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Ill-posed inverse problems have wide applications in many fields such as oceanography, signal processing, machine learning, biomedical imaging, remote sensing, geophysics, and others. In this dissertation, we address the problem of solving unstable operator equations with iteratively regularized Newton-type algorithms. Important practical questions such as selection of regularization parameters, construction of generating (filtering) functions based on a priori information available for different models, algorithms for stopping rules and error estimates are investigated with equal attention given to theoretical study and numerical experiments.

The ill-posed problem of aerosol distribution determination from a small number of backscatter and extinction lidar measurements was solved successfully via a hybrid method by a variable dimension of projection with B-Splines. Numerical simulation results with noisy data at different measurement situations show that it is possible to derive a reconstruction of the aerosol distribution only with 4 measurements.

In this paper we consider the regularization of the Cauchy problem for a system of second order differential equations with constant coefficients.

Die Untersuchung mikrogelinster astronomischer Objekte ermöglicht es, Informationen über die Größe und Struktur dieser Objekte zu erhalten. Im ersten Teil dieser Arbeit werden die Spektren von drei gelinsten Quasare, die mit dem Potsdamer Multi Aperture Spectrophotometer (PMAS) erhalten wurden, auf Anzeichen für Mikrolensing untersucht. In den Spektren des Vierfachquasares HE 0435-1223 und des Doppelquasares HE 0047-1756 konnten Hinweise für Mikrolensing gefunden werden, während der Doppelquasar UM 673 (Q 0142--100) keine Anzeichen für Mikrolensing zeigt. Die Invertierung der Lichtkurve eines Mikrolensing-Kausik-Crossing-Ereignisses ermöglicht es, das eindimensionale Helligkeitsprofil der gelinsten Quelle zu rekonstruieren. Dies wird im zweiten Teil dieser Arbeit untersucht. Die mathematische Beschreibung dieser Aufgabe führt zu einer Volterra'schen Integralgleichung der ersten Art, deren Lösung ein schlecht gestelltes Problem ist. Zu ihrer Lösung wird in dieser Arbeit ein lokales Regularisierungsverfahren angewendet, das an die kausale Strukture der Volterra'schen Gleichung besser angepasst ist als die bisher verwendete Tikhonov-Phillips-Regularisierung. Es zeigt sich, dass mit dieser Methode eine bessere Rekonstruktion kleinerer Strukturen in der Quelle möglich ist. Weiterhin wird die Anwendbarkeit der Regularisierungsmethode auf realistische Lichtkurven mit irregulärem Sampling bzw. größeren Lücken in den Datenpunkten untersucht. / The study of microlensed astronomical objects can reveal information about the size and the structure of these objects. In the first part of this thesis we analyze the spectra of three lensed quasars obtained with the Potsdam Multi Aperture Spectrophotometer (PMAS). The spectra of the quadrupole quasar HE 0435--1223 and the double quasar HE 0047--1756 show evidence for microlensing whereas in the double quasar UM 673 (Q 0142--100) no evidence for microlensing could be found. By inverting the lightcurve of a microlensing caustic crossing event the one dimensional luminosity profile of the lensed source can be reconstructed. This is investigated in the second part of this thesis.The mathematical formulation of this problem leads to a Volterra integral equation of the first kind, whose solution is an ill-posed problem. For the solution we use a local regularization method which is better adapted to the causal structure of the Volterra integral equation compared to the so far used Tikhonov-Phillips regularization. Furthermore we show that this method is more robust on reconstructing small structures in the source profile. We also study the influence of irregular sampled data and gaps in the lightcurve on the result of the inversion.

The ill-posed problem of aerosol size distribution determination from a small number of backscatter and extinction measurements was solved successfully with a mollifier method which is advantageous since the ill-posed part is performed on exactly given quantities, the points r where n(r) is evaluated may be freely selected. A new twodimensional model for the troposphere is proposed.

Introducing a new method for measureing ultra-short laser pulses, the research group "Solid State Light Sources" of the Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Berlin, encountered a new type of autoconvolution problem. The so called SD-SPIDER method aims for the reconstruction of the real valued phase of a complex valued laser pulse from noisy measurements. The measurements are also complex valued and additionally influenced by a device-related kernel function. Although the autoconvolution equation has been examined intensively in the context of inverse problems, results for complex valued functions occurring as solutions and right-hand sides of the autoconvolution equation and for nontrivial kernels were missing. The thesis is a first step to bridge this gap. In the first chapter, the physical background is explained and especially the autoconvolution effect is pointed out. From this, the mathematical model is derived, leading to the final autoconvolution equation. Analytical results are given in the second chapter. It follows the numerical treatment of the problem in chapter three. A regularization approach is presented and tested with artificial data. In particular, a new parameter choice rule making use of a specific property of the SD-SPIDER method is proposed and numerically verified. / Bei der Entwicklung einer neuen Methode zur Messung ultra-kurzer Laserpulse stieß die Forschungsgruppe "Festkörper-Lichtquellen" des Max-Born-Institutes für Nichtlineare Optik und Kurzzeitspektroskopie, Berlin, auf ein neuartiges Selbstfaltungsproblem. Die so genannte SD-SPIDER-Methode dient der Rekonstruktion der reellen Phase eines komplexwertigen Laserpulses mit Hilfe fehlerbehafteter Messungen. Die Messwerte sind ebenfalls komplexwertig und zusätzlich beeinflusst von einer durch das Messprinzip erzeugten Kernfunktion. Obwohl Selbstfaltungsgleichungen intensiv im Kontext Inverser Probleme untersucht wurden, fehlen Resultate für komplexwertige Lösungen und rechte Seiten ebenso wie für nichttriviale Kernfunktionen. Die Diplomarbeit stellt einen ersten Schritt dar, diese Lücke zu schließen. Im ersten Kapitel wird der physikalische Hintergrund erläutert und insbesondere der Selbstfaltungseffekt erklärt. Davon ausgehend wird das mathematische Modell aufgestellt. Kapitel zwei befasst sich mit der Analysis der Gleichung. Es folgt die numerische Behandlung des Problems in Kapitel drei. Eine Regularisierungsmethode wird vorgestellt und an künstlichen Daten getestet. Insbesondere wird eine neue Regel zur Wahl des Regularisierungsparameters vorgeschlagen und numerisch bestätigt, welche auf einer speziellen Eigenschaft des SD-SPIDER Verfahrens beruht.

