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Chen, Xudong 1977. "Inverse problems in electromagnetics." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/33933.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Vita.
Includes bibliographical references (p. 155-164).
Two inverse problems in electromagnetics are investigated in this thesis. The first is the retrieval of the effective constitutive parameters of metamaterials from the measurement of the reflection and the transmission coefficients. A robust method is proposed for the retrieval of metamaterials as isotropic media, and four improvements over the existing methods make the retrieval results more stable. Next, a new scheme is presented for the retrieval of a specific bianisotropic metamaterial that consists of split-ring resonators, which signifies that the cross polarization terms of the metamaterial are quantitatively investigated for the first time. Finally, an optimization approach is designed to achieve the retrieval of general bianisotropic media with 72 unknown parameters. The hybrid algorithm combining the differential evolution (DE) algorithm and the simplex method steadily converges to the exact solution. The second inverse problem deals with the detection and the classification of buried metallic objects using electromagnetic induction (EMI). Both the exciting and the induced magnetic fields are expanded as a linear combination of basic modes in the spheroidal coordinate system. Due to the orthogonality and the completeness of the spheroidal basic modes, the scattering coefficients are uniquely determined and are characteristics of the object.
(cont.) The scattering coefficients are retrieved from the knowledge of the induced fields, where both synthetic and measurement data are used. The ill-conditioning issue is dealt with by mode truncation and Tikhonov regularization technique. Stored in a library, the scattering coefficients can produce fast forward models for use in pattern matching. In addition, they can be used to train support vector machine (SVM) to sort objects into generic classes.
by Xudong Chen.
Ph.D.
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Deolmi, Giulia. "Computational Parabolic Inverse Problems." Doctoral thesis, Università degli studi di Padova, 2012. http://hdl.handle.net/11577/3423351.
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This thesis presents a general approach to solve numerically parabolic Inverse Problems, whose underlying mathematical model is discretized using the Finite Element method. The proposed solution is based upon an adaptive parametrization and it is applied specically to a geometric conduction inverse problem of corrosion estimation and to a boundary convection inverse problem of pollution rate estimation.In questa tesi viene presentato un approccio numerico volto alla risoluzione di problemi inversi parabolici, basato sull'utilizzo di una parametrizzazione adattativa. L'algoritmo risolutivo viene descritto per due specici problemi: mentre il primo consiste nella stima della corrosione di una faccia incognita del dominio, il secondo ha come scopo la quanticazione di inquinante immesso in un fiume.
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Debarnot, Valentin. "Microscopie computationnelle." Thesis, Toulouse 3, 2020. http://www.theses.fr/2020TOU30156.
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Les travaux présentés de cette thèse visent à proposer des outils numériques et théoriques pour la résolution de problèmes inverses en imagerie. Nous nous intéressons particulièrement au cas où l'opérateur d'observation (e.g. flou) n'est pas connu. Les résultats principaux de cette thèse s'articulent autour de l'estimation et l'identification de cet opérateur d'observation. Une approche plébiscitée pour estimer un opérateur de dégradation consiste à observer un échantillon contenant des sources ponctuelles (microbilles en microscopie, étoiles en astronomie). Une telle acquisition fournit une mesure de la réponse impulsionnelle de l'opérateur en plusieurs points du champ de vue. Le traitement de cette observation requiert des outils robustes pouvant utiliser rapidement les données rencontrées en pratique. Nous proposons une boîte à outils qui estime un opérateur de dégradation à partir d'une image contenant des sources ponctuelles. L'opérateur estimé à la propriété qu'en tout point du champ de vue, sa réponse impulsionnelle s'exprime comme une combinaison linéaire de fonctions élémentaires. Cela permet d'estimer des opérateurs invariants (convolutions) et variants (développement en convolution-produit) spatialement. Une spécificité importante de cette boîte à outils est son caractère automatique : seul un nombre réduit de paramètres facilement accessibles permettent de couvrir une grande majorité des cas pratiques. La taille de la source ponctuelle (e.g. bille), le fond et le bruit sont également pris en compte dans l'estimation. Cet outil se présente sous la forme d'un module appelé PSF-Estimator pour le logiciel Fiji, et repose sur une implémentation parallélisée en C++. En réalité, les opérateurs modélisant un système optique varient d'une expérience à une autre, ce qui, dans l'idéal, nécessite une calibration du système avant chaque acquisition. Pour pallier à cela, nous proposons de représenter un système optique non pas par un unique opérateur de dégradation, mais par un sous-espace d'opérateurs. Cet ensemble doit permettre de représenter chaque opérateur généré par un microscope. Nous introduisons une méthode d'estimation d'un tel sous-espace à partir d'une collection d'opérateurs de faible rang (comme ceux estimés par la boîte à outils PSF-Estimator). Nous montrons que sous des hypothèses raisonnables, ce sous-espace est de faible dimension et est constitué d'éléments de faible rang. Dans un second temps, nous appliquons ce procédé en microscopie sur de grands champs de vue et avec des opérateurs variant spatialement. Cette mise en œuvre est possible grâce à l'utilisation de méthodes complémentaires pour traiter des images réelles (e.g. le fond, le bruit, la discrétisation de l'observation). La construction d'un sous-espace d'opérateurs n'est qu'une étape dans l'étalonnage de systèmes optiques et la résolution de problèmes inverses. [...]The contributions of this thesis are numerical and theoretical tools for the resolution of blind inverse problems in imaging. We first focus in the case where the observation operator is unknown (e.g. microscopy, astronomy, photography). A very popular approach consists in estimating this operator from an image containing point sources (microbeads or fluorescent proteins in microscopy, stars in astronomy). Such an observation provides a measure of the impulse response of the degradation operator at several points in the field of view. Processing this observation requires robust tools that can rapidly use the data. We propose a toolbox that estimates a degradation operator from an image containing point sources. The estimated operator has the property that at any location in the field of view, its impulse response is expressed as a linear combination of elementary estimated functions. This makes it possible to estimate spatially invariant (convolution) and variant (product-convolution expansion) operators. An important specificity of this toolbox is its high level of automation: only a small number of easily accessible parameters allows to cover a large majority of practical cases. The size of the point source (e.g. bead), the background and the noise are also taken in consideration in the estimation. This tool, coined PSF-estimator, comes in the form of a module for the Fiji software, and is based on a parallelized implementation in C++. The operators generated by an optical system are usually changing for each experiment, which ideally requires a calibration of the system before each acquisition. To overcome this, we propose to represent an optical system not by a single operator (e.g. convolution blur with a fixed kernel for different experiments), but by subspace of operators. This set allows to represent all the possible states of a microscope. We introduce a method for estimating such a subspace from a collection of low rank operators (such as those estimated by the toolbox PSF-Estimator). We show that under reasonable assumptions, this subspace is low-dimensional and consists of low rank elements. In a second step, we apply this process in microscopy on large fields of view and with spatially varying operators. This implementation is possible thanks to the use of additional methods to process real images (e.g. background, noise, discretization of the observation).The construction of an operator subspace is only one step in the resolution of blind inverse problems. It is then necessary to identify the degradation operator in this set from a single observed image. In this thesis, we provide a mathematical framework to this operator identification problem in the case where the original image is constituted of point sources. Theoretical conditions arise from this work, allowing a better understanding of the conditions under which this problem can be solved. We illustrate how this formal study allows the resolution of a blind deblurring problem on a microscopy example.[...]
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Kitic, Srdan. "Cosparse regularization of physics-driven inverse problems." Thesis, Rennes 1, 2015. http://www.theses.fr/2015REN1S152/document.
