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.

L'évaluation et le contrôle non destructifs (E.C.N.D.) des matériaux et des structures sont une problématique industrielle très importante dans les domaines du transport, de l'aéronautique et du spatial, et dans le milieu médical. La thermographie infrarouge active est une technique d'E.C.N.D qui consiste à apporter une excitation extérieure afin d'entraîner une élévation de température dans le matériau, puis à évaluer le champ de température résultant à la surface. Cependant, les excitateurs thermiques utilisés (lampes flash, halogènes, lasers) agissent uniquement sur la surface du matériau. Plusieurs systèmes de conversion d'énergie peuvent en revanche mener à l'apparition de sources volumiques : on peut citer en particulier les phénomènes de thermo-acoustique, de thermo-induction, de thermomécanique ou de thermochimie. Par exemple, une excitation par ondes ultrasonores peut entraîner des sources thermiques volumiques si le matériau est viscoélastique ou s'il y a présence de défaut. La reconstruction de ces sources est donc la première étape permettant de remonter aux paramètres responsables de l'échauffement. Caractériser une source thermique consiste à reconstruire sa géométrie et la puissance qu'elle génère. Cependant, l'identification de sources thermiques volumiques par la mesure des champs de température de surface est un problème mathématiquement mal posé. Le caractère diffusif de la température en est le principal responsable. Dans ce travail, la reconstruction 3D des sources volumiques à partir du champ de température résultant à la surface, mesuré par InfraRouge, est étudié. Tout d'abord, une analyse du problème physique permet de spécifier les limites de la reconstruction. En particulier, un critère sur la résolution spatiale atteignable est défini et une limitation de reconstruction pour les sources en profondeur est mise en lumière. Ensuite, une méthode de reconstruction par approche probabiliste est proposée et comparée aux méthodes d'inversions existantes. Le temps d'exécution et la sensibilité au bruit de mesure sont étudiés pour chacune de ces méthodes. Des applications numériques et expérimentales seront enfin présentées pour illustrer les résultats. / Non Destructive Testing (N.D.T.) of materials and structures is a very important industrial issue in the fields of transport, aeronautics and space and in the medical domain. Active infrared thermography is a N.D.T. method that consists in providing an external excitation to cause an elevation of temperature field in the material and then to evaluate the resulting temperature field at the surface. However, thermal exciters used (flash lamps, halogen, lasers) act only on the surface of the sample. Several energy conversion systems can on the other hand lead to the generation of volumetric sources: the phenomena of thermo-acoustic, thermo-induction, thermomechanic or thermochemistry can be cited. For example, ultrasonic waves can generate volumetric heat sources if the material is viscoelastic or if there is a defect. The reconstruction of these sources is the first step for the quantification of parameters responsible of the heating. Characterizing a heat source means reconstructing its geometry and the power it generates. For example, a defect in a structure and / or the viscoelasticity of a material can be detected and quantified by this technique if it acts directly on temperature field. However, identification of volumetric heat sources from surface temperature fields is a mathematical ill-posed problem. The diffusive nature of the temperature is the main cause. In this work, the 3D reconstruction of the volumetric heat sources from the resulting surface temperature field, measured by InfraRed, is studied. First, an analysis of the physical problem enables to specify the limits of the reconstruction. In particular, a criterion on achievable spatial resolution is defined and a reconstruction limitation for in-depth sources is highlighted. Then, a probabilistic approach for the reconstruction is proposed and compared to existing inverse methods. The computation time and noise sensitivity are studied for each of these methods. Numerical and experimental applications will thus be presented to illustrate the results.

