# Browsing by Subject "Inverse problem"

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• (Helsingin yliopisto, 2020)
This thesis considers certain mathematical formulation of the scattering phenomena. Scattering is a common physical process, where some initial wave is disturbed, producing a scattered wave. If the direct problem is to determine the scattered wave from the knowledge of the object that causes the scattering as well and the initial wave, then the inverse problem would be to determine the object from the knowledge on how different waves scatter from it. In this thesis we consider direct and inverse scattering problems governed by Helmholtz equation $\Delta u + k^2 \eta u = 0$ in $\mathbb{R}^d$ with $d = 3$. The positive function $\eta \in L^\infty(\mathbb{R}^d)$ is considered to be such that $\eta(x) = 1$ outside of some ball. In particular the function $\eta$ models the physical properties of the scattering object and in a certain physical setting, the function $n = +\sqrt{\eta}$ is the index of refraction. The initial motivation for this thesis was the inverse scattering problem and its uniqueness. However, for any inverse problem, one first has to understand the corresponding direct problem. In the end, the balance between treating the direct and inverse problem is left fairly even. This thesis closely follows books by Colton and Kress, and Kirsch. The first chapter is the introduction, in which the overview of the thesis is presented and the working assumptions are made. The second chapter treats the needed preliminaries, such as compact operators, Sobolev spaces, Fredholm alternative, spherical harmonics and spherical Bessel functions. In particular these are needed in various results of chapter three, in which the direct scattering problem is considered. After motivating and defining the direct scattering problem, the main goal is to prove its well-posedness. The uniqueness of the problem is proved by two results, Rellich's lemma and unique continuation principle. The Fredholm alternative is applied to prove existence of the solution on the basis of uniqueness. Equipped with the understanding of the direct scattering problem, the inverse scattering problem can be considered in the fourth chapter. After defining the inverse scattering problem, the uniqueness of the solution is considered. The proof is contrasted to the historically important paper by Calderón considering another kind of inverse problem. The proof consists of three lemmas, from which the second and third are directly used in proving the uniqueness of the inverse problem. The uniqueness of the inverse problem can be considered as the main result of this thesis.
• (2019)
An inverse boundary value problem for the 1+1 dimensional wave equation (partial derivative(2)(t) - c(x)(2)partial derivative(2)(x))u(x,t) = 0, x is an element of R+ is considered. We give a discrete regularization strategy to recover wave speed c(x) when we are given the boundary value of the wave, u(0,t), that is produced by a single pulse-like source. The regularization strategy gives an approximative wave speed (c) over tilde, satisfying a Holder type estimate parallel to (c) over tilde - c parallel to
• (2012)
• (2018)
Given a smooth non-trapping compact manifold with strictly convex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. These data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of the metric, the scattering data measured on the boundary determine the Riemannian manifold up to isometry.
• (2018)
Abstract A standard inverse problem is to determine a source which is supported in an unknown domain D from external boundary measurements. Here we consider the case of a time-independent situation where the source is equal to unity in an unknown subdomain D of a larger given domain Ω and the boundary of D has the star-like shape, i.e. ∂ D = { q ( θ ) ( cos ⁡ θ , sin ⁡ θ ) ⊤ : θ ∈ [ 0 , 2 π ] } . Overposed measurements consist of time traces of the solution or its flux values on a set of discrete points on the boundary ∂Ω. The case of a parabolic equation was considered in [6]. In our situation we extend this to cover the subdiffusion case based on an anomalous diffusion model and leading to a fractional order differential operator. We will show a uniqueness result and examine a reconstruction algorithm. One of the main motives for this work is to examine the dependence of the reconstructions on the parameter α, the exponent of the fractional operator which controls the degree of anomalous behaviour of the process. Some previous inverse problems based on fractional diffusion models have shown considerable differences between classical Brownian diffusion and the anomalous case.