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Les problèmes inverses liés à des processus physiques sont d'une grande importance dans la plupart des domaines liés au traitement du signal, tels que la tomographie, l'acoustique, les communications sans fil, le radar, l'imagerie médicale, pour n'en nommer que quelques uns. Dans le même temps, beaucoup de ces problèmes soulèvent des défis en raison de leur nature mal posée. Par ailleurs, les signaux émanant de phénomènes physiques sont souvent gouvernées par des lois s'exprimant sous la forme d'équations aux dérivées partielles (EDP) linéaires, ou, de manière équivalente, par des équations intégrales et leurs fonctions de Green associées. De plus, ces phénomènes sont habituellement induits par des singularités, apparaissant comme des sources ou des puits d'un champ vectoriel. Dans cette thèse, nous étudions en premier lieu le couplage entre de telles lois physiques et une hypothèse initiale de parcimonie des origines du phénomène physique. Ceci donne naissance à un concept de dualité des régularisations, formulées soit comme un problème d'analyse coparcimonieuse (menant à la représentation en EDP), soit comme une parcimonie à la synthèse équivalente à la précédente (lorsqu'on fait plutôt usage des fonctions de Green). Nous dédions une part significative de notre travail à la comparaison entre les approches de synthèse et d'analyse. Nous défendons l'idée qu'en dépit de leur équivalence formelle, leurs propriétés computationnelles sont très différentes. En effet, en raison de la parcimonie héritée par la version discrétisée de l'EDP (incarnée par l'opérateur d'analyse), l'approche coparcimonieuse passe bien plus favorablement à l'échelle que le problème équivalent régularisé par parcimonie à la synthèse. Nos constatations sont illustrées dans le cadre de deux applications : la localisation de sources acoustiques, et la localisation de sources de crises épileptiques à partir de signaux électro-encéphalographiques. Dans les deux cas, nous vérifions que l'approche coparcimonieuse démontre de meilleures capacités de passage à l'échelle, au point qu'elle permet même une interpolation complète du champ de pression dans le temps et en trois dimensions. De plus, dans le cas des sources acoustiques, l'optimisation fondée sur le modèle d'analyse \emph{bénéficie} d'une augmentation du nombre de données observées, ce qui débouche sur une accélération du temps de traitement, plus rapide que l'approche de synthèse dans des proportions de plusieurs ordres de grandeur. Nos simulations numériques montrent que les méthodes développées pour les deux applications sont compétitives face à des algorithmes de localisation constituant l'état de l'art. Pour finir, nous présentons deux méthodes fondées sur la parcimonie à l'analyse pour l'estimation aveugle de la célérité du son et de l'impédance acoustique, simultanément à l'interpolation du champ sonore. Ceci constitue une étape importante en direction de la mise en œuvre de nos méthodes en en situation réelleInverse problems related to physical processes are of great importance in practically every field related to signal processing, such as tomography, acoustics, wireless communications, medical and radar imaging, to name only a few. At the same time, many of these problems are quite challenging due to their ill-posed nature. On the other hand, signals originating from physical phenomena are often governed by laws expressible through linear Partial Differential Equations (PDE), or equivalently, integral equations and the associated Green’s functions. In addition, these phenomena are usually induced by sparse singularities, appearing as sources or sinks of a vector field. In this thesis we primarily investigate the coupling of such physical laws with a prior assumption on the sparse origin of a physical process. This gives rise to a “dual” regularization concept, formulated either as sparse analysis (cosparse), yielded by a PDE representation, or equivalent sparse synthesis regularization, if the Green’s functions are used instead. We devote a significant part of the thesis to the comparison of these two approaches. We argue that, despite nominal equivalence, their computational properties are very different. Indeed, due to the inherited sparsity of the discretized PDE (embodied in the analysis operator), the analysis approach scales much more favorably than the equivalent problem regularized by the synthesis approach. Our findings are demonstrated on two applications: acoustic source localization and epileptic source localization in electroencephalography. In both cases, we verify that cosparse approach exhibits superior scalability, even allowing for full (time domain) wavefield interpolation in three spatial dimensions. Moreover, in the acoustic setting, the analysis-based optimization benefits from the increased amount of observation data, resulting in a speedup in processing time that is orders of magnitude faster than the synthesis approach. Numerical simulations show that the developed methods in both applications are competitive to state-of-the-art localization algorithms in their corresponding areas. Finally, we present two sparse analysis methods for blind estimation of the speed of sound and acoustic impedance, simultaneously with wavefield interpolation. This is an important step toward practical implementation, where most physical parameters are unknown beforehand. The versatility of the approach is demonstrated on the “hearing behind walls” scenario, in which the traditional localization methods necessarily fail. Additionally, by means of a novel algorithmic framework, we challenge the audio declipping problemregularized by sparsity or cosparsity. Our method is highly competitive against stateof-the-art, and, in the cosparse setting, allows for an efficient (even real-time) implementation
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Baysal, Arzu. "Inverse Problems For Parabolic Equations." Master's thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/12605623/index.pdf.
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In this thesis, we study inverse problems of restoration of the unknown function in a boundary condition, where on the boundary of the domain there is a convective heat exchange with the environment. Besides the temperature of the domain, we seek either the temperature of the environment in Problem I and II, or the coefficient of external boundary heat emission in Problem III and IV. An additional information is given, which is the overdetermination condition, either on the boundary of the domain (in Problem III and IV) or on a time interval (in
Problem I and II). If solution of inverse problem exists, then the temperature can be defined everywhere on the domain at all instants.
The thesis consists of six chapters. In the first chapter, there is the introduction where the definition and applications of inverse problems are given and definition of the four inverse problems, that we will analyze in this thesis, are stated. In the second chapter, some definitions and theorems which we will use to obtain some conclusions about the corresponding direct problem of our four inverse problems are stated, and the conclusions about direct problem are obtained.
In the third, fourth, fifth and sixth chapters we have the analysis of inverse problems I, II, III and IV, respectively.
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Connolly, T. John. "Nonlinear methods for inverse problems." Thesis, University of Canterbury. Mathematics, 1989. http://hdl.handle.net/10092/8563.
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The general inverse problem is formulated as a nonlinear operator equation. The solution of this via the Newton-Kantorovich method is outlined. Fréchet differentiability of the operator is given by the implicit function theorem. We also consider questions such as uniqueness, stability and regularization of the inverse problem.
This general theory is then applied to a number of different inverse problems. The Newton-Kantorovich method is derived for each example and Fréchet differentiability examined. In some cases numerical results are provided, for others our work provides a theoretical basis for results obtained by different authors.
The problems considered include an interior measurement inverse problem from steady-state diffusion, and a boundary measurement problem for electrical conductivity imaging. We also examine the determination of refractive indices and scattering boundaries for the Helmholtz equation from measurements of the farfield. In addition an inverse problem from geometric optics is investigated.
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Stewart, K. A. "Inverse problems in signal processing." Thesis, University of Strathclyde, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.382448.
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Aljohani, Hassan Musallam S. "Wavelet methods and inverse problems." Thesis, University of Leeds, 2017. http://etheses.whiterose.ac.uk/18830/.
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Archaeological investigations are designed to acquire information without damaging the archaeological site. Magnetometry is one of the important techniques for producing a surface grid of readings, which can be used to infer underground features. The inversion of this data, to give a fitted model, is an inverse problem. This type of problem can be ill-posed or ill-conditioned, making the estimation of model parameters less stable or even impossible. More precisely, the relationship between archaeological data and parameters is expressed by a likelihood. It is not possible to use the standard regression estimate obtained through the likelihood, which means that no maximum likelihood estimate exists. Instead, various constraints can be added through a prior distribution with an estimate produced using the posterior distribution. Current approaches incorporate prior information describing smoothness, which is not always appropriate. The biggest challenge is that the reconstruction of an archaeological site as a single layer requires various physical features such as depth and extent to be assumed. By applying a smoothing prior in the analysis of stratigraphy data, however, these features are not easily estimated. Wavelet analysis has proved to be highly efficient at eliciting information from noisy data. Additionally, complicated signals can be explained by interpreting only a small number of wavelet coefficients. It is possible that a modelling approach, which attempts to describe an underlying function in terms of a multi-level wavelet representation will be an improvement on standard techniques. Further, a new method proposed uses an elastic-net based distribution as the prior. Two methods are used to solve the problem, one is based on one-stage estimation and the other is based on two stages. The one-stage considers two approaches a single prior for all wavelet resolution levels and a level-dependent prior, with separate priors at each resolution level. In a simulation study and a real data analysis, all these techniques are compared to several existing methods. It is shown that the methodology using a single prior provides good reconstruction, comparable even to several established wavelet methods that use mixture priors.
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Marroquin, J. L. (Jose Luis). "Probabilistic solution of inverse problems." Thesis, Massachusetts Institute of Technology, 1985. http://hdl.handle.net/1721.1/15286.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1985.MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING.
Bibliography: p. 195-200.
by Jose Luis Marroquin.
Ph.D.
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Agapiou, Sergios. "Aspects of Bayesian inverse problems." Thesis, University of Warwick, 2013. http://wrap.warwick.ac.uk/60138/.
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The goal of this thesis is to contribute to the formulation and understanding of the Bayesian approach to inverse problems in function space. To this end we examine two important aspects of this approach: the frequentist asymptotic properties of the posterior, and the extraction of information from the posterior via sampling. We work in a separable Hilbert space setting and consider Gaussian priors on the unknown in conjugate Gaussian models. In the first part of this work we consider linear inverse problems with Gaussian additive noise and study the contraction in the small noise limit of the Gaussian posterior distribution to a Dirac measure centered on the true parameter underlying the data. In a wide range of situations, which include both mildly and severely ill-posed problems, we show how carefully calibrating the scaling of the prior as a function of the size of the noise, based on a priori known information on the regularity of the truth, yields optimal rates of contraction. In the second part we study the implementation in RN of hierarchical Bayesian linear inverse problems with Gaussian noise and priors, and with hyper-parameters introduced through the scalings of the prior and noise covariance operators. We use function space intuition to understand the large N behaviour of algorithms designed to sample the posterior and show that the two scaling hyper-parameters evolve under these algorithms in contrasting ways: as N grows the prior scaling slows down while the noise scaling speeds up. We propose a reparametrization of the prior scaling which is robust with respect to the increase in dimension. Our theory on the slowing down of the evolution of the prior scaling extends to hierarchical approaches in more general conjugate Gaussian settings, while our intuition covers other parameters of the prior covariance operator as well. Throughout the thesis we use a blend of results from measure theory and probability theory with tools from the theory of linear partial differential equations and numerical analysis.
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Lian, Duan. "Bayesian methods for inverse problems." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:b4000e98-7d56-4274-8210-a22b04be436c.