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Inversion and Depth Range of Dipole Electromagnetic Induction Measurements in Geophysics Fernando César Moura de Andrade Institute of Hydrogeology, Engineering Geology and Applied Geophysics Faculty of Science, Charles University Electromagnetic induction geophysical methods are, basically, composed by a transmitter which produces a magnetic field and a set of receivers which measure the primary magnetic field, from the transmitter, superimposed by secondary magnetic fields inducted in the subsurface. Equipment operating at, relatively, low frequencies and with short distances between the transmitter and the receivers are usually called conductivity meters and operate at low inductions numbers. The depth of investigation, in such kind of equipment, depends mainly on the transmitter-receiver distance, on the orientations of the magnetic dipoles and the height of the instrument from the ground, in order that a depth sounding can be done changing these parameters in a single measurement location. Making a series of these multi-configuration measurements, two-dimensional, or even three-dimensional surveys, can be performed and, subsequently, inverted in order to produce an image of the subsurface of the earth. Forward modelling and inversion of multi-configuration electromagnetic induction data can be made...

Seismic hazard analysis is concerned with estimating risk to human populations due to earthquakes and the other natural disasters that they cause. In many parts of the world, earthquake-generated tsunamis are especially dangerous. Assessing the risk for seismic disasters relies on historical data that indicate which fault zones are capable of supporting significant earthquakes. Due to the nature of geologic time scales, the era of seismological data collection with modern instruments has captured only a part of the Earth's seismic hot zones. However, non-instrumental records, such as anecdotal accounts in newspapers, personal journals, or oral tradition, provide limited information on earthquakes that occurred before the modern era. Here, we introduce a method for reconstructing the source earthquakes of historical tsunamis based on anecdotal accounts. We frame the reconstruction task as a Bayesian inference problem by making a probabilistic interpretation of the anecdotal records. Utilizing robust models for simulating earthquakes and tsunamis provided by the software package GeoClaw, we implement a Metropolis-Hastings sampler for the posterior distribution on source earthquake parameters. In this work, we present our analysis of the 1852 Banda Arc earthquake and tsunami as a case study for the method. Our method is implemented as a Python package, which we call tsunamibayes. It is available, open-source, on GitHub: https://github.com/jwp37/tsunamibayes.

<div>Structural health monitoring (SHM) has immense potential to improve the safety of aerospace, mechanical, and civil structures because it allows for continuous, real-time damage prognostication. However, conventional SHM methodologies are limited by factors such as the need for extensive external sensor arrays, inadequate sensitivity to small-sized damage, and poor spatial damage localization. As such, widespread implementation of SHM in engineering structures has been severely restricted. These limitations can be overcome through the use of multi-functional materials with intrinsic self-sensing capabilities. In this area, composite materials with nanofiller-modified polymer matrices have received considerable research interest. The electrical conductivity of these materials is affected by mechanical stimuli such as strain and damage. This is known as the piezoresistive effect and it has been leveraged extensively for SHM in self-sensing materials. However, prevailing conductivity-based SHM modalities suffer from two critical limitations. The first limitation is that the mechanical state of the structure must be indirectly inferred from conductivity changes. Since conductivity is not a structurally relevant property, it would be much more beneficial to know the displacements, strains, and stresses as these can be used to predict the onset of damage and failure. The second limitation is that the precise shape and size of damage cannot be accurately determined from conductivity changes. From a SHM point of view, knowing the precise shape and size of damage would greatly aid in-service inspection and nondestructive evaluation (NDE) of safety-critical structures. The underlying cause of these limitations is that recovering precise mechanics from conductivity presents an under determined and multi-modal inverse problem. Therefore, commonly used inversion schemes such as gradient-based optimization methods fail to produce physically meaningful solutions. Instead, metaheuristic search algorithms must be used in conjunction with physics-based damage models and realistic constraints on the solution search space. To that end, the overarching goal of this research is to address the limitations of conductivity-based SHM by developing metaheuristic algorithm-enabled methodologies for recovering precise mechanics from conductivity changes in self-sensing composites.</div><div><div><br></div><div>Three major scholarly contributions are made in this thesis. First, a piezoresistive inversion methodology is developed for recovering displacements, strains, and stresses in an elastically deformed self-sensing composite based on observed conductivity changes. For this, a genetic algorithm (GA) is integrated with an analytical piezoresistivity model and physics-based constraints on the search space. Using a simple stress based failure criterion, it is demonstrated that this approach can be used to accurately predict material failure. Second, the feasibility of using other widely used metaheuristic algorithms for piezoresistive inversion is explored. Specifically, simulated annealing (SA) and particle swarm optimization (PSO) are used and their performances are compared to the performance of the GA. It is concluded that while SA and PSO can certainly be used to solve the piezoresistive inversion problem, the GA is the best algorithm based on solution accuracy, consistency, and efficiency. Third, a novel methodology is developed for precisely determining damage shape and size from observed conductivity changes in self-sensing composites. For this, a GA is integrated with physics-based geometric models for damage and suitable constraints on the search space. By considering two specific damage modes —through-holes and delaminations —it is shown that this method can be used to precisely reconstruct the shape and size of damage. </div><div><br></div><div>In achieving these goals, this thesis advances the state of the art by addressing critical limitations of conductivity-based SHM. The methodologies developed herein can enable unprecedented NDE capabilities by providing real-time information about the precise mechanical state (displacements, strains, and stresses) and damage shape in self-sensing composites. This has incredible potential to improve the safety of structures in a myriad of engineering venues.</div></div>

<p>Applications in materials and biological imaging are currently limited by the ability to collect high-resolution data over large areas in practical amounts of time. One possible solution to this problem is to collect low-resolution data and apply a super-resolution interpolation algorithm to produce a high-resolution image. However, state-of-the-art super-resolution algorithms are typically designed for natural images, require aligned pairing of high and low-resolution training data for optimal performance, and do not directly incorporate a data-fidelity mechanism.</p><p><br></p><p>We present a Multi-Resolution Data Fusion (MDF) algorithm for accurate interpolation of low-resolution SEM and TEM data by factors of 4x and 8x. This MDF interpolation algorithm achieves these high rates of interpolation by first learning an accurate prior model denoiser for the TEM sample from small quantities of unpaired high-resolution data and then balancing this learned denoiser with a novel mismatched proximal map that maintains fidelity to measured data. The method is based on Multi-Agent Consensus Equilibrium (MACE), a generalization of the Plug-and-Play method, and allows for interpolation at arbitrary resolutions without retraining. We present electron microscopy results at 4x and 8x super resolution that exhibit reduced artifacts relative to existing methods while maintaining fidelity to acquired data and accurately resolving sub-pixel-scale features.</p>

In this note we explain the use of the Akiake Information Criterion and its related model comparison indices (usually derived for maximum likelihood estimator inverse problem formulations) in the context of least squares (ordinary, weighted, iterative weighted or “generalized”, etc.) based inverse problem formulations. The ideas are illustrated with several examples of interest in biology.

Seismic hazard analysis is concerned with estimating risk to human populations due to earthquakes and the other natural disasters that they cause. In many parts of the world, earthquake-generated tsunamis are especially dangerous. Assessing the risk for seismic disasters relies on historical data that indicate which fault zones are capable of supporting significant earthquakes. Due to the nature of geologic time scales, the era of seismological data collection with modern instruments has captured only a part of the Earth's seismic hot zones. However, non-instrumental records, such as anecdotal accounts in newspapers, personal journals, or oral tradition, provide limited information on earthquakes that occurred before the modern era. Here, we introduce a method for reconstructing the source earthquakes of historical tsunamis based on anecdotal accounts. We frame the reconstruction task as a Bayesian inference problem by making a probabilistic interpretation of the anecdotal records. Utilizing robust models for simulating earthquakes and tsunamis provided by the software package GeoClaw, we implement a Metropolis-Hastings sampler for the posterior distribution on source earthquake parameters. In this work, we present our analysis of the 1852 Banda Arc earthquake and tsunami as a case study for the method. Our method is implemented as a Python package, which we call tsunamibayes. It is available, open-source, on GitHub: https://github.com/jwp37/tsunamibayes.

No description available.