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  Variational Methods using Transport Metrics and Applications
Marzena Franek, 2011 (Reviewer: Daniel Matthes, TU München)

The goal of this thesis is to introduce a new variational approach based on the optimal transport theory. In particular, we want to investigate the solution of an optimisation problem consisting of a data term, which is here the quadratic Wasserstein distance between probability measures and a regularisation term. This novel formulation leads to a method, which can handle unied discrete measures as well as continuous probability measures. Moreover, with a special kind of regularisation, namely the TVregularisation, discontinuities in the data can be preserved. Further, if we consider the Wasserstein distance in the data term we obtain mass conservation. This is an improvement to existing models, since the mass conservation must be usually implemented as an additional constraint.
The second but equivalent method we derive in this thesis is in the spirit of the uid dynamic formulation of Benamou and Brenier. By introducing an artificial time variable we obtain a regularised time dependent optimal transport problem. In addition, there is a time interpolant between masses given.
Furthermore, the connection to the gradient floow theory is developed in this thesis and several evolution equations are identified as Wasserstein gradient flow equations. This connection yields a new understanding of the partial differential equations and to an improved study of existence, uniqueness and the asymptotic behaviour of solutions.
Especially, we study an energy functional which consists of a nonlocal interaction potential and an internal energy. The associated gradient ow equation is an aggregation equation with diffusion term which has applications in chemotaxis or swarming. We provide a detailed analysis for this aggregation equation. In particular, we give existence and uniqueness results of nontrivial stationary solutions. The existence versus nonexistence of such solutions is ruled by a threshold phenomenon, namely nontrivial stationary solutions exist if and only if the diusivity constant is strictly smaller than the total mass of the interaction kernel. Furthermore, we prove that nontrivial stationary solutions in one space dimension with xed mass and center of mass are unique. We also provide numerical results.
The central core of this thesis is the discussion of the variational problem with several standard regularisation functionals, e.g. the logarithmic entropy, the L2-regularisation, the Dirichlet-regularisation, the Fisher information and the TV-regularisation. Besides existence and uniqueness results for each regularisation energy we present numerical results which illustrate the different impact on the data. We calculate the self-similar solutions for the different regularisation functionals to give an idea of the structure of the solutions. We apply the numerical algorithms to synthetic data as well as on real life data.
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  Mathematical Models for Particle Transport: Crowded Motion
Bärbel Schlake, 2011 (Reviewer: Ansgar Jüngel, TU Wien)

This thesis is concerned with problems arising in the mathematical modelling of crowded particles. A stochastic, self-consistent model with volume-exclusion is developed, which asymptotically leads to a nonlinear system of drift-diffusion equations.
We present an entropy for the model, compute equilibria, and show that the system has a formal gradient flow structure. We analyse the time-dependent system for two species and compute the entropy dissipation and equilibria. Linearisation around equilibrium is examined. We investigate the system with respect to well-posedness close to equilibrium, global existence and asymptotic behaviour at different diffusion scales.
The stationary model coupled with the Poisson equation to compute the potential (drift) is discussed. We use the model to compute densities as well as electrical potentials for an ion channel, the L-type calcium selective channel. Existence and uniqueness of the stationary problem is analysed. We simulate this channel and compare the numerical results with the commonly used Poisson-Nernst-Planck model, which does not include volume-exclusion. As a result, we observe that the current in the model with volume-exclusion lies significantly below the current without volumeexclusion. This is due to the fact that ions inside the channel are crowded and cannot move uninterrupted.
Next, we extend the model for particles with different radii. We recover basically all the above mentioned results for the model. We also simulate the L-type calcium channel with the model including different radii. As ion channels do not only select over charge, but also over radii, interesting biological phenomena such as the Anomalous Mole Fraction Effect can be observed.
Moreover, we examine the Brazil Nut Effect, which is an effect generated at the mixing of particles with different radii and masses. Mixed particles segregate on shaking. At equal masses, larger particles go to the top, while smaller particles sink to the bottom. We investigate in which cases this effect arises in our model and carry out some simulations on this effect, as well as on the Reverse Brazil Nut Effect: In case that one species is larger, but also more heavy, this species sinks to the bottom.
Finally, we apply the problem to human motion, more precisely to a typical pedestrian setup with two directions. We investigate linear stability. Arising instabilities for pedestrians with opposite walking directions mean the forming of lanes of pedestrians with the same desired direction. The model is able to reproduce lane formation, and we are able to predict the number of lanes dependent on the width of the domain and on the density.
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  (Nonlocal) Total Variation in Medical Imaging
Alex Sawatzky, 2011 (Reviewer: Gabriele Steidl, TU Kaiserslautern)

This thesis deals with the reconstruction of images where the measured data are corrupted by Poisson noise and a specific signal-dependent speckle noise, which occurs e.g in medical ultrasound images. Since both noise types fundamentally differ from the frequently studied additive Gaussian noise in mathematical image processing, adapted variational models are required to handle these types of noise accurately.
The first part of this thesis introduces variational regularization frameworks for inverse problems with data corrupted by Poisson and ultrasound speckle noise. Due to the strong nonlinearity of both data delity terms, a forward-backward splitting approach is used to provide efficient numerical schemes allowing the use of arbitrary convex regularization energies, in particular singular ones. Moreover, analytical results such as the well-posedness of the variational problems as well as the positivity preservation and convergence of the proposed iteration methods are proved. Finally, an iterative extension of both frameworks is proposed in order to refine the systematic errors of variational regularization techniques, using inverse scale space methods and Bregman distance iterations.
The second part of this thesis considers the use of the (nonlocal) total variation functional as regularization energy in both previously developed frameworks. In particular, a modified version of the projected gradient descent algorithm of Chambolle and an augmented Lagrangian method are presented to solve the weighted (nonlocal) ROF model arising in both previously developed frameworks. In the case of the total variation regularization strategy, analytical results obtained previous in the general context of a convex regularization functional are carried over to the TV seminorm. In the case of the nonlocal regularization approach, a continuous framework of nonlocal derivative operators on directed graphs is introduced. This framework generalizes the nonlocal operators on undirected graphs in continuous and discrete setting and is consistent to the discrete local derivative operators.
Finally, the performance of the proposed algorithms is illustrated by 2D and 3D synthetic and real data reconstructions. To validate the method proposed for inverse problems with data corupted by Poisson noise, simulated PET data (2D) and real cardiac H2 15O (2D) and 18F-FDG (3D) PET measurements with low count rates are used. Additionally, a denoising and reconstruction comparison between TV and nonlocal TV regularization is presented using 2D synthetic Poisson data. In the case of denoising problems in medical US imaging, results on real patient data (2D) are illustrated.
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  Singular Regularization of Inverse Problems
Martin Benning, 2011 (Reviewer: Elena Resmerita, JKU Linz)

This thesis comments on the use of Bregman distances in the context of singular regularization schemes for inverse problems. According to previous works the use of Bregman distances in combination with variational frameworks, based on singular regularization energies, leads to improved approximations of inverse problems solutions. The Bregman distance has become a powerful tool for the analysis of these frameworks, and has brought iterative algorithms to life that enhance the quality of solutions of existing frameworks significantly. However, most works have yet considered Bregman distances in the context of variational frameworks with quadratic fidelity only.
One of the goals of this thesis is to extend analytical results to more general, nonlinear fidelity terms arising from applications as e.g. medical imaging. Moreover, the concept of Eigenfunctions of linear operators is transferred to nonlinear operators arising from the optimality conditions of the variational frameworks.
From a computational perspective, a novel compressed sensing algorithm based on an inverse scale space formulation is introduced. Furthermore, important concepts related to Bregman distances are carried over to non-quadratic frameworks arising from the applications of dynamic Positron Emission Tomography and Bioluminescence Tomography.  [pdf-file]

  4D Imaging in Tomography and Optical Nanoscopy
Christoph Brune, 2010 (Reviewer: Stanley Osher, UCLA)