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This thesis describes two novel Bayesian methods: the Iterative Ensemble Square Filter (IEnSRF) and the Warp Ensemble Square Root Filter (WEnSRF) for solving the barcode detection problem, the deconvolution problem in well testing and the history matching problem of facies patterns. For the barcode detection problem, at the expanse of overestimating the posterior uncertainty, the IEnSRF efficiently achieves successful detections with very challenging real barcode images which the other considered methods and commercial software fail to detect. It also performs reliable detection on low-resolution images under poor ambient light conditions. For the deconvolution problem in well testing, the IEnSRF is capable of quantifying estimation uncertainty, incorporating the cumulative production data and estimating the initial pressure, which were thought to be unachievable in the existing well testing literature. The estimation results for the considered real benchmark data using the IEnSRF significantly outperform the existing methods in the commercial software. The WEnSRF is utilised for solving the history matching problem of facies patterns. Through the warping transformation, the WEnSRF performs adjustment on the reservoir features directly and is thus superior in estimating the large-scale complicated facies patterns. It is able to provide accurate estimates of the reservoir properties robustly and efficiently with reasonably reliable prior reservoir structural information.
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Li, Jing. "Inverse Problems in Structural Mechanics." Diss., Virginia Tech, 2005. http://hdl.handle.net/10919/30075.
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This dissertation deals with the solution of three inverse problems in structural mechanics. The first one is load updating for finite element models (FEMs). A least squares fitting is used to identify the load parameters. The basic studies are made for geometrically linear and nonlinear FEMs of beams or frames by using a four-noded curved beam element, which, for a given precision, may significantly solve the ill-posed problem by reducing the overall number of degrees of freedom (DOF) of the system, especially the number of the unknown variables to obtain an overdetermined system. For the basic studies, the unknown applied load within an element is represented by a linear combination of integrated Legendre polynomials, the coefficients of which are the parameters to be extracted using measured displacements or strains. The optimizer L-BFGS-B is used to solve the least squares problem.
The second problem is the placement optimization of a distributed sensing fiber optic sensor for a smart bed using Genetic Algorithms (GA), where the sensor performance is maximized. The sensing fiber optic cable is represented by a Non-uniform Rational B-Splines (NURBS) curve, which changes the placement of a set of infinite number of the infinitesimal sensors to the placement of a set of finite number of the control points. The sensor performance is simplified as the integration of the absolute curvature change of the fiber optic cable with respect to a perturbation due to the body movement of a patient. The smart bed is modeled as an elastic mattress core, which supports a fiber optic sensor cable. The initial and deformed geometries of the bed due to the body weight of the patient are calculated using MSC/NASTRAN for a given body pressure. The deformation of the fiber optic cable can be extracted from the deformation of the mattress. The performance of the fiber optic sensor for any given placement is further calculated for any given perturbation.
The third application is stiffened panel optimization, including the size and placement optimization for the blade stiffeners, subject to buckling and stress constraints. The present work uses NURBS for the panel and stiffener representation. The mesh for the panel is generated using DistMesh, a triangulation algorithm in MATLAB. A NASTRAN/MATLAB interface is developed to automatically transfer the data between the analysis and optimization processes respectively. The optimization consists of minimizing the weight of the stiffened panel with design variables being the thickness of the plate and height and width of the stiffener as well as the placement of the stiffeners subjected to buckling and stress constraints under in-plane normal/shear and out-plane pressure loading conditions.Ph. D.
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Flenner, Arjuna. "Lévy processes in inverse problems /." free to MU campus, to others for purchase, 2004. http://wwwlib.umi.com/cr/mo/fullcit?p3144416.
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Wokiyi, Dennis. "Non-linear inverse geothermal problems." Licentiate thesis, Linköpings universitet, Matematik och tillämpad matematik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-143031.
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The inverse geothermal problem consist of estimating the temperature distribution below the earth’s surface using temperature and heat-flux measurements on the earth’s surface. The problem is important since temperature governs a variety of the geological processes including formation of magmas, minerals, fosil fuels and also deformation of rocks. Mathematical this problem is formulated as a Cauchy problem for an non-linear elliptic equation and since the thermal properties of the rocks depend strongly on the temperature, the problem is non-linear. This problem is ill-posed in the sense that it does not satisfy atleast one of Hadamard’s definition of well-posedness. We formulated the problem as an ill-posed non-linear operator equation which is defined in terms of solving a well-posed boundary problem. We demonstrate existence of a unique solution to this well-posed problem and give stability estimates in appropriate function spaces. We show that the operator equation is well-defined in appropriate function spaces. Since the problem is ill-posed, regularization is needed to stabilize computations. We demostrate that Tikhonov regularization can be implemented efficiently for solving the operator equation. The algorithm is based on having a code for solving a well- posed problem related to the operator equation. In this study we demostrate that the algorithm works efficiently for 2D calculations but can also be modified to work for 3D calculations.
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Hofmann, Bernd. "Chemnitz Symposium on Inverse Problems 2014." Technische Universität Chemnitz, 2014. https://monarch.qucosa.de/id/qucosa%3A20125.
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Our symposium will bring together experts from the German and international 'Inverse Problems Community' and young scientists. The focus will be on ill-posedness phenomena, regularization theory and practice, and on the analytical, numerical, and stochastic treatment of applied inverse problems in natural sciences, engineering, and finance.
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Foroozan, Farshad. "Discrete inverse conductivity problems on networks." College Park, Md. : University of Maryland, 2006. http://hdl.handle.net/1903/3542.
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Thesis (Ph. D.) -- University of Maryland, College Park, 2006.Thesis research directed by: Applied Mathematics and Scientific Computation Program. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.
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Lahmer, Tom. "Forward and inverse problems in piezoelectricity." kostenfrei, 2008. http://www.opus.ub.uni-erlangen.de/opus/volltexte/2008/958/.
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Meadows, Leslie J. "Iteratively Regularized Methods for Inverse Problems." Digital Archive @ GSU, 2013. http://digitalarchive.gsu.edu/math_diss/13.
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We are examining iteratively regularized methods for solving nonlinear inverse problems. Of particular interest for these types of methods are application problems which are unstable. For these application problems, special methods of numerical analysis are necessary, since classical algorithms tend to be divergent.
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Smit, Denis John. "Simulation and inverse problems in aggregation." Thesis, University of Cambridge, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.320058.
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Attalla, Atia May Ramsis. "Inverse eigenvalue problems : theory and algorithms." Thesis, Imperial College London, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.299725.
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Broky, John. "Inverse Problems in Multiple Light Scattering." Doctoral diss., University of Central Florida, 2013. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/5608.
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The interaction between coherent waves and material systems with complex optical properties is a complicated, deterministic process. Light that scatters from such media gives rise to random fields with intricate properties. It is common perception that the randomness of these complex fields is undesired and therefore is to be removed, usually through a process of ensemble averaging. However, random fields emerging from light matter interaction contain information about the properties of the medium and a thorough analysis of the scattered light allows solving specific inverse problems. Traditional attempts to solve these kinds of inverse problems tend to rely on statistical average quantities and ignore the deterministic interaction between the optical field and the scattering structure. Thus, because ensemble averaging inherently destroys specific characteristics of random processes, one can only recover limited information about the medium. This dissertation discusses practical means that go beyond ensemble averaging to probe complex media and extract additional information about a random scattering system. The dissertation discusses cases in which media with similar average properties can be differentiated by detailed examination of fluctuations between different realizations of the random process of multiple scattering. As a different approach to this type of inverse problems, the dissertation also includes a description of how higher-order field and polarization correlations can be used to extract features of random media and complex systems from one single realization of the light-matter interaction. Examples include (i) determining the level of multiple scattering, (ii) identifying non-stationarities in random fields, and (iii) extracting underlying correlation lengths of random electromagnetic fields that result from basic interferences. The new approaches introduced and the demonstrations described in this dissertation represent practical means to extract important material properties or to discriminate between media with similar characteristics even in situations when experimental constraints limit the number of realizations of the complex light-matter interaction.Ph.D.
Doctorate
Optics and Photonics
Optics and Photonics
Optics
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Lim, Sean Wei Xinq. "Bayesian inverse problems and seismic inversion." Thesis, University of Oxford, 2016. https://ora.ox.ac.uk/objects/uuid:ed60b058-3957-4414-bcb2-db6b5b3c0593.
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The Bayesian formulation for inverse problems gives a way of making inferences about unknown quantities not directly observable. The application of Bayes' Theorem combines the prior information and the observation to give a posterior measure, which contains information about the quantity we are trying to estimate. In this thesis, we review a particular formulation, conventionally known as the strong constraint formulation of inverse problems. We describe methods to obtain summaries of information from the posterior measure. We also describe how prior measures are constructed using linear differential operators, to quantify as accurately as possible our knowledge of the parameters, independent of any observations. Then, we note that the strong constraint formulation of inverse problems makes it hard to obtain summaries of information of the posterior measure, typically obtained through an optimization of a misfit functional. Therefore, we introduce the weak constraint formulation in a Bayesian context for inverse problems, which eases the task of optimization. We use this formulation to perform sampling of the posterior measure. This method is tested on some simple test problems. We also compare the results between a strong and weak constraint formulation of inverse problems by studying a one-dimensional example. Finally, we apply the weak constraint formulation to the problem of full waveform inversion, which is a common problem in seismology. The forward problem we use here is the Laplace transform of the acoustic wave equation, and the inverse problem is solved in several frequencies. There are two approaches when observations at several frequencies are available. First is the sequential method, which processes the observation at different frequencies individually. The second method, which is the simultaneous method, processes the observations at all frequencies at once. We use the simultaneous method here, and used a non-trivial model problem, which yields promising results.
