Lechleiter, Armin
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Lechleiter, Armin
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Lechleiter, Armin
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Item type:Publication, Advanced Inverse Modeling of Sediment Thermal Diffusion Processes : Reconstructing Temporal Variant Boundary Conditions for the One-Dimensional Heat Equation(2018-07-05); ; ; Temperatures in marine sediments are driven by the geothermal heat flow from the Earth's crust and the evolution of the bottom water temperature. Mathematically, the temperature field can be modeled with the heat equation, a Robin boundary condition at the sediment-water interface, and a Neumann condition at the lower boundary. Given the thermal properties of the sediment and a model for the bottom water temperature function the forward problem is well-posed. The inverse problem, i.e. reconstructing the bottom water temperature function from measurements of the sediment temperature, on the other hand is ill-posed; the parameterized model is non-linear but low-dimensional. Different Newton-linke methods, as well as a linear fitting approach with Tikhonov minimization, and a Markov Chain Monte Carlo method are shown and their performances for this problem are compared. The algorithms work differently well on this problem, and regularising methods are not necessarily better. The heuristic linear fitting has the best accuracy in reasonable computing time, while the Markov Chain Monte Carlo method has proven convergence for enlarging ensembles.doctoral thesis513 220 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Effective and Efficient Reconstruction Schemes for the Inverse Medium Problem in Scattering(2019-08-19); ; ; This thesis challenges with the development of a computational framework facilitating the solution for the inverse medium problem in time-independent scattering in two- and three-dimensional setting. This includes three main application cases: the simulation of the scattered field for a given transmitter-receiver geometry; the generation of simulated data as well as the handling of real-world data; the reconstruction of the refractive index of a penetrable medium from several measured, scattered fields. We focus on an effective and efficient reconstruction algorithm. Therefore we set up a variational reconstruction scheme. The underlying paradigm is to minimize the discrepancy between the predicted data based on the reconstructed refractive index and the given data while taking into account various structural a priori information via suitable penalty terms, which are designed to promote information expected in real-world environments. Finally, the scheme relies on a primal-dual algorithm. In addition, information about the obstacle's shape and position obtained by the factorization method can be used as a priori information to increase the overall effectiveness of the scheme. An implementation is provided as MATLAB toolbox IPscatt. It is tailored to the needs of practitioners, e.g. a heuristic algorithm for an automatic, data-driven choice of the regularization parameters is available. The effectiveness and efficiency of the proposed approach are demonstrated for simulated as well as real-world data by comparisons with existing software packages.doctoral thesis912 381 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, The Inside-Outside Duality in Inverse Scattering Theory(2016-05-20); ; ; In this thesis we investigate a connection between far field data that arises from time-harmonic scattering problems and interior eigenvalues of corresponding scattering objects. This connection has been used to develop the so-called ``inside-outside duality`` method, which can be used to detect the interior eigenvalues from far field data. In this method a particular focus lies on the behavior of certain eigenvalues of the far field operator, which characterizes the interior eigenvalues. This thesis is separated into two parts. In the first part, we consider acoustic, time-harmonic scattering from impenetrable and penetrable scattering objects. We start by considering acoustic scattering from impenetrable objects and subsequently outline the principle arguments for the derivation of the inside-outside duality. In this context we also show how to work with near field data instead of far field data. In the remainder of the first part, the arguments are then adapted to scattering from penetrable scattering objects that may contain cavities. For all scattering scenarios under investigation, numerical examples for the verification of the theoretical results are provided. In the second part of this thesis we consider elastic and electromagnetic scattering problems. In the case of elastic scattering, we assume an isotropic background medium in which either a rigid or a penetrable scattering object is embedded. For electromagnetic scattering, we consider penetrable objects that may contain cavities. The main challenge in this part lies in adapting the preceding arguments for the different scattering equations. Therefore we focus on theoretical results, which can potentially be used to detect interior eigenvalues from corresponding far field data.doctoral thesis504 480 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Integral Equation Methods for Ocean Acoustics with Depth-Dependent Background Sound Speed(2015-09-30); ; ; Time-harmonic acoustic wave propagation in an ocean with depth-dependent background sound speed can be described by the Helmholtz equation in an infinite, two- or three-dimensional waveguide of finite height. A crucial subproblem for the anayltic and numeric treatment of associated wave propagation problems is a Liouville eigenvalue problem that involves the depth-dependent contrast. For different types of background sound speed profiles, we discuss discretization schemes for the Liouville eigenvalue problem arising in the vertical variable. Due to variational theory in Sobolev spaces, we then show well-posedness of weak solutions to the corresponding scattering problem from a bounded inhomogeneity inside such an ocean: We introduce an exterior Dirichlet-to-Neumann operator for depth-dependent sound speed and prove boundedness, coercivity, and holomorphic dependence of this operator in suitable function spaces adapted to our weak solution theory. Analytic Fredholm theory then implies existence and uniqueness of solution for the scattering problem for all but a countable sequence of frequencies. Introducing the Green's function of the waveguide, we prove equivalence of the source problem for the Helmholtz equation with depth-dependent sound speed profile, Neumann boundary condition on the bottom and Dirichlet boundary condition on the top surface, to the Lippmann-Schwinger integral equation in dimensions two and three. Next, we periodize the Lippmann-Schwinger integral equation in dimensions two and three. The periodized version of the Lippmann-Schwinger integral equation and an interpolation projection onto a space spanned by finitely many eigenfunctions in the vertical variable and trigonometric polynomials in the horizontal variables, two different collocation schemes are derived. A result of Sloan [J.Approx Theory, 39:97-117,1983] on non-polynomial interpolation yields both converge and algebraic convergence rates depending on the smoothness of the inhomogeneity and the source of both schemes. Using one collocation scheme we present numerical results in dimension two. We further present an optimization technique of the vertical transform process, when the height of the obstacle is small compared to the finite height of the ocean, which makes computation in dimension three possible. If several scatters are present in the waveguide, this discretization technique leads to one computational domain containing all scatterers. For a three dimensional waveguide, we reformulate the Lippmann-Schwinger integral equation as a coupled system in an union of several boxes, each containing one part of the scatter.doctoral thesis526 137 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Regularization Methods in Banach Spaces applied to Inverse Medium Scattering Problems(2017-06-26); ; ; This work handles inverse scattering problems for both acoustic and electromagnetic waves. That is to reconstruct the irradiated media from measurements of the scattered felds by regularization methods. As a particular feature, the contrasts of the scattering objects are assumed to be supported within a small region, hence called sparse. To apply sparsity regularization schemes it becomes crucial to model the problems in Banach spaces. Traditionally, they are given in a Hilbert space setting, such that reformulation in an L p-sense becomes a key point. Contrasts are linked to the data by forward operators, basing on beforehand stated solution operators and their continuity properties. Thereby, appropriate regularization techniques providing sparsity are given. As the case of scalar-valued contrast functions is already covered in the literature, mainly inverse scattering problems for anisotropic media are shown. In the case where electromagnetic waves are considered, a distinction is made between magnetic and non-magnetic media, since the latter is less complex. Finally, the case of inverse acoustic backscattering is handled, which is rarely seen in literature.doctoral thesis539 275