# 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
• (Helsingin yliopisto, 2022)
Ultrasonic guided lamb waves can be used to monitor structural conditions of pipes and other equipment in industry. An example is to detect accumulated precipitation on the surface of pipes in a non-destructive and non-invasive way. The propagation of Lamb waves in a pipe is influenced by the fouling on its surface, which makes the fouling detection possible. In addition, multiple helical propagation paths around pipe structure provides rich information that allows the spatial localization of the fouled area. Gaussian Processes (GP) are widely used tools for estimating unknown functions. In this thesis, we propose machine learning models for fouling detection and spatial localization of potential fouled pipes based on GPs. The research aims to develop a systematic machine learning approach for ultrasonic detection, interpret fouling observations from wave signals, as well as reconstruct fouling distribution maps from the observations. The lamb wave signals are generated in physics experiments. We developed a Gaussian Process Regression model as a detector, to determine whether each propagation path is going across the fouling or not, based on comparison with clean pipe. This binary classification can be regarded as one case of the different fouling observations. Latent variable Gaussian Process models are deployed to model the observations over the unknown fouling map. Then Hamiltonian Monte Carlo sampling is utilized to perform full Bayesian inference for the GP hyper-parameters. Thus, the fouling map can be reconstructed based on the estimated parameters. We investigate different latent variable GP models for different fouling observation cases. In this thesis, we present the first unsupervised machine learning methods for fouling detection and localization on the surface of pipe based on guided lamb waves. In these thesis we evaluate the performance of our methods with a collection of synthetic data. We also study the effect of noise on the localization accuracy.
• (Helsingin yliopisto, 2022)
Ultrasonic guided lamb waves can be used to monitor structural conditions of pipes and other equipment in industry. An example is to detect accumulated precipitation on the surface of pipes in a non-destructive and non-invasive way. The propagation of Lamb waves in a pipe is influenced by the fouling on its surface, which makes the fouling detection possible. In addition, multiple helical propagation paths around pipe structure provides rich information that allows the spatial localization of the fouled area. Gaussian Processes (GP) are widely used tools for estimating unknown functions. In this thesis, we propose machine learning models for fouling detection and spatial localization of potential fouled pipes based on GPs. The research aims to develop a systematic machine learning approach for ultrasonic detection, interpret fouling observations from wave signals, as well as reconstruct fouling distribution maps from the observations. The lamb wave signals are generated in physics experiments. We developed a Gaussian Process Regression model as a detector, to determine whether each propagation path is going across the fouling or not, based on comparison with clean pipe. This binary classification can be regarded as one case of the different fouling observations. Latent variable Gaussian Process models are deployed to model the observations over the unknown fouling map. Then Hamiltonian Monte Carlo sampling is utilized to perform full Bayesian inference for the GP hyper-parameters. Thus, the fouling map can be reconstructed based on the estimated parameters. We investigate different latent variable GP models for different fouling observation cases. In this thesis, we present the first unsupervised machine learning methods for fouling detection and localization on the surface of pipe based on guided lamb waves. In these thesis we evaluate the performance of our methods with a collection of synthetic data. We also study the effect of noise on the localization accuracy.
• (2012)
• (2020)
Given a connected compact Riemannian manifold (M, g) without boundary, dim M >= 2, we consider a space-time fractional diffusion equation with an interior source that is supported on an open subset Vof the manifold. The time-fractional part of the equation is given by the Caputo derivative of order alpha is an element of(0, 1], and the space fractional part by (-Delta(g))(beta), where beta is an element of(0, 1] and Delta(g) is the Laplace-Beltrami operator on the manifold. The case alpha= beta= 1, which corresponds to the standard heat equation on the manifold, is an important special case. We construct a specific source such that measuring the evolution of the corresponding solution on Vdetermines the manifold up to a Riemannian isometry. (c) 2020 Published by Elsevier Inc.
• (2021)
Online optimisation revolves around new data being introduced into a problem while it is still being solved; think of deep learning as more training samples become available. We adapt the idea to dynamic inverse problems such as video processing with optical flow. We introduce a corresponding predictive online primal-dual proximal splitting method. The video frames now exactly correspond to the algorithm iterations. A user-prescribed predictor describes the evolution of the primal variable. To prove convergence we need a predictor for the dual variable based on (proximal) gradient flow. This affects the model that the method asymptotically minimises. We show that for inverse problems the effect is, essentially, to construct a new dynamic regulariser based on infimal convolution of the static regularisers with the temporal coupling. We finish by demonstrating excellent real-time performance of our method in computational image stabilisation and convergence in terms of regularisation theory.
• (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.