Ultrasound-modulated optical tomography is modeled by a linear integral equation and an inverse problem involving a diffusion equation in n spatial dimensions, n=2, 3. Based on measured data, the optical absorption coefficient μ is reconstructed inside of a given domain. We make a two-step mathematical model. First, we solve a linear integral equation. Assuming the energy fluence rate has been recovered from the previous equation, the absorption coefficient μ is then reconstructed by solving an inverse problem. Numerical experiments are presented for the case n=2. Two methods are used for the numerical experiments, gradient descent and levelset. At the end, advantages and disadvantages of those two methods are mentioned.

La pharmacognosie, science multidisciplinaire étudiant les produits naturels, n'a eu de cesse de s'adapter et d'évoluer parallèlement aux autres disciplines, en particulier les techniques d'analyse chimique et les technologies de l'information. De nouvelles approches complémentaires, comme la pharmacognosie inverse allant des molécules vers les organismes sