Stability of coupled-physics inverse problems with one internal measurement

Abstract

In this paper, we develop a general approach to prove stability for the non linear second step of hybrid inverse problems. We work with general functionals of the form σ|∇u| p , 0 < p ≤ 1, where u is the solution of the elliptic partial differential equation ∇ · σ∇u = 0 on a bounded domain Ω with boundary conditions u| ∂Ω = f . We prove stability of the linearization and Hölder conditional stability for the non-linear problem of recovering σ from the internal measurement.

Analysis of the Linearized Problem of Quantitative Photoacoustic Tomography

“…Single-stage qPAT with multiple illuminations has been studied in [27,38,42,9,60]. Moreover, multiple illuminations have been proposed in several mathematical works to stabilize various inverse problems with internal data (e.g., [12,11,7,10,13,57,64,81]). For each illumination and measurement, we denote by H i := H f i ,q i and W i := W Λ i ,T the corresponding heating and acoustic operator and by F i = W i • H i the resulting forward operator.…”

Stable reconstruction of regular 1-Harmonic maps with a given trace at the boundary

“…The numerical experiments below, see Figure 7, show better numerical stability behavior, equivalent to the loss of one derivative in the solution u. In the case of interior data |J | occurring as the magnitude of the current density field of some sufficiently small perturbation in conductivity (in the sense that σ −σ C 2 ( ) < for some small ,), this numerical behavior can be explained using the local stability result in [13]. However, for perturbation in the interior data |J | that are due to measurement errors, (and not coming from a perturbed conductivity), their arguments do not cover the conditional stability in Proposition 3.2.…”

“…In the recent work in [13], a local stability result of recovering σ from one magnitude |J | without using the 1-Laplacian was shown. However, this approach ignores boundary considerations, and the absence of singular points is implicitly postulated by a restricted neighborhood of the conductivity where stability is shown.…”

A regularized weighted least gradient problem for conductivity imaging

“…Reconstruction algorithms based on the minimization problem were proposed in [24] and [20], and based on level set methods in [23,24,33]. Continuous dependence on σ on a (for a given unperturbed Dirichlet data) can be found in [18], and, for partial data in [19]. For further references on determining the isotropic conductivity based on measurements of current densities see [35,11,13,14,10,15,12], and for reconstructions on anisotropic conductivities from multiple measurements see [9,6,1,2].…”