In this thesis, we present the O(n(log n)^2) superfast linear least squares Schur algorithm (ssschur). The algorithm we will describe illustrates a fast way of solving linear equations or linear least squares problems with low displacement rank. This program is based on the O(n^2) Schur algorithm speeded up via FFT. The algorithm solves a ill-conditioned Toeplitz-like system using Tikhonov regularization. The regularized system is Toeplitz-like of displacement rank 4. We also show the effect of choice of the regularization parameter on the quality of the image reconstructed.

This work is about solving an inverse problem for an elliptic equation. An inverse problem is often ill-posed, which means that a small measurement error in data can yield a vigorously perturbed solution. Regularization is a way to make an ill-posed problem well-posed and thus solvable. Two important tools to determine if a problem is well-posed or not are norms and convergence. With help from these concepts, the error of the reg- ularized function can be calculated. The error between this function and the exact function is depending on two error terms. By solving the problem with an elliptic equation, a linear operator is eval- uated. This operator maps a given function to another function, which both can be found in the solution of the problem with an elliptic equation. This opera- tor can be seen as a mapping from the given function’s Fourier-sine coefficients onto the other function’s Fourier-sine coefficients, since these functions are com- pletely determined by their Fourier-sine series. The regularization method in this thesis, uses a chosen number of Fourier-sine coefficients of the function, and the rest are set to zero. This regularization method is first illustrated for a simpler problem with Laplace’s equation, which can be solved analytically and thereby an explicit parameter choice rule can be given. The goal with this work is to show that the considered method is a reg- ularization of a linear operator, that is evaluated when the problem with an elliptic equation is solved. In the tests in Chapter 3 and 4, the ill-posedness of the inverse problem is illustrated and that the method does behave like a regularization is shown. Also in the tests, it can be seen how many Fourier-sine coefficients that should be considered in the regularization in different cases, to make a good approximation. / Det här arbetet handlar om att lösa ett inverst problem för en elliptisk ekvation. Ett inverst problem är ofta illaställt, vilket betyder att ett litet mätfel i data kan ge en kraftigt förändrad lösning. Regularisering är ett tillvägagångssätt för att göra ett illaställt problem välställt och således lösbart. Två viktiga verktyg för att bestämma om ett problem är välställt eller inte är normer och konvergens. Med hjälp av dessa begrepp kan felet av den regulariserade lösningen beräknas. Felet mellan den lösningen och den exakta är beroende av två feltermer. Genom att lösa problemet med den elliptiska ekvationen, så är en linjär operator evaluerad. Denna operator avbildar en given funktion på en annan funktion, vilka båda kan hittas i lösningen till problemet med en elliptisk ekva- tion. Denna operator kan ses som en avbildning från den givna funktions Fouri- ersinuskoefficienter på den andra funktionens Fouriersinuskoefficienter, eftersom dessa funktioner är fullständigt bestämda av sina Fouriersinusserier. Regularise- ringsmetoden i denna rapport använder ett valt antal Fouriersinuskoefficienter av funktionen, och resten sätts till noll. Denna regulariseringsmetod illustreras först för ett enklare problem med Laplaces ekvation, som kan lösas analytiskt och därmed kan en explicit parametervalsregel anges. Målet med detta arbete är att visa att denna metod är en regularisering av den linjära operator som evalueras när problemet med en elliptisk ekvation löses. I testerna i kapitel 3 och 4, illustreras illaställdheten av det inversa problemet och det visas att metoden beter sig som en regularisering. I testerna kan det också ses hur många Fouriersinuskoefficienter som borde betraktas i regulariseringen i olika fall, för att göra en bra approximation.

The aim of this report is to study how machine learning can be used to reconstruct 2 dimensional computed tomography images from limited angle data. This could be used in a variety of applications where either the space or timeavailable for the CT scan limits the acquired data.In this study, three different types of models are considered. The first model uses filtered back projection (FBP) with a single learned filter, while the second uses a combination of multiple FBP:s with learned filters. The last model instead uses an FNO (Fourieer Neural Operator) layer to both inpaint and filter the limited angle data followed by a backprojection layer. The quality of the reconstructions are assessed both visually and statistically, using PSNR and SSIM measures.The results of this study show that while an FBP-based model using one or more trainable filter(s) can achieve better reconstructions than ones using an analytical Ram-Lak filter, their reconstructions still fail for small angle spans. Better results in the limited angle case can be achieved using the FNO-basedmodel.