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 réelle. / Inverse 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.

[pt] Este trabalho apresenta inicialmente uma revisão bibliográfica sobre a aplicabilidade da teoria da equação de onda e fórmulas dinâmicas na estimativa da capacidade de carga em estacas. Como parte experimental deste trabalho, reuniram-se diversos relatórios de resultados de ensaios de carregamento dinâmico (CAPWAP) realizados em 29 estacas do tipo: Hélice contínua, raiz, estacas pré-fabricadas de concreto armado e metálicas. Foi analisada a eficiência da energia transferida (ETH), a máxima tensão de cravação gerada durante o ensaio, a resistência máxima mobilizada, a integridade da estaca assim como a influência do damping de amortecimento (Jc). Além disso, foram utilizadas as chamadas fórmulas dinâmicas para calcular a previsão da resistência mobilizada das estacas, cujos resultados foram comparados com os valores do CAPWAP. Estas fórmulas dinâmicas analisadas foram: A fórmula de Weisbach, Hiley, Dinamarqueses, Janbu, Brix, Holandeses e Engineering News Record. Com o propósito de validar uma possível aplicabilidade das fórmulas dinâmicas em estacas escavadas, foi calculada a previsão da resistência mobilizada seguindo o mesmo procedimento das estacas cravadas. Estes resultados foram optimizados comparando-os com os valores do CAPWAP, através de uma análise estatística simples. Neste estudo constatou-se que através da revisão dos coeficientes (incluindo a aplicação da ferramenta solver para a fórmula de Hiley), que as fórmulas dinamarquesas, Hiley e Weisbach apresentam uma melhor aproximação com os valores do CAPWAP para o caso de estacas cravadas, ao mesmo tempo em que para o caso de estacas escavadas a previsão obteve valores dispersos. / [en] This work presents, initially a review about the theoretical basis of the applicability of the wave equation and of dynamic formulas in the estimate of load capacity in piles. As experimental part of this work, it was used, results obtained from twenty nine dynamic loading tests (CAPWAP), carried in out on several types of piles: Continuous flight auger, grout intruded in place; reinforced concrete and steel piles. Was analyzed the efficiency of the energy transferred (ETH), the maximum value of the tension, the maximum value of the mobilized pile resistance, the integrity of the pile and the influence of damping (Jc). Furthermore, dynamic formulas were used to calculate the prediction of the resistance mobilized of the piles and the results were compared with the values of the CAPWAP. These dynamic formulas were analyzed: The formula of Weisbach, Hiley, Danes, Janbu, Brix, Holandeses and Engineering News Record. For the purpose to validate a possible applicability of dynamic formulas in the prediction of bored piles, mobilized resistance was calculated following the same procedure of driven piles. These results were optimized by comparing them with the values of CAPWAP through a statistical analysis simple. In this study it was found that by reviewing the coefficients (including the application of the tool solver for the Hiley formula), that the Danish, Hiley and Weisbach formulas have a better approximation to the CAPWAP values for the case of piles driven, at the same time as in the case of bored piles prediction values obtained dispersed.

Laplacian differential operator -- Vibrations of plucked strings and Hollow cylinders.

On considère dans cette thèse l'équation des ondes posée sur un domaine $\Omega$ supposé régulier. Si $\Gamma$ désigne une surface supposée observable, on peut définir la constante d'observabilité associée à $\Gamma$. L'intérêt de cette constante est de rendre compte de la qualité de la reconstruction dans le problème inverse qui consiste à reconstruire les données initiales à partir de la mesure de la solution sur $\Gamma$. Ainsi l'étude de cette constante s'applique entre autres à la détermination de la forme et du placement optimaux de capteurs, pour la mesure de toute sorte de phénomènes ondulatoires. Le but du premier chapire est de caractériser de manière théorique les domaines $\Gamma$ de surface prescrite qui maximisent cette constante d'observabilité, ou plus exactement une version "randomizée" de ce critère. Dans le second chapitre il s'agit d'appliquer les résulats obtenus au placement optimal de capteurs pour la tomographie photoacoustique. La tomographie photoacoustique est un procédé d'imagerie médicale ultra-sonore, non invasif encore peu développé qui est une alternative précise et plus économique à l'imagerie X. C'est dans ce cadre que l'on propose une modélisation de l'influence de la forme et de la disposition des capteurs dans le problème de reconstruction de la densité des tissus. Plus particulièrement, il s'agira de construire une fonctionnelle de la forme des capteurs, rendant compte de la qualité de l'image obtenue. / In the first part of this thesis, we consider the wave equation on a regular bounded domain $\Omega$. We investigate the problem of optimizing, in some appropriate sense, the shape and location of sensors spread on an arbitrary measurable subdomain $\Gamma$ of the boundary of $\Omega$. We introduce a spectral quantity called randomized observability constant, corresponding to the best constant in an average of the classical observability inequality, over random initial data. The pupose of the first chapter is to investigate optimal domains, maximizing the new objective function. The second part consists in applying the previous results to medical imaging, and more precisely to photoacoustic tomography. This imaging technique, constitutes a cutting-edge technology that has drawn considerable attention in the medical imaging area. Firstly because it is non-ionizing and non-invasive, and also because it constitutes a precise and cheap alternative to X imaging. In this framework, we propose here to model the influence of the shape and position of sensors in the inverse problem consisting in the reconstruction of the imaged body. In a nutshell, we build a functional of the shape of the sensors, providing an account for the reconstructed image quality.

