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

TL;DR

This paper introduces a novel iterative regularization method for solving the inverse Helmholtz problem using Cauchy data, achieving stable reconstruction of wave speed with adaptive domain partitioning in 3D.

Contribution

It develops a new misfit functional and a hierarchical subspace approach for stable inversion from partial Cauchy data, improving prior methods by avoiding eigenfrequency issues.

Findings

Demonstrates effective 3D reconstructions with Cauchy data.

Achieves Lipschitz stability on hierarchical subspaces.

Shows improved robustness over previous methods.

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.