This thesis contributes to the field of mathematical image processing and inverse problems. An inverse problem is a task, where the values of some model parameters (in our case images) must be computed from observed data. Such problems arise in a wide variety of applications in sciences and engineering, such as medical imaging, biophysics, geophysics, remote sensing, ocean acoustic tomography, nondestructive testing or astronomy. Here, we mainly consider reconstruction problems with Poisson noise in tomography and optical nanoscopy. In optical nanoscopy the task is to reconstruct images from blurred and noisy measurements, whereas e.g. in positron emission tomography the task is to visualize biochemical and physiological processes of a patient by measurements from outside the body. In the literature there are several models and algorithms for 3D static image reconstruction. However, standard methods do not incorporate time-dependent information or dynamics, e.g. heart beat or breathing in medical imaging or cell motion in microscopy. This can lead to de?cient accuracy particularly at object boundaries, e.g. at cardiac walls in medical imaging
This dissertation contains a treatise on models, analysis and e?cient algorithms to solve 3D static and 4D time-dependent inverse problems. In the first part of this thesis the main goal is to present an accurate, robust and fast 3D static reconstruction framework based on total variation for inverse problems with non-standard noise models. We will provide a detailed analysis including existence, uniqueness and convergence proofs.
In the second part our main goal is to study different transport and motion models and to combine them with the ideas of the first part, to build a joint 4D model for simultaneous reconstruction, total variation regularization and optimal transport. The fundamental concepts are based on non-standard noise models, sparsity regularization techniques, Bregman distances, splitting techniques and motion estimation. In the course of this thesis, topics of various areas in applied mathematics and computer science are brought together, e.g. static and time dependent inverse problems, regularization of ill-posed problems, applied functional analysis, error estimation, convex optimization theory, numerical algorithms, computational science (parallelization, GPU programming), continuum mechanics, computer vision, motion estimation or optimal transport. The performance of our main concepts is illustrated by experimental data in tomography and optical nanoscopy  [pdf-file]

  Mathematical Modelling and Simulation of Ion Channels
Kattrin Arning, 2009 (Reviewer: Christoph Romanin, JKU Linz)

This work is concerned with the mathematical modelling and simulation of ion channels. Ion channels are of major interest and form an area of intensive research in the fields of biophysics and medicine, since they control many vital physiological functions. As certain aspects with respect to ion channel structure and function are hard or impossible to address in experimental investigations, mathematical models form a useful completion and alternative to these studies.
This thesis will mainly deal with two aspects of the channel behaviour, namely ion conduction and gating.
The first part is dedicated to the description of ion transport across single open channels, focusing on a macroscopic model composed of a system of coupled nonlinear partial differential equations (PDEs), known as the Poisson-Nernst-Planck (PNP) system in biological context. A one-dimensional approximation of this PDE system is derived by introducing additional potentials to account for the channel protein geometry, and furthermore a computationally efficient way to include size-exclusion effects, which become important in narrow geometries, is developed.
Since for most ion channels the structure of the selectivity filter (the region where the specificity of the ion channel is determined) cannot be resolved in experiments yet, the PNP model is subsequently used to address questions from the field of inverse problems. It is investigated if electrophysiological measurements like current-voltage curves can be used to characterise the underlying channel structure. A special focus is put on the employment of surrogate models in the identification procedure.
The second part of the thesis deals with the opening and closing of ion channels, a process known as gating. Different modelling approaches that can be used to simulate the behaviour of voltage-gated ion channels are presented, and a model of Fokker-Planck type is derived to describe the gating currents and open probabilities on a macroscopic, i.e. whole cell, level. This model is then used to analyse certain characteristic features arising in gating current data, like the existence of a rising phase under certain conditions. As above in the case of ion conduction, the derived gating model is subsequently employed to address inverse problems from the field of parameter identification. Macroscopic current data are used to investigate what can be inferred about the underlying physical system.  [pdf-file]

  Forward and Inverse Solvers for Electrodiffusion Problems
Marie-Therese Wolfram, 2008 (Reviewer: Christian Schmeiser, University Vienna)