No description available.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第22863号 / 情博第742号 / 新制||情||127(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 矢ヶ崎 一幸, 教授 中村 佳正, 准教授 柴山 允瑠, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

The discretization of time-dependent wave propagation is plagued with dispersion in which the wavefield is perceived to travel with an erroneous velocity. To remediate the problem, simulations are run on dense and computationally expensive grids yielding plausible approximate solutions. This work introduces an error analysis tool which can be used to obtain optimal simulation parameters that account for mesh size, orders of spatial and temporal discretizations, angles of propagation, temporal stability conditions (usually referred to as CFL conditions), and time of propagation. The classical criteria of 10-15 nodes per wavelength for second-order finite differences, and 4-5 nodes per wavelength for fourth-order spectral elements are shown to be unrealistic and overly-optimistic simulation parameters for different propagation times. This work analyzes finite differences, spectral elements, optimally-blended spectral elements, and isogeometric analysis.

I present a general methodology for inverting seismic data in either the data or image domains. It partially overcomes one of the most serious problems with current waveform inversion methods, which is the tendency to converge to models far from the actual one. The key idea is to develop a multiscale misfit function that is composed of both a simplified version of the data and one associated with the complex part of the data. Misfit functions based on simple data are characterized by many fewer local minima so that a gradient optimization method can make quick progress in getting to the general vicinity of the actual model. Once we are near the actual model, we then use the gradient based on the more complex data. Below, we describe two implementations of this multiscale strategy: wave equation traveltime inversion in the data domain and generalized differential semblance optimization in the image domain. • Wave Equation Traveltime Inversion in the Data Domain (WT): The main difficulty with iterative waveform inversion is that it tends to get stuck in local minima associated with the waveform misfit function. To mitigate this problem and avoid the need to fit amplitudes in the data, we present a waveequation method that inverts the traveltimes of reflection events, and so is less prone to the local minima problem. Instead of a waveform misfit function, the penalty function is a crosscorrelation of the downgoing direct wave and the upgoing reflection wave at the trial image point. The time lag which maximizes the crosscorrelation amplitude represents the reflection-traveltime residual that is back-projected along the reflection wavepath to update the velocity. Shot- and angle-domain crosscorrelation functions are introduced to estimate the reflection-traveltime residual by semblance analysis and scanning. In theory, only the traveltime information is inverted and there is no need to precisely fit the amplitudes or assume a high-frequency approximation. Results with both synthetic data and field records reveal both the benefits and limitations of WT. • Generalized Differental Semblance Optimization in the Image Domain (GDSO): We now extend the multiscale physics approach to differential semblance optimization (DSO) in the image domain. That is, we identify the space-lag offset H(x, z, h) in the subsurface-offset domain as an implicit function of velocity. It describes the smoothly varying moveout H(x, z, h) of the migration image m(x, z, h) in the subsurface-offset domain, which is analogous to the smoothly varying traveltime residual ∆τ(x) of a reflection event in a shot gather. The velocity model is found that minimizes the objective function ∑x,z,h H(x, z, h)2m(x, z, h)2, where coherent noise is eliminated everywhere except along the picked curve H(x, z, h). This method is denoted as generalized DSO (GDSO) and mitigates the coherent noise problem with DSO. Numerical examples are presented that empirically demonstrate its effectiveness in providing more accurate velocity models compared to conventional DSO.

The aim of the work is to study the electromagnetic field theory, finite element method and the interaction of electromagnetic field with tissues. Gained knowledge is then used to calculate spreading of the electromagnetic field in the microwave field and to create a temperature profile of spreading the electromagnetic fields in human tissue. The finite element method was implemented in the Matlab programming environment, where the 1D model was created in the frequency and time domain and a simple 2D model created in time domain. The program was developed to analyze spreading electromagnetic wave. Another part of work was done in the programming environment of COMSOL Multiphysics. In this case was the human leg exposed to electromagnetic fields. The analysis determined the changes of temperature in these biological tissues for six minutes.

This thesis concerns the applications and analysis of the Heterogeneous Multiscale methods (HMM) for Multiscale Elliptic and Hyperbolic Partial Differential Equations. We have gathered the main contributions in two papers. The first paper deals with the cell-boundary error which is present in multi-scale algorithms for elliptic homogenization problems. Typical multi-scale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. Solving the micro model requires, on the other hand, imposing boundary conditions on the boundary of the microscopic domain. Imposing a naive boundary condition leads to $O(\varepsilon/\eta)$ error in the computation, where $\varepsilon$ is the size of the microscopic variations in the media and $\eta$ is the size of the micro-domain. Until now, strategies were proposed to improve the convergence rate up to fourth-order in $\varepsilon/\eta$ at best. However, the removal of this error in multi-scale algorithms still remains an important open problem. In this paper, we present an approach with a time-dependent model which is general in terms of dimension. With this approach we are able to obtain $O((\varepsilon/\eta)^q)$ and $O((\varepsilon/\eta)^q  + \eta^p)$ convergence rates in periodic and locally-periodic media respectively, where $p,q$ can be chosen arbitrarily large.      In the second paper, we analyze a multi-scale method developed under the Heterogeneous Multi-Scale Methods (HMM) framework for numerical approximation of wave propagation problems in periodic media. In particular, we are interested in the long time $O(\varepsilon^{-2})$ wave propagation. In the method, the microscopic model uses the macro solutions as initial data. In short-time wave propagation problems a linear interpolant of the macro variables can be used as the initial data for the micro-model. However, in long-time multi-scale wave problems the linear data does not suffice and one has to use a third-degree interpolant of the coarse data to capture the $O(1)$ dispersive effects apperaing in the long time. In this paper, we prove that through using an initial data consistent with the current macro state, HMM captures this dispersive effects up to any desired order of accuracy in terms of $\varepsilon/\eta$. We use two new ideas, namely quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic hyperbolic PDEs. As a byproduct, these ideas naturally reveal the role of consistency for high accuracy approximation of homogenized quantities. / <p>QC 20130926</p>