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

Abstract

In this paper, we study the performance of Full Waveform Inversion (FWI) from timeharmonic 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.

Tangential cone condition and Lipschitz stability for the full waveform forward operator in the acoustic regime

“…Note that Lipschitz stability is known for a variety of semi-discrete inverse problems. We refer, e.g., to [1,2,3,4,5].…”

“…We will demonstrate this implication under rather general assumptions. Semi-discrete Lipschitz estimates and conditional well-posedness for various inverse problems have already been derived, e.g., in [1,2,3,4,5] and we add time-domain FWI in the acoustic regime to this list. In fact, if we confine v p and to suitable finite dimensional spaces, the F-derivative of ΨS is one-to-one.…”

Time-harmonic diffuse optical tomography: Hölder stability of the derivatives of the optical properties of a medium at the boundary

“…Our result also provides a first step towards a reconstruction procedure of µ a by boundary measurements based on a Landweber iterative method for non-linear problems studied in [24], where the authors provided an analysis of the convergence of such algorithm in terms of either a Hölder or Lipschitz global stability estimates (see also [2], [18], [26], [27]). We recall the important results of [9] and [19] of global stability estimates for Calderón's inverse conductivity problem in the case of real and complex isotropic conductivities, respectively, and refer to the subsequent papers [4], [5], [6], [14], [15], [16], [17], [20], [28], [31] for an overview of the issue of stability estimates in related inverse problems.…”

Lipschitz stability at the boundary for time-harmonic diffuse optical tomography

“…The initial step of their induction argument relies on Lipschitz (or Hölder) stability estimates at the boundary of the physical parameter that one wants to estimate in terms of the boundary measurements, which is the subject of the current manuscript. Our paper also provides a first step towards a reconstruction procedure of μ a by boundary measurements based on a Landweber iterative method for nonlinear problems studied in [23], where the authors provided an analysis of the convergence of such algorithm in terms of either a Hölder or Lipschitz global stability estimates (see also [24]). We also refer to [25] and [32] for further reconstruction techniques of the optical properties of a medium.…”