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Klimmek, Martin. "On inverse problems in mathematical finance." Thesis, University of Warwick, 2012. http://wrap.warwick.ac.uk/55821/.
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We consider two inverse problems motivated by questions in mathematical finance. In the first two chapters (Part 1) we recover processes consistent with given perpetual American option prices. In the third and fourth chapters (Part 2) we construct model-independent bounds for prices of contracts based on the realized variance of an asset price process. The two parts are linked by the question of how to recover information about asset price dynamics from option prices: in part one we assume knowledge of perpetual American option prices while in the second part we will assume knowledge of European call and put option prices. Mathematically, the first part of the thesis presents a framework for constructing generalised diffusions consistent with optimal stopping values. The second part aims at constructing bounds for path-dependent functionals of martingales given their terminal distribution.
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Alfowzan, Mohammed Fowzan, and Mohammed Fowzan Alfowzan. "Solutions to Space-Time Inverse Problems." Diss., The University of Arizona, 2016. http://hdl.handle.net/10150/621791.
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Two inverse problems are investigated in this dissertation, taking into account both the spatial and temporal aspects. The first problem addresses the under determined image reconstruction problem for dynamic SPECT. The quality of the reconstructed image is often limited due to having fewer observations than the number of voxels. The proposed algorithms make use of the generalized α-divergence function to improve the estimation performance. The first algorithm is based on an alternating minimization framework to minimize a regularized α-divergence objective function. We demonstrate that selecting an adaptive α policy depending on the time evolution of the voxels gives better performance than a fixed α assignment. The second algorithm is based on Newton's method. A regularized approach has been taken to avoid stability issues. Newton's method is generally computationally demanding due to the complexity associated with inverting the Hessian matrix. A fast Newton-based method is proposed using majorization-minimization techniques that diagonalize the Hessian matrix. In dynamically evolving systems, the prediction matrix plays an important role in the estimation process. An estimation technique is proposed to estimate the prediction matrix using the α-divergence function. The simulation results show that our algorithms provide better performance than the techniques based on the Kullback-Leibler distance. The second problem is the recovery of data transmitted over free-space optical communication channels using orbital angular momentum (OAM). In the presence of atmospheric turbulence, crosstalk occurs among OAM optical modes resulting in an error floor at a relatively high bit error rate. The modulation format considered for the underlying problem is Q-ary pulse position modulation (PPM). We propose and evaluate three joint detection strategies to overcome the OAM crosstalk problem: i) maximum likelihood sequence estimation (MLSE). ii) Q-PPM factor graph detection. iii) branch-and-bound detection. We compare the complexity and the bit-error-rate performance of these strategies in realistic scenarios.
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Van, Cong Tuan Son. "Numerical solutions to some inverse problems." Diss., Kansas State University, 2017. http://hdl.handle.net/2097/38248.
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Doctor of PhilosophyDepartment of Mathematics
Alexander G. Ramm
In this dissertation, the author presents two independent researches on inverse problems: (1) creating materials in which heat propagates a long a line and (2) 3D inverse scattering problem with non-over-determined data. The theories of these methods were developed by Professor Alexander Ramm and are presented in Chapters 1 and 3. The algorithms and numerical results are taken from the papers of Professor Alexander Ramm and the author and are presented in Chapters 2 and 4.
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Trucu, Dumitru. "Inverse problems for blood perfusion identification." Thesis, University of Leeds, 2009. http://etheses.whiterose.ac.uk/21104/.
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In this thesis we investigate a sequence of important inverse problems associated with the bio-heat transient flow equation which models the heat transfer within the human body. Given the physical importance of the blood perfusion coefficient that appears in the bio-heat equation, attention is focused on the inverse problems concerning the accurate recovery of this information when exact and noisy measurements are considered in terms of the mass, flux, or temperature, which we sampled over the specific regions of the media under investigation. Five different cases are considered for the retrieval of the perfusion coefficient, namely when this parameter is assumed to be either constant, or dependent on time, space, temperature, or on both space and time. Theanalytica:l and numerical techniques that arc used to investigate the existence and uniqueness of the solution for this inverse coefficient identification are embedded in an extensiveú computational approach for the retrieval of the perfusion coefficient. Boundary integral methods, for the constant and the time-dependent cases, or Crank-Nicolson-type global schemes or local methods based on solutions of the first-kind integral equations, in the space, temperature, or space and time cases, are used in conjunction either with Gaussian mollification or with Tikhonov regularization methods, which arc coupled with optimization techniques. Analytically, a number of uniqueness and existence criteria and structural results are formulated and proved.
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Yang, Zhaoqing. "Variational inverse methods for transport problems." W&M ScholarWorks, 1996. https://scholarworks.wm.edu/etd/1539616917.
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Variational inverse data assimilation schemes are developed for three types of parameter identification problems in transport models: (1) the tracer inverse for the Lagrangian mean transport velocity in a long-term advection-diffusion transport model; (2) determination of inflow salinity open boundary condition in an intra-tidal salinity transport model; and (3) determination of settling velocity and resuspension rate for a cohesive sediment transport model. The gradient of the cost function with respect to the control variables is obtained by the adjoint model. A series of twin experiments are conducted to test the inverse models for the three types of problems. Results show that variational data assimilation can successfully retrieve poorly known parameters in transport models. The first problem is associated with the long-term advective transport, represented by the Lagrangian mean transport velocity which can be decomposed into two parts: the Eulerian transport velocity and the curl of a 3-D vector potential A. The optimal long-term advective transport field is obtained through adjusting the vector potential using a variational data assimilation method. Experiments are performed in an idealized estuary. Results show that the variational data assimilation method can successfully retrieve the effective Lagrangian mean transport velocity in a long-term transport model. Results also show that the smooth best fit model state can still be retrieved using a penalty method when observations are too sparse or contain noisy signals. A variational inverse model for optimally determining open boundary condition is developed and tested in a 3-D intra-tidal salinity transport model. The maximum inflow salinity open boundary value and its recovery time from outflow condition are treated as control variables. Effects of scaling, preconditioning, and penalty are investigated. It is shown that proper scaling and preconditioning can greatly speed up the convergence rate of the minimization process. The spatial oscillations in the recovery time of the inflow boundary condition can be effectively eliminated by an penalty technique. A variational inverse model is developed to estimate the settling velocity and resuspension constant. The settling velocity &w\sb{lcub}s{rcub}& and resuspension constant &M\sb{lcub}o{rcub}& are assumed to be constant in the whole model domain. The inverse model is tested in an idealized 3-D estuary and the James River, a tributary of the Chesapeake Bay. Experimental results demonstrate that the variational inverse model can be used to identify the poorly known parameters in cohesive sediment transport modeling.
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Fu, Shuai. "Inverse problems occurring in uncertainty analysis." Thesis, Paris 11, 2012. http://www.theses.fr/2012PA112208/document.
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Ce travail de recherche propose une solution aux problèmes inverses probabilistes avec des outils de la statistique bayésienne. Le problème inverse considéré est d'estimer la distribution d'une variable aléatoire non observée X à partir d'observations bruitées Y suivant un modèle physique coûteux H. En général, de tels problèmes inverses sont rencontrés dans le traitement des incertitudes. Le cadre bayésien nous permet de prendre en compte les connaissances préalables d'experts en particulier lorsque peu de données sont disponibles. Un algorithme de Metropolis-Hastings-within-Gibbs est proposé pour approcher la distribution a posteriori des paramètres de X avec un processus d'augmentation des données. A cause d'un nombre élevé d'appels, la fonction coûteuse H est remplacée par un émulateur de krigeage (métamodèle). Cette approche implique plusieurs erreurs de natures différentes et, dans ce travail,nous nous attachons à estimer et réduire l'impact de ces erreurs. Le critère DAC a été proposé pour évaluer la pertinence du plan d'expérience (design) et le choix de la loi apriori, en tenant compte des observations. Une autre contribution est la construction du design adaptatif adapté à notre objectif particulier dans le cadre bayésien. La méthodologie principale présentée dans ce travail a été appliquée à un cas d'étude en ingénierie hydrauliqueThis thesis provides a probabilistic solution to inverse problems through Bayesian techniques.The inverse problem considered here is to estimate the distribution of a non-observed random variable X from some noisy observed data Y explained by a time-consuming physical model H. In general, such inverse problems are encountered when treating uncertainty in industrial applications. Bayesian inference is favored as it accounts for prior expert knowledge on Xin a small sample size setting. A Metropolis-Hastings-within-Gibbs algorithm is proposed to compute the posterior distribution of the parameters of X through a data augmentation process. Since it requires a high number of calls to the expensive function H, the modelis replaced by a kriging meta-model. This approach involves several errors of different natures and we focus on measuring and reducing the possible impact of those errors. A DAC criterion has been proposed to assess the relevance of the numerical design of experiments and the prior assumption, taking into account the observed data. Another contribution is the construction of adaptive designs of experiments adapted to our particular purpose in the Bayesian framework. The main methodology presented in this thesis has been applied to areal hydraulic engineering case-study
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Zhang, Wenlong. "Forward and Inverse Problems Under Uncertainty." Thesis, Paris Sciences et Lettres (ComUE), 2017. http://www.theses.fr/2017PSLEE024/document.
