Inverse problem for the Helmholtz equation with Cauchy data: reconstruction with conditional well-posedness driven iterative regularization

¹ Dipartimento di Matematica e Geoscienze, Università di Trieste, Trieste Italy
² Department of Computational and Applied Mathematics and Department of Earth Science, Rice University, Houston, TX 77005, USA
³ Project-Team Magique-3D, Inria Bordeaux Sud-Ouest Research Center, Laboratoire de Mathématiques et de leurs Applications, Université de Pau et des Pays de l’Adour, UMR CNRS 5142 Pau, France
⁴ Department of Mathematics and Statistics, Health Research Institute (HRI), University of Limerick, Limerick, Ireland

^* Corresponding author: florian.faucher@inria.fr

Received: 11 May 2018
Accepted: 30 January 2019

Abstract

In this paper, we study the performance of Full Waveform Inversion (FWI) from time-harmonic Cauchy data via conditional well-posedness driven iterative regularization. The Cauchy data can be obtained with dual sensors measuring the pressure and the normal velocity. We define a novel misfit functional which, adapted to the Cauchy data, allows the independent location of experimental and computational sources. The conditional well-posedness is obtained for a hierarchy of subspaces in which the inverse problem with partial data is Lipschitz stable. Here, these subspaces yield piecewise linear representations of the wave speed on given domain partitions. Domain partitions can be adaptively obtained through segmentation of the gradient. The domain partitions can be taken as a coarsening of an unstructured tetrahedral mesh associated with a finite element discretization of the Helmholtz equation. We illustrate the effectiveness of the iterative regularization through computational experiments with data in dimension three. In comparison with earlier work, the Cauchy data do not suffer from eigenfrequencies in the configurations.