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On uniqueness, stability and reconstruction in anisotropic inverse problems
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Date
2025-12
Abstract
Inverse problems aim to determine unknown properties of the interior or inaccessible part of an object from exterior or far field measurements, and are central to many scientific and engineering applications, including geophysics, medical imaging, and material science. A critical challenge in inverse problems is accurately modelling the physical properties of the underlying media. A major distinction in inverse problem and imaging is that between isotropic and anisotropic materials. A medium is said to be isotropic if the physical property of interest is directionally independent or invariant under spatial rotations, as opposed to anisotropic media, in which the physical property of interest depends on direction. If one is interested in imaging biological tissue within the human body or mapping the subsurface of the Earth, its electrical conductivity distribution can provide extremely valuable information about its state. For the case of anisotropic conductivity, this means the response of the material to an applied electric field depends on the direction of the field, and the conductivity is described by a symmetric, matrix-valued function rather than a scalar. Anisotropy naturally arises in many real world applications, such as biological tissues, layered geological structures, or composite materials where internal structure gives rise to direction-dependent behaviour. Neglecting anisotropy can lead to significant modelling inaccuracies and unreliable reconstructions. Here, we consider two inverse problems by including certain types of anisotropy in their forward model.
In the first part of this thesis, we consider the issues of stability and reconstruction of the electrical anisotropic conductivity of biological tissues in a domain, an open connected set which is simply connected, Ω ⊂ R3 by means of the hybrid inverse problem known as Magneto-Acoustic Tomography with Magnetic Induction (MAT-MI), which consists of two steps. The first step in MAT-MI involves solving a well-posed problem involving a high resolution-low contrast modality from knowledge of boundary measurements, which corresponds to retrieving the so-called internal functional, F(σ). The functional F(σ) corresponds to the acoustic source data produced inside the biological tissue during the first step of MAT-MI. Then the second step consists of the reconstruction of the distribution of electrical conductivity, σ from the acoustic source, F(σ). This second step is the focus of the first part of this thesis. The class of anisotropic conductivities considered here are of type σ(・) = A(γ(・)) and σ(・) = A(・, γ(・)) in Ω, where [λ−1, λ] ∋ t 7→ A(t) and [λ−1, λ] ∋ t 7→ A(・, t), are known one-parameter families of matrix-valued functions which are C1,β , for β ∈ (0, 1) and γ ∈ C1,β(Ω) is a unknown scalar valued function to be determined. Under these assumptions, we are able to stably reconstruct the scalar function γ, and in turn, σ in Ω in terms of an internal functional F(σ). Our results also extend previous results in MAT-MI where σ(・) = γ(・)D(・), with D an a-priori known 3×3 symmetric matrix valued function on Ω. The case σ(・) = A(・, γ(・)) considered here allows, instead, for σ to depend non-linearly on the unknown scalar function γ to be reconstructed. Our stability results are then extended to fully anisotropic conductivities, σ in Ω when σ is a C1,β , symmetric, matrix-valued function satisfying the uniform ellipticity condition where no structure is assumed on σ.
In the second part of the thesis, we consider the inverse boundary value problem of the simultaneous determination of the coefficients σ and q in the Schrodinger-type equation Lu = −div(σ∇u) + qu = 0 from knowledge of the so-called Neumann-to-Dirichlet (N-D) map, given locally on a non-empty curved portion Σ of the boundary ∂Ω of a domain Ω ⊂ Rn, with n ≥ 3. We assume that σ and q are a-priori known to be a piecewise constant matrixvalued and scalar function, respectively, on a given partition of Ω, {Dj}Nj =1, with curved interfaces. We prove that σ and q can be uniquely determined in Ω from the knowledge of the local N-D map, NΣ σ,q. We also quantify this result by establishing a H¨older-type stability estimate of σ and q in terms of NΣ σ,q at the boundary. The proof of these uniqueness and stability results rely on an accurate inspection of NΣ σ,q, which is also an integral operator whose kernel K contains information on the tangential part of the metric g related to σ via g = (detσ) 1 n−2 σ−1, when n ≥ 3 where we consider the Riemannian manifold {Ω, g}. By exploiting the non-flatness of Σ ⊂ ∂Ω, the full metric g can be recovered uniquely and stably from three sufficiently distinct tangential spaces to ∂Ω. This allows for the unique and stable determination of σ on Σ. Once σ is determined on Σ, we combine the method of singular solutions together with the asymptotic behavior of the Neumann Kernel of L in Ω near its pole, chosen on Σ, allowing us to uniquely and stably recover q on Σ from knowledge of σ and NΣ σ,q. We can then uniquely determine σ on a subdomain of the partition, say D1, which boundary intersects Σ on a C1,α curved portion. Once σ is identified in D1, we combine the method of singular solutions with the asymptotic behavior of the Neumann kernel of L in Ω near its pole, chosen on Σ, allowing for the unique determination of q in D1 from knowledge of σ and NΣ σ,q. The proof is completed by an induction argument and the unique continuation property, allowing us to uniquely determine, domain by domain, σ and q in Ω. We also show that when the interfaces of the partition of Ω are C1,α boundaries of an unknown nested family of subdomains invading Ω, the knowledge of NΣ σ,q allows us to uniquely determine σ, q in Ω, and the interfaces of the unknown partition of Ω. We conclude with an extension of this latter result to the case when Ω is occupied only in some local subregion ˜Ω by stratified material. We show that, even this case, where no stratification assumption is made outside ˜Ω, we are able to uniquely identify both σ and q in ˜Ω by a local N-D map, NΣ σ,q.
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University of Limerick
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Attribution-NonCommercial-ShareAlike 4.0 International