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

Abstract

We study the inverse problem in Optical Tomography of determining the optical properties of a medium ⊂ R n , with n ≥ 3, under the so-called diffusion approximation. We consider the time-harmonic case where is probed with an input field that is modulated with a fixed harmonic frequency ω = k/c, where c is the speed of light and k is the wave number. We prove a result of Lipschitz stability of the absorption coefficient μ a at the boundary ∂ in terms of the measurements in the case when the scattering coefficient μ s is assumed to be known and k belongs to certain intervals depending on some a-priori bounds on μ a , μ s .

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

“…In the time-harmonic case, where the medium s probed with an input field which is modulated with a fixed harmonic frequency ω = k c , with k = 0, the forward model (1.1) is a complex elliptic equation. A result of Lipschitz stability of the boundary values of µ a in terms of the D-N map, when µ s is again assumed known, was established in the time-harmonic anisotropic case by some of the authors in [25]. In this paper, we consider the anisotropic time-harmonic case and extend the result in [25], by stably determining the derivatives of the absorption coefficient µ a , D h µ a , for any h ≥ 1, at the boundary of an anisotropic medium Ω ⊂ R n , n ≥ 3, whose scattering coefficient µ s is assumed to be known.…”

“…A result of Lipschitz stability of the boundary values of µ a in terms of the D-N map, when µ s is again assumed known, was established in the time-harmonic anisotropic case by some of the authors in [25]. In this paper, we consider the anisotropic time-harmonic case and extend the result in [25], by stably determining the derivatives of the absorption coefficient µ a , D h µ a , for any h ≥ 1, at the boundary of an anisotropic medium Ω ⊂ R n , n ≥ 3, whose scattering coefficient µ s is assumed to be known. More precisely, we show that, under suitable conditions, D h µ a at the boundary ∂Ω, depends upon the D-N map of (1.1), Λ K,µa , with a modulus of continuity of Hölder type, if k is chosen in certain intervals that depend on a-priori bounds on µ a , µ s and on the ellipticity constant of I − B (Theorem 2.5).…”

“…More precisely, we show that, under suitable conditions, D h µ a at the boundary ∂Ω, depends upon the D-N map of (1.1), Λ K,µa , with a modulus of continuity of Hölder type, if k is chosen in certain intervals that depend on a-priori bounds on µ a , µ s and on the ellipticity constant of I − B (Theorem 2.5). The intervals of variability for k were determined in [25]. The case where µ a is assumed to be known and the scattering coefficient µ s is to be determined, can be treated in a similar manner.…”

“…The case where µ a is assumed to be known and the scattering coefficient µ s is to be determined, can be treated in a similar manner. The choice in this paper, as well as in [25], of focusing on the determination of µ a rather than the one of µ s is driven by the medical application of OT we have in mind. While µ s varies from tissue to tissue, it is the absorption coefficient µ a that carries the more interesting physiological information as it is related to the global concentrations of certain metabolites in their oxygenated and deoxygenated states [21].…”

Inversion of a SIR-based model: A critical analysis about the application to COVID-19 epidemic

“…The other relevant topic for ill-posedness is the lack of stability, i.e., the lack of continuous dependence of the parameters to be identified on the data, so that small errors in the data can lead to large discrepancies in the parameters one is trying to identify via the inverse problem. We do not provide a full review here on these topics, but we mention [24] for a general-purpose description, [25] , [26] for a deep discussion on the instability issue in the context of the so-called inverse conductivity problem and [27] for recent results about optical tomography.…”