Convolution and, as a special case, autoconvolution of functions are important in many branches of mathematics and have found lots of applications, such as in physics, statistics, image processing and others. While it is a relatively easy task to determine the autoconvolution of a function (at least from the numerical point of view), the inverse problem, which consists in reconstructing a function from its autoconvolution is an ill-posed problem. Hence there is no possibility to solve such an inverse autoconvolution problem with a simple algebraic operation. Instead the problem has to be regularized, which means that it is replaced by a well-posed problem, which is close to the original problem in a certain sense. The outline of this thesis is as follows: In the first chapter we give an introduction to the type of inverse problems we consider, including some basic definitions and some important examples of regularization methods for these problems. At the end of the introduction we shortly present some general results about the convergence theory of Tikhonov-regularization. The second chapter is concerned with the autoconvolution of square integrable functions defined on the interval [0, 1]. This will lead us to the classical autoconvolution problems, where the term “classical” means that no kernel function is involved in the autoconvolution operator. For the data situation we distinguish two cases, namely data on [0, 1] and data on [0, 2]. We present some well-known properties of the classical autoconvolution operators. Moreover, we investigate nonlinearity conditions, which are required to show applicability of certain regularization approaches or which lead convergence rates for the Tikhonov regularization. For the inverse autoconvolution problem with data on the interval [0, 1] we show that a convergence rate cannot be shown using the standard convergence rate theory. If the data are given on the interval [0, 2], we can show a convergence rate for Tikhonov regularization if the exact solution satisfies a sparsity assumption. After these theoretical investigations we present various approaches to solve inverse autoconvolution problems. Here we focus on a discretized Lavrentiev regularization approach, for which even a convergence rate can be shown. Finally, we present numerical examples for the regularization methods we presented. In the third chapter we describe a physical measurement technique, the so-called SD-Spider, which leads to an inverse problem of autoconvolution type. The SD-Spider method is an approach to measure ultrashort laser pulses (laser pulses with time duration in the range of femtoseconds). Therefor we first present some very basic concepts of nonlinear optics and after that we describe the method in detail. Then we show how this approach, starting from the wave equation, leads to a kernel-based equation of autoconvolution type. The aim of chapter four is to investigate the equation and the corresponding problem, which we derived in chapter three. As a generalization of the classical autoconvolution we define the kernel-based autoconvolution operator and show that many properties of the classical autoconvolution operator can also be shown in this new situation. Moreover, we will consider inverse problems with kernel-based autoconvolution operator, which reflect the data situation of the physical problem. It turns out that these inverse problems may be locally well-posed, if all possible data are taken into account and they are locally ill-posed if one special part of the data is not available. Finally, we introduce reconstruction approaches for solving these inverse problems numerically and test them on real and artificial data.

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.

Convolution and, as a special case, autoconvolution of functions are important in many branches of mathematics and have found lots of applications, such as in physics, statistics, image processing and others. While it is a relatively easy task to determine the autoconvolution of a function (at least from the numerical point of view), the inverse problem, which consists in reconstructing a function from its autoconvolution is an ill-posed problem. Hence there is no possibility to solve such an inverse autoconvolution problem with a simple algebraic operation. Instead the problem has to be regularized, which means that it is replaced by a well-posed problem, which is close to the original problem in a certain sense. The outline of this thesis is as follows: In the first chapter we give an introduction to the type of inverse problems we consider, including some basic definitions and some important examples of regularization methods for these problems. At the end of the introduction we shortly present some general results about the convergence theory of Tikhonov-regularization. The second chapter is concerned with the autoconvolution of square integrable functions defined on the interval [0, 1]. This will lead us to the classical autoconvolution problems, where the term “classical” means that no kernel function is involved in the autoconvolution operator. For the data situation we distinguish two cases, namely data on [0, 1] and data on [0, 2]. We present some well-known properties of the classical autoconvolution operators. Moreover, we investigate nonlinearity conditions, which are required to show applicability of certain regularization approaches or which lead convergence rates for the Tikhonov regularization. For the inverse autoconvolution problem with data on the interval [0, 1] we show that a convergence rate cannot be shown using the standard convergence rate theory. If the data are given on the interval [0, 2], we can show a convergence rate for Tikhonov regularization if the exact solution satisfies a sparsity assumption. After these theoretical investigations we present various approaches to solve inverse autoconvolution problems. Here we focus on a discretized Lavrentiev regularization approach, for which even a convergence rate can be shown. Finally, we present numerical examples for the regularization methods we presented. In the third chapter we describe a physical measurement technique, the so-called SD-Spider, which leads to an inverse problem of autoconvolution type. The SD-Spider method is an approach to measure ultrashort laser pulses (laser pulses with time duration in the range of femtoseconds). Therefor we first present some very basic concepts of nonlinear optics and after that we describe the method in detail. Then we show how this approach, starting from the wave equation, leads to a kernel-based equation of autoconvolution type. The aim of chapter four is to investigate the equation and the corresponding problem, which we derived in chapter three. As a generalization of the classical autoconvolution we define the kernel-based autoconvolution operator and show that many properties of the classical autoconvolution operator can also be shown in this new situation. Moreover, we will consider inverse problems with kernel-based autoconvolution operator, which reflect the data situation of the physical problem. It turns out that these inverse problems may be locally well-posed, if all possible data are taken into account and they are locally ill-posed if one special part of the data is not available. Finally, we introduce reconstruction approaches for solving these inverse problems numerically and test them on real and artificial data.

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.

We revisit in L2-spaces the autoconvolution equation x ∗ x = y with solutions which are real-valued or complex-valued functions x(t) defined on a finite real interval, say t ∈ [0,1]. Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low order convergence rates for regularized solutions of nonlinear ill-posed operator equations F(x)=y with solutions x† in a Hilbert space setting. So for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of F (x† )∗ fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on [0,2] and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex L2-functions with a fixed and uniformly bounded modulus function.

We revisit in L2-spaces the autoconvolution equation x ∗ x = y with solutions which are real-valued or complex-valued functions x(t) defined on a finite real interval, say t ∈ [0,1]. Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low order convergence rates for regularized solutions of nonlinear ill-posed operator equations F(x)=y with solutions x† in a Hilbert space setting. So for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of F (x† )∗ fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on [0,2] and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex L2-functions with a fixed and uniformly bounded modulus function.