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Cette thèse contient deux matières différentes. Dans la première partie, deux cas sont considérés. L'un est le modèle plus lisse de la plaque mince et l'autre est les équations des limites elliptiques avec des données limites incertaines. Dans cette partie, les convergences stochastiques des méthodes des éléments finis sont prouvées pour chaque problème.Dans la deuxième partie, nous fournissons une analyse mathématique du problème inverse linéarisé dans la tomographie d'impédance électrique multifréquence. Nous présentons un cadre mathématique et numérique pour une procédure d'imagerie du tenseur de conductivité électrique anisotrope en utilisant une nouvelle technique appelée Tentomètre de diffusion Magnéto-acoustographie et proposons une approche de contrôle optimale pour reconstruire le facteur de propriété intrinsèque reliant le tenseur de diffusion au tenseur de conductivité électrique anisotrope. Nous démontrons la convergence et la stabilité du type Lipschitz de l'algorithme et présente des exemples numériques pour illustrer sa précision. Le modèle cellulaire pour Electropermécanisme est démontré. Nous étudions les paramètres efficaces dans un modèle d'homogénéisation. Nous démontrons numériquement la sensibilité de ces paramètres efficaces aux paramètres microscopiques critiques régissant l'électropermécanismeThis thesis contains two different subjects. In first part, two cases are considered. One is the thin plate spline smoother model and the other one is the elliptic boundary equations with uncertain boundary data. In this part, stochastic convergences of the finite element methods are proved for each problem.In second part, we provide a mathematical analysis of the linearized inverse problem in multifrequency electrical impedance tomography. We present a mathematical and numerical framework for a procedure of imaging anisotropic electrical conductivity tensor using a novel technique called Diffusion Tensor Magneto-acoustography and propose an optimal control approach for reconstructing the cross-property factor relating the diffusion tensor to the anisotropic electrical conductivity tensor. We prove convergence and Lipschitz type stability of the algorithm and present numerical examples to illustrate its accuracy. The cell model for Electropermeabilization is demonstrated. We study effective parameters in a homogenization model. We demonstrate numerically the sensitivity of these effective parameters to critical microscopic parameters governing electropermeabilization
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Isaev, Mikhail. "Stability and instability in inverse problems." Palaiseau, Ecole polytechnique, 2013. https://pastel.hal.science/docs/00/91/22/98/PDF/these.pdf.
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Dans cette thèse nous nous intéressons aux questions de stabilité et d'instabilité dans certains problèmes inverses classiques pour l'équation de Schrödinger et l'équation acoustique en dimension d>=2. Les problèmes considérés sont le problème inverse de Gel'fand de valeurs au bord et les problèmes inverses de diffusion en champ proche et en champ lointain. Les résultats de stabilité et d'instabilité présentés dans cette thèse se complètent mutuellement et contribuent à une meilleure compréhension de la nature des problèmes précités. En particulier, nous démontrons des nouvelles estimations de stabilité globale qui dépendent explicitement de la régularité du coefficient et de l'énergie. En outre, nous considérons le problème inverse de valeurs au bord pour l'équation de Schrödinger à l'énergie fixée avec des mesures frontières représentées comme l'opérateur frontière d'impédance (ou l'opérateur Robin-Robin). Nous démontrons des estimations de stabilité globale pour détermination du potentiel à partir de mesures frontières dans cette représentation d'impédance. De plus, des techniques similaires donnent aussi une procédure de reconstruction globale pour ce problèmeIn this thesis we focus on stability and instability issues in some classical inverse problems for the Schrödinger equation and the acoustic equation in dimension d>=2. The problems considered are the Gel'fand inverse boundary value problem, the nearfield and the far-field inverse scattering problems. Stability and instability results presented in the thesis complement each other and contribute to a better understanding of the nature of the aforementioned problems. In particular, we prove new global stability estimates which explicitly depend on coefficient regularity and energy. In addition, we consider the inverse boundary value problem for the Schrödinger equation at fixed energy with boundary measurements represented as the impedance boundary map (or Robin-to-Robin map). We prove global stability estimates for determining potential from boundary measurements in this impedance representation. Moreover, similar techniques also give a global reconstruction procedure for this problem
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Kempton, Mark Condie. "The Minimum Rank, Inverse Inertia, and Inverse Eigenvalue Problems for Graphs." BYU ScholarsArchive, 2010. https://scholarsarchive.byu.edu/etd/2182.
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For a graph G we define S(G) to be the set of all real symmetric n by n matrices whose off-diagonal zero/nonzero pattern is described by G. We show how to compute the minimum rank of all matrices in S(G) for a class of graphs called outerplanar graphs. In addition, we obtain results on the possible eigenvalues and possible inertias of matrices in S(G) for certain classes of graph G. We also obtain results concerning the relationship between two graph parameters, the zero forcing number and the path cover number, related to the minimum rank problem.
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Rachele, Lizabeth. "An inverse problem in elastodynamics /." Thesis, Connect to this title online; UW restricted, 1996. http://hdl.handle.net/1773/5735.
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Bürger, Steven. "Inverse Autoconvolution Problems with an Application in Laser Physics." Doctoral thesis, Universitätsbibliothek Chemnitz, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-211850.
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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.
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Nguyen, Thi Phong. "Direct and inverse solvers for scattering problems from locally perturbed infinite periodic layers." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLX004/document.
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Nous sommes intéressés dans cette thèse par l'analyse de la diffraction directe et inverse des ondes par des couches infinies périodiques localement perturbées à une fréquence fixe. Ce problème a des connexions avec le contrôle non destructif des structures périodiques telles que des structures photoniques, des fibres optiques, des réseaux, etc. Nous analysons d'abord le problème direct et établissons certaines conditions sur l'indice de réfraction pour lesquelles il n'existe pas de modes guidés. Ce type de résultat est important car il montre les cas pour lesquels les mesures peuvent être effectuées par exemple sur une couche au dessus de la structure périodique sans perdre des informations importantes dans la partie propagative de l'onde. Nous proposons ensuite une méthode numérique pour résoudre le problème de diffraction basée sur l'utilisation de la transformée de Floquet-Bloch dans les directions de périodicité. Nous discrétisons le problème de manière uniforme dans la variable de Floquet-Bloch et utilisons une méthode spectrale dans la discrétisation spatiale. La discrétisation en espace exploite une reformulation volumétrique du problème dans une cellule (équation intégrale de Lippmann-Schwinger) et une périodisation du noyau dans la direction perpendiculaire à la périodicité. Cette dernière transformation permet d'utiliser des techniques de type FFT pour accélérer le produit matrice-vecteur dans une méthode itérative pour résoudre le système linéaire. On aboutit à un système d'équations intégrales couplées (à cause de la perturbation locale) qui peuvent être résolues en utilisant une décomposition de Jacobi. L'analyse de la convergence est faite seulement dans le cas avec absorption et la validation numérique est réalisées sur des exemples 2D. Pour le problème inverse, nous étendons l'utilisation de trois méthodes d'échantillonnage pour résoudre le problème de la reconstruction de la géométrie du défaut à partir de la connaissance de données mutistatiques associées à des ondes incidentes planes en champ proche (c.à.d incluant certains modes évanescents). Nous analysons ces méthodes pour le problème semi-discrétisée dans la variable Floquet-Bloch. Nous proposons ensuite une nouvelle méthode d'imagerie capable de visualiser directement la géométrie du défaut sans savoir ni les propriétés physiques du milieux périodique, ni les propriétés physiques du défaut. Cette méthode que l'on appelle imagerie-différentielle est basée sur l'analyse des méthodes d'échantillonnage pour un seul mode de Floquet-Bloch et la relation avec les solutions de problèmes de transmission intérieurs d'un type nouveau. Les études théoriques sont corroborées par des expérimentations numériques sur des données synthétiques. Notre analyse est faite d'abord pour l'équation d'onde scalaire où le contraste est sur le terme d'ordre inférieur de l'opérateur de Helmholtz. Nous esquissons ensuite l'extension aux cas où la le contraste est également présent dans l'opérateur principal. Nous complémentons notre travail par deux résultats sur l'analyse du problème de diffraction pour des matériaux périodiques ayant des indices négatifs. Nous établissons en premier le caractère bien posé du problème en 2D dans le cas d'un contraste est égal à -1. Nous montrons également le caractère Fredholm de la formulation Lipmann-Schwinger du problème en utilisant l'approche de T-coercivité dans le cas d'un contraste différent de -1We are interested in this thesis by the analysis of scattering and inverse scattering problems for locally perturbed periodic infinite layers at a fixed frequency. This problem has connexions with non destructive testings of periodic media like photonics structures, optical fibers, gratings, etc. We first analyze the forward scattering problem and establish some conditions under which there exist no guided modes. This type of conditions is important as it shows that measurements can be done on a layer above the structure without loosing substantial informations in the propagative part of the wave. We then propose a numerical method that solves the direct scattering problem based on Floquet-Bloch transform in the periodicity directions of the background media. We discretize the problem uniformly in the Floquet-Bloch variable and use a spectral method in the space variable. The discretization in space exploits a volumetric reformulation of the problem in a cell (Lippmann-Schwinger integral equation) and a periodization of the kernel in the direction orthogonal to the periodicity. The latter allows the use of FFT techniques to speed up Matrix-Vector product in an iterative to solve the linear system. One ends up with a system of coupled integral equations that can be solved using a Jacobi decomposition. The convergence analysis is done for the case with absorption and numerical validating results are conducted in 2D. For the inverse problem we extend the use of three sampling methods to solve the problem of retrieving the defect from the knowledge of mutistatic data associated with incident near field plane waves. We analyze these methods for the semi-discretized problem in the Floquet-Bloch variable. We then propose a new method capable of retrieving directly the defect without knowing either the background material properties nor the defect properties. This so-called differential-imaging functional that we propose is based on the analysis of sampling methods for a single Floquet-Bloch mode and the relation with solutions toso-called interior transmission problems. The theoretical investigations are corroborated with numerical experiments on synthetic data. Our analysis is done first for the scalar wave equation where the contrast is the lower order term of the Helmholtz operator. We then sketch the extension to the cases where the contrast is also present in the main operator. We complement our thesis with two results on the analysis of the scattering problem for periodic materials with negative indices. Weestablish the well posedness of the problem in 2D in the case of a contrast equals -1. We also show the Fredholm properties of the volume potential formulation of the problem using the T-coercivity approach in the case of a contrast different from -1
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Agaltsov, Alexey. "Reconstruction methods for inverse problems for Helmholtz-type equations." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLX099/document.
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La présente thèse est consacrée à l'étude de quelques problèmes inverses pour l'équation de Helmholtz jauge-covariante, dont des cas particuliers comprennent l'équation de Schrödinger pour une particule élémentaire chargée dans un champ magnétique et l'équation d'onde harmonique en temps qui décrive des ondes acoustiques dans un fluide en écoulement. Ces problèmes ont comme motivation des applications dans des tomographies différentes, qui comprennent la tomographie acoustique, la tomographie qui utilise des particules élémentaires et la tomographie d'impédance électrique. En particulier, nous étudions des problèmes inverses motivés par des applications en tomographie acoustique de fluide en écoulement. Nous proposons des formules et équations qui permettent de réduire le problème de tomographie acoustique à un problème de diffusion inverse approprié. En suivant, nous développons un algorithme fonctionnel-analytique pour la résolution de ce problème de diffusion inverse. Cependant, en général, la solution de ce problème n'est unique qu'à une transformation de jauge appropriée près. À cet égard, nous établissons des formules qui permettent de se débarrasser de cette non-unicité de jauge et retrouver des paramètres du fluide, en mesurant des ondes acoustiques à des plusieurs fréquences. Nous présentons également des exemples des fluides qui ne sont pas distinguable dans le cadre de tomographie acoustique considérée. En suivant, nous considérons le problème de diffusion inverse sans information de phase. Ce problème est motivé par des applications en tomographie qui utilise des particules élémentaires, où seulement le module de l'amplitude de diffusion peut être mesuré facilement. Nous établissons des estimations dans l'espace de configuration pour les reconstructions sans phase de type Borne, qui sont requises pour le développement des méthodes de diffusion inverse précises. Finalement, nous considérons le problème de détermination d'une surface de Riemann dans le plan projectif à partir de son bord. Ce problème survient comme une partie du problème de Dirichlet-Neumann inverse pour l'équation de Laplace sur une surface inconnue, qui est motivé par des applications en tomographie d'impédance électriqueThis work is devoted to study of some inverse problems for the gauge-covariant Helmholtz equation, whose particular cases include the Schrödinger equation for a charged elementary particle in a magnetic field and the time-harmonic wave equation describing sound waves in a moving fluid. These problems are mainly motivated by applications in different tomographies, including acoustic tomography, tomography using elementary particles and electrical impedance tomography. In particular, we study inverse problems motivated by applications in acoustic tomography of moving fluid. We present formulas and equations which allow to reduce the acoustic tomography problem to an appropriate inverse scattering problem. Next, we develop a functional-analytic algorithm for solving this inverse scattering problem. However, in general, the solution to the latter problem is unique only up to an appropriate gauge transformation. In this connection, we give formulas and equations which allow to get rid of this gauge non-uniqueness and recover the fluid parameters, by measuring acoustic fields at several frequencies. We also present examples of fluids which are not distinguishable in this acoustic tomography setting. Next, we consider the inverse scattering problem without phase information. This problem is motivated by applications in tomography using elementary particles, where only the absolute value of the scattering amplitude can be measured relatively easily. We give estimates in the configuration space for the phaseless Born-type reconstructions, which are needed for the further development of precise inverse scattering algorithms. Finally, we consider the problem of determination of a Riemann surface in the complex projective plane from its boundary. This problem arises as a part of the inverse Dirichlet-to-Neumann problem for the Laplace equation on an unknown 2-dimensional surface, and is motivated by applications in electrical impedance tomography
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Tian, Wenyi. "Numerical study on some inverse problems and optimal control problems." HKBU Institutional Repository, 2015. https://repository.hkbu.edu.hk/etd_oa/193.
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In this thesis, we focus on the numerical study on some inverse problems and optimal control problems. In the first part, we consider some linear inverse problems with discontinuous or piecewise constant solutions. We use the total variation to regularize these inverse problems and then the finite element technique to discretize the regularized problems. These discretized problems are treated from the saddle-point perspective; and some primal-dual numerical schemes are proposed. We intensively investigate the convergence of these primal-dual type schemes, establishing the global convergence and estimating their worst-case convergence rates measured by the iteration complexity. We test these schemes by some experiments and verify their efficiency numerically. In the second part, we consider the finite difference and finite element discretization for an optimal control problem which is governed by time fractional diffusion equation. The prior error estimate of the discretized model is analyzed, and a projection gradient method is applied for iteratively solving the fully discretized surrogate. Some numerical experiments are conducted to verify the efficiency of the proposed method. Overall speaking, the thesis has been mainly inspired by some most recent advances developed in optimization community, especially in the area of operator splitting methods for convex programming; and it can be regarded as a combination of some contemporary optimization techniques with some relatively mature inverse and control problems. Keywords: Total variation minimization, linear inverse problem, saddle-point problem, finite element method, primal-dual method, convergence rate, optimal control problem, time fractional diffusion equation, projection gradient method.
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Flemming, Jens. "Quadratic Inverse Problems and Sparsity Promoting Regularization." Doctoral thesis, Universitätsbibliothek Chemnitz, 2018. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-232402.
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Ill-posed inverse problems with quadratic structure are introduced, studied and solved. As an example an inverse problem appearing in laser optics is solved numerically based on a new regularized inversion algorithm. In addition, the theory of sparsity promoting regularization is extended to situations in which sparsity cannot be expected and also to equations with non-injective operators.
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Lafargue, Thomas. "Comportement de matériaux illuminés par des sources laser multi-kW." Electronic Thesis or Diss., Paris, HESAM, 2023. http://www.theses.fr/2023HESAE021.
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Résumé : Ce sujet de thèse se place dans le cadre d’un projet à moyen/long terme derecherche et développement d’armes laser de fortes puissances. En effet, suite à la diversificationdes menaces (cibles/modes opératoires) et aux risques futurs d’illuminationslaser, MBDA France, en coopération avec ALPhANOV, développe des essais autour d’unlaser continu haute puissance. L’objectif final de MBDA France est de pouvoir quantifierla vulnérabilité de cibles. Compte tenu de la variété des familles de matériaux d’intérêt,les mécanismes de dégradation intervenants dans ce type d’applications sont nombreux(fort couplage thermique, mécanique et chimique). De plus, ils sont peu documentés dufait de la rareté de moyens d’essais équipés de laser continu multi-kW. Le travail doctoralpermettront d’acquérir de la connaissance et de la compréhension de ces phénomènesd’interaction laser-matière grâce à des approches expérimentales et analyses numériquesThis thesis subject is part of a medium/long-term project for the researchand development of high-power laser weapons. Indeed, following the diversification ofthreats (targets/operating modes) and the future risks of laser illuminations, MBDA France,in cooperation with ALPhANOV, is developing trials around a high-power continuouslaser. The final objective of MBDA France is to be able to quantify the vulnerability oftargets. Given the variety of families of materials of interest, the degradation mechanismsinvolved in this type of application are numerous (strong thermal, mechanical and chemicalcoupling). In addition, they are poorly documented due to the scarcity of test facilitiesequipped with multi-kW continuous lasers. The doctoral work will make it possible toacquire knowledge and understanding of these laser-matter interaction phenomena thanksto experimental approaches and numerical analyzes
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Szasz, Teodora. "Advanced beamforming techniques in ultrasound imaging and the associated inverse problems." Thesis, Toulouse 3, 2016. http://www.theses.fr/2016TOU30221/document.
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L'imagerie ultrasonore (US) permet de réaliser des examens médicaux non invasifs avec des méthodes d'acquisition rapides à des coûts modérés. L'imagerie cardiaque, abdominale, fœtale, ou mammaire sont quelques-unes des applications où elle est largement utilisée comme outil de diagnostic. En imagerie US classique, des ondes acoustiques sont transmises à une région d'intérêt du corps humain. Les signaux d'écho rétrodiffusés, sont ensuite formés pour créer des lignes radiofréquences. La formation de voies (FV) joue un rôle clé dans l'obtention des images US, car elle influence la résolution et le contraste de l'image finale. L'objectif de ce travail est de modéliser la formation de voies comme un problème inverse liant les données brutes aux signaux RF. Le modèle de formation de voies proposé ici améliore le contraste et la résolution spatiale des images échographiques par rapport aux techniques de FV existants. Dans un premier temps, nous nous sommes concentrés sur des méthodes de FV en imagerie US. Nous avons brièvement passé en revue les techniques de formation de voies les plus courantes, en commencent par la méthode par retard et somme standard puis en utilisant les techniques de formation de voies adaptatives. Ensuite, nous avons étudié l'utilisation de signaux qui exploitent une représentation parcimonieuse de l'image US dans le cadre de la formation de voies. Les approches proposées détectent les réflecteurs forts du milieu sur la base de critères bayésiens. Nous avons finalement développé une nouvelle façon d'aborder la formation de voies en imagerie US, en la formulant comme un problème inverse linéaire liant les échos réfléchis au signal final. L'intérêt majeur de notre approche est la flexibilité dans le choix des hypothèses statistiques sur le signal avant la formation de voies et sa robustesse dans à un nombre réduit d'émissions. Finalement, nous présentons une nouvelle méthode de formation de voies pour l'imagerie US basée sur l'utilisation de caractéristique statistique des signaux supposée alpha-stableUltrasound (US) allows non-invasive and ultra-high frame rate imaging procedures at reduced costs. Cardiac, abdominal, fetal, and breast imaging are some of the applications where it is extensively used as diagnostic tool. In a classical US scanning process, short acoustic pulses are transmitted through the region-of-interest of the human body. The backscattered echo signals are then beamformed for creating radiofrequency(RF) lines. Beamforming (BF) plays a key role in US image formation, influencing the resolution and the contrast of final image. The objective of this thesis is to model BF as an inverse problem, relating the raw channel data to the signals to be recovered. The proposed BF framework improves the contrast and the spatial resolution of the US images, compared with the existing BF methods. To begin with, we investigated the existing BF methods in medical US imaging. We briefly review the most common BF techniques, starting with the standard delay-and-sum BF method and emerging to the most known adaptive BF techniques, such as minimum variance BF. Afterwards, we investigated the use of sparse priors in creating original two-dimensional beamforming methods for ultrasound imaging. The proposed approaches detect the strong reflectors from the scanned medium based on the well-known Bayesian Information Criteria used in statistical modeling. Furthermore, we propose a new way of addressing the BF in US imaging, by formulating it as a linear inverse problem relating the reflected echoes to the signal to be recovered. Our approach offers flexibility in the choice of statistical assumptions on the signal to be beamformed and it is robust to a reduced number of pulse emissions. At the end of this research, we investigated the use of the non-Gaussianity properties of the RF signals in the BF process, by assuming alpha-stable statistics of US images
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Okamoto, Kei. "Optimal numerical methods for inverse heat conduction and inverse design solidification problems." Online access for everyone, 2005. http://www.dissertations.wsu.edu/Dissertations/Fall2005/k%5Fokamoto%5F120905.pdf.
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Rullgård, Hans. "Topics in geometry, analysis and inverse problems." Doctoral thesis, Stockholm University, Department of Mathematics, 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-15.
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The thesis consists of three independent parts.
Part I: Polynomial amoebas
We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1.
Part II: Differential equations in the complex plane
We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform.
Part III: Radon transforms and tomography
This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.
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Choi, Kerkil. "Minimum I-divergence Methods for Inverse Problems." Diss., Georgia Institute of Technology, 2005. http://hdl.handle.net/1853/7543.
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Problems of estimating nonnegative functions from nonnegative data induced by nonnegative mappings are ubiquitous in science and engineering. We address such problems by minimizing an information-theoretic discrepancy measure, namely Csiszar's I-divergence, between the collected data and hypothetical data induced by an estimate.
Our applications can be summarized along the following three lines:
1) Deautocorrelation: Deautocorrelation involves
recovering a function from its autocorrelation. Deautocorrelation can be interpreted as phase retrieval in that recovering a function from its autocorrelation is equivalent to retrieving
Fourier phases from just the corresponding Fourier magnitudes.
Schulz and Snyder invented an minimum I-divergence algorithm for phase retrieval. We perform a numerical study concerning the convergence of their algorithm to local minima.
X-ray crystallography is a method for finding the interatomic structure of a crystallized molecule. X-ray crystallography problems can be viewed as deautocorrelation problems from aliased autocorrelations, due to the periodicity of the crystal structure. We derive a modified version of the Schulz-Snyder algorithm for application to crystallography. Furthermore, we prove that our tweaked version can theoretically preserve special
symmorphic group symmetries that some crystals possess.
We quantify noise impact via several error metrics as the signal-to-ratio changes.
Furthermore, we propose penalty methods using Good's roughness and total variation for alleviating roughness in estimates caused by
noise.
2) Deautoconvolution: Deautoconvolution involves finding a function from its autoconvolution.
We derive an iterative algorithm that attempts to recover a function from its autoconvolution via minimizing I-divergence. Various theoretical properties of our deautoconvolution algorithm are
derived.
3) Linear inverse problems: Various linear inverse
problems can be described by the Fredholm integral equation of the first kind. We address two such problems via minimum I-divergence
methods, namely the inverse blackbody radiation problem, and the problem of estimating an input distribution to a communication channel
(particularly Rician channels) that would create a desired output.
Penalty methods are proposed for dealing with the ill-posedness of the inverse blackbody problem.
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Tsering-xiao, Basang. "Electromagnetic inverse problems for nematic liquid crystals." Thesis, University of Oxford, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.540262.
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Berntsson, Fredrik. "Numerical methods for inverse heat conduction problems /." Linköping : Univ, 2001. http://www.bibl.liu.se/liupubl/disp/disp2001/tek723s.pdf.
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Rullgård, Hans. "Topics in geometry, analysis and inverse problems /." Stockholm : Matematiska institutionen, Univ, 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-15.
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Jalalzai, Khalid. "Regularization of inverse problems in image processing." Phd thesis, Ecole Polytechnique X, 2012. http://pastel.archives-ouvertes.fr/pastel-00787790.
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Les problèmes inverses consistent à retrouver une donnée qui a été transformée ou perturbée. Ils nécessitent une régularisation puisque mal posés. En traitement d'images, la variation totale en tant qu'outil de régularisation a l'avantage de préserver les discontinuités tout en créant des zones lisses, résultats établis dans cette thèse dans un cadre continu et pour des énergies générales. En outre, nous proposons et étudions une variante de la variation totale. Nous établissons une formulation duale qui nous permet de démontrer que cette variante coïncide avec la variation totale sur des ensembles de périmètre fini. Ces dernières années les méthodes non-locales exploitant les auto-similarités dans les images ont connu un succès particulier. Nous adaptons cette approche au problème de complétion de spectre pour des problèmes inverses généraux. La dernière partie est consacrée aux aspects algorithmiques inhérents à l'optimisation des énergies convexes considérées. Nous étudions la convergence et la complexité d'une famille récente d'algorithmes dits Primal-Dual.
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Johnston, John C. "Bayesian analysis of inverse problems in physics." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.337737.
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Leung, Wun Ying Valerie. "Inverse problems in astronomical and general imaging." Thesis, University of Canterbury. Electrical and Computer Engineering, 2002. http://hdl.handle.net/10092/7513.
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The resolution and the quality of an imaged object are limited by four contributing factors. Firstly, the primary resolution limit of a system is imposed by the aperture of an instrument due to the effects of diffraction. Secondly, the finite sampling frequency, the finite measurement time and the mechanical limitations of the equipment also affect the resolution of the images captured. Thirdly, the images are corrupted by noise, a process inherent to all imaging systems. Finally, a turbulent imaging medium introduces random degradations to the signals before they are measured. In astronomical imaging, it is the atmosphere which distorts the wavefronts of the objects, severely limiting the resolution of the images captured by ground-based telescopes. These four factors affect all real imaging systems to varying degrees.
All the limitations imposed on an imaging system result in the need to deduce or reconstruct the underlying object distribution from the distorted measured data. This class of problems is called inverse problems. The key to the success of solving an inverse problem is the correct modelling of the physical processes which give rise to the corresponding forward problem. However, the physical processes have an infinite amount of information, but only a finite number of parameters can be used in the model. Information loss is therefore inevitable. As a result, the solution to many inverse problems requires additional information or prior knowledge. The application of prior information to inverse problems is a recurrent theme throughout this thesis.
An inverse problem that has been an active research area for many years is interpolation, and there exist numerous techniques for solving this problem. However, many of these techniques neither account for the sampling process of the instrument nor include prior information in the reconstruction. These factors are taken into account in the proposed optimal Bayesian interpolator. The process of interpolation is also examined from the point of view of superresolution, as these processes can be viewed as being complementary.
Since the principal effect of atmospheric turbulence on an incoming wavefront is a phase distortion, most of the inverse problem techniques devised for this seek to either estimate or compensate for this phase component. These techniques are classified into computer post-processing methods, adaptive optics (AO) and hybrid techniques.
Blind deconvolution is a post-processing technique which uses the speckle images to estimate both the object distribution and the point spread function (PSF), the latter of which is directly related to the phase. The most successful approaches are based on characterising the PSF as the aberrations over the aperture. Since the PSF is also dependent on the atmosphere, it is possible to constrain the solution using the statistics of the atmosphere. An investigation shows the feasibility of this approach. Bispectrum is also a post-processing method which reconstructs the spectrum of the object. The key component for phase preservation is the property of phase closure, and its application as prior information for blind deconvolution is examined.
Blind deconvolution techniques utilise only information in the image channel to estimate the phase which is difficult. An alternative method for phase estimation is from a Shack-Hartmann (SH) wavefront sensing channel. However, since phase information is present in both the wavefront sensing and the image channels simultaneously, both of these approaches suffer from the problem that phase information from only one channel is used. An improved estimate of the phase is achieved by a combination of these methods, ensuring that the phase estimation is made jointly from the data in both the image and the wavefront sensing measurements. This formulation, posed as a blind deconvolution framework, is investigated in this thesis. An additional advantage of this approach is that since speckle images are imaged in a narrowband, while wavefront sensing images are captured by a charge-coupled device (CCD) camera at all wavelengths, the splitting of the light does not compromise the light level for either channel. This provides a further incentive for using simultaneous data sets.
The effectiveness of using Shack-Hartmann wavefront sensing data for phase estimation relies on the accuracy of locating the data spots. The commonly used method which calculates the centre of gravity of the image is in fact prone to noise and is suboptimal. An improved method for spot location based on blind deconvolution is demonstrated.
Ground-based adaptive optics (AO) technologies aim to correct for atmospheric turbulence in real time. Although much success has been achieved, the space- and time-varying nature of the atmosphere renders the accurate measurement of atmospheric properties difficult. It is therefore usual to perform additional post-processing on the AO data. As a result, some of the techniques developed in this thesis are applicable to adaptive optics.
One of the methods which utilise elements of both adaptive optics and post-processing is the hybrid technique of deconvolution from wavefront sensing (DWFS). Here, both the speckle images and the SH wavefront sensing data are used. The original proposal of DWFS is simple to implement but suffers from the problem where the magnitude of the object spectrum cannot be reconstructed accurately. The solution proposed for overcoming this is to use an additional set of reference star measurements. This however does not completely remove the original problem; in addition it introduces other difficulties associated with reference star measurements such as anisoplanatism and reduction of valuable observing time. In this thesis a parameterised solution is examined which removes the need for a reference star, as well as offering a potential to overcome the problem of estimating the magnitude of the object.
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Bernauer, Moritz. "Reducing non-uniqueness in seismic inverse problems." Diss., Ludwig-Maximilians-Universität München, 2014. http://nbn-resolving.de/urn:nbn:de:bvb:19-171638.
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The scientific investigation of the solid Earth's complex processes, including their interactions with the oceans and the atmosphere, is an interdisciplinary field in which seismology has one key role. Major contributions of modern seismology are (1) the development of high-resolution tomographic images of the Earth's structure and (2) the investigation of earthquake
source processes. In both disciplines the challenge lies in solving a seismic inverse
problem, i.e. in obtaining information about physical parameters that are not directly observable.
Seismic inverse studies usually aim to find realistic models through the minimization of the
misfit between observed and theoretically computed (synthetic) ground motions. In general, this approach depends on the numerical simulation of seismic waves propagating in a specified Earth model (forward problem) and the acquisition of illuminating data. While the former is routinely solved using spectral-element methods, many seismic inverse problems still suffer from the lack of information typically leading to ill-posed inverse problems with multiple solutions and trade-offs between the model parameters. Non-linearity in forward modeling and the non-convexity of misfit functions aggravate
the inversion for structure and source.
This situation requires an efficient exploitation of the available data. However, a careful analysis of whether individual models can be considered a reasonable approximation of the true solution (deterministic approach) or if single models should be replaced with statistical distributions of model parameters (probabilistic or Bayesian approach) is inevitable.
Deterministic inversion attempts to find the model that provides the best explanation of the data, typically using iterative optimization techniques. To prevent the inversion process from being trapped in a meaningless local minimum an accurate initial low frequency model is indispensable. Regularization, e.g. in terms of smoothing or damping, is necessary to avoid artifacts from the mapping of high frequency information. However, regularization increases parameter trade-offs and is subjective to some degree, which means that resolution estimates tend to be biased.
Probabilistic (or Bayesian) inversions overcome the drawbacks of the deterministic approach by using a global model search that provides unbiased measures of resolution and trade-offs. Critical aspects are computational costs, the appropriate incorporation of prior knowledge and the difficulties in interpreting and processing
the results.
This work studies both the deterministic and the probabilistic approach. Recent observations of rotational ground motions, that complement translational ground motion measurements from conventional seismometers, motivated the research.
It is investigated if alternative seismic observables, including rotations and dynamic strain, have the potential to reduce non-uniqueness and parameter trade-offs in seismic inverse problems.
In the framework of deterministic full waveform inversion a novel approach to seismic tomography
is applied for the first time to (synthetic) collocated measurements of translations, rotations and strain. The concept is based on the definition of new observables combining translation and rotation, and translation and strain measurements, respectively. Studying the corresponding sensitivity kernels assesses the capability of the new observables to constrain various aspects of a three-dimensional Earth structure. These observables are generally sensitive only to small-scale near-receiver structures. It follows, for example, that knowledge of deeper Earth structure are not required in tomographic inversions for local structure based on the new observables.
Also in the context of deterministic full waveform inversion a new method for the design of seismic observables with focused sensitivity to a target model parameter class, e.g. density structure, is developed. This is achieved through the optimal linear combination of fundamental observables that can be any scalar measurement extracted from seismic recordings. A series of examples illustrate that the resulting optimal observables are able to minimize inter-parameter trade-offs that result from regularization in ill-posed multi-parameter inverse problems.
The inclusion of alternative and the design of optimal observables in seismic tomography also affect more general objectives in geoscience. The investigation of the history and the dynamics of tectonic plate motion benefits, for example, from the detailed knowledge of small-scale heterogeneities in the crust and the upper mantle. Optimal observables focusing on density help to independently constrain the Earth's temperature and composition and provide information on convective flow.
Moreover, the presented work analyzes for the first time if the inclusion of rotational ground motion measurements enables a more detailed description of earthquake source processes. The complexities of earthquake rupture suggest a probabilistic (or Bayesian) inversion approach. The results of the synthetic study indicate that the incorporation of rotational ground motion recordings can significantly reduce the non-uniqueness in finite source inversions, provided that measurement uncertainties are similar to or below the uncertainties of translational velocity recordings. If this
condition is met, the joint processing of rotational and translational ground motion provides more detailed information about earthquake dynamics, including rheological fault properties and friction law parameters. Both are critical e.g. for the reliable assessment of seismic hazards.
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Tregidgo, Henry. "Inverse problems and control for lung dynamics." Thesis, University of Manchester, 2018. https://www.research.manchester.ac.uk/portal/en/theses/inverse-problems-and-control-for-lung-dynamics(0f3224e6-7449-4417-bd2b-8e48ec88e2bf).html.
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Mechanical ventilation is vital for the treatment of patients in respiratory intensive care and can be life saving. However, the risks of regional pressure gradients and over-distension must be balanced with the need to maintain function. For these reasons mechanical ventilation can benefit from the regional information provided by bedside imaging such as electrical impedance tomography (EIT). In this thesis we develop and test methods to retrieve clinically meaningful measures of lung function from EIT and examine the feasibility of closing the feedback loop to enable EIT-guided control of mechanical ventilation. Working towards this goal we develop a reconstruction algorithm capable of providing fast absolute values of conductivity from EIT measurements. We couple the resulting conductivity time series to a compartmental ordinary differential equation (ODE) model of lung function in order to recover regional parameters of elastance and airway resistance. We then demonstrate how these parameters may be used to generate optimised pressure controls for mechanical ventilation that expose the lungs to minimal gradients of pressure and are stable with respect to EIT measurement errors. The EIT reconstruction algorithm we develop is capable of producing low dimensional absolute values of conductivity in real time after a limited additional setup time. We show that this algorithm retains the ability to give fast feedback on regional lung changes. We also describe methods of improving computational efficiency for general Gauss-Newton type EIT algorithms. In order to couple reconstructed conductivity time series to our ODE model we describe and test the recovery of regional ventilation distributions through a process of regularised differentiation. We prove that the parameters of our ODE model are recoverable from these ventilation distributions apart from the degenerate case where all compartments have the same parameters. We then test this recovery process under varying levels of simulated EIT measurement and modelling errors. Finally we examine the ODE lung model using control theory. We prove that the ODE model is controllable for a wide range of parameter values and link controllability to observable ventilation patterns in the lungs. We demonstrate the generation and optimisation of pressure controls with minimal time gradients and provide a bound on the resulting magnitudes of these pressures. We then test the control generation process using ODE parameter values recovered through EIT simulations at varying levels of measurement noise. Through this work we have demonstrated that EIT reconstructions can be of benefit to the control of mechanical ventilation.