This work is concerned with forward and inverse solvers for electro-diffusion systems. Electro-diffusion equations serve as a mathematical model for different phenomena in physics and biology. Classical examples include the transport of charged particles in semiconductors or biological channels.
This thesis contributes to three different, yet loosely connected areas. The first part deals with the adaptation of well-known semiconductor discretization techniques, namely exponential fitting schemes and stabilized hybrid discontinuous Galerkin methods, to different biological applications. Numerical discretization techniques for semiconductor devices are a highly developed field due to the high demand in semiconductor industry. It was not until the last few years, that researchers got more and more interested in biological applications. Therefore numerical methods for these applications have hardly been researched up to now. In particular we are interested in transport of ions in biological and synthetic channels as well as the chemotactical movement of cells.
Electro-diffusion equations can also be formulated in the context of optimal transport problems. In this field we strive to find an optimal transportation map that carries one density to another while minimizing the transportation cost and energy. This formulation provides a basis for a new numerical discretization method, which can be applied to various optimal transportation problems like the porous medium equations.
The optimal design of semiconductor poses another challenge. The performance of a semiconductor is mainly determined by its doping profile. Therefore industry is interested in efficient numerical methods to obtain a desired behavior, like maximizing the current flow over a contact by changing the doping profile. Here computational cost is the limiting factor in optimization schemes. By interpreting the potential V as an optimal design variable we obtain a decoupled scheme of electro-diffusion type equations. The resulting system can be solved more efficiently than conventional schemes, enabling the application of this technique to more sophisticated semiconductor devices, like a MOSFET.  [pdf-file]

  Shape Variations, Level Set and Phase-field Methods for Perimeter Regularized Geometric Inverse Problems
Benjamin Hackl, 2006 (Reviewer: Karl Kunisch, KFU Graz)

Geometric inverse problems are in general ill-posed problems whose variable is a shape. A problem is ill-posed if the solution to the problem depend in a stable way on the data, i.e. small perturbations of the data may result into arbitrarily large deviations in the solutions. In order to achieve meaningful solutions, one has to regularize the problem. Usually one considers the problem in a least squares form and e.g. adds a regularization term to the least squares functional (Tikhonov regularization).
A geometric equivalent to the Tikhonov regularization is the perimeter regularization, where one adds a multiple of the perimeter (arclength or surface area of the boundary of the shape) of the shape to the least squares functional. Therefore heavily oscillating shapes or even micro-structures are avoided. For a cavity detection problem in linear elasticity we prove, mathematically rigorous, that perimeter regularization is indeed a convergent regularization method.
Motivated by the asymptotic regularization for inverse problems in function spaces, we suggest the terminated velocity method as an alternative regularization method. Numerous numerical experiments with cavity detection in linear elasticity and thermo-elasticity illustrate the regularizing behavior of this method.
Next we study the level set method for geometric inverse problems, which is supposed to be able to handle topology changes easily. In the level set method the shape, more precisely the boundary of the shape, is represented by the zero level set of a function. The shape evolves in a velocity field, where the velocity field is chosen such that an appropriately chosen objective function is non-increasing. Usually the velocity field is chosen due to a shape sensitivity analysis of the objective function (shape derivatives).
In some cases the level set method using shape derivatives to determine the velocity gets stuck in local minima and does not perform the desired topology change. Therefore we develop modifications of the level set method to overcome such local minima and force topology changes.
One modification incorporates the topological derivative into the level set method. For the topological derivative one considers the variation of the objective functional with respect to an additional small cavity. Hence the topological derivative indicates whether it is desirable to perform a topology change or not. Unfortunately the perimeter is not topologically differentiable and therefore this modification of the level set method applies just to the terminated velocity method.
Another modification we suggest is based on a topological expansion of the objective function with respect to the volume and the perimeter of the topology change. The expansion is similar to a Taylor expansion and allows to provide estimates of the change of the objective functional to first and second order. Due to the fact that the topological expansion is also an expansion in the perimeter, it is possible to treat perimeter regularized problems. Following the construction ideas of steepest descent- and Newton- type methods, it is possible to enforce reliable topology changes by solving a sub-minimization problem.
As a final solution method for perimeter regularized problems, we discuss the phase-field method. In the phase-field method one relaxes the perimeter and represents the shape by a function in a function space. This relaxation allows for classical function space optimization. For non-linear source reconstruction problem we compare the classical and newly developed algorithms.  [pdf-file]

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