approach. We study the two- and three-dimensional case where the reconstruction is only considered in an open set strictly contained in the hold-all domain. For this case, under additional regularity and compatibility assumptions on the data, we are able to prove second-order optimality conditions yielding local optimality in an -neighborhood. A numerical test based on a synthetic configuration with nonsmooth data is provided, illustrating the performance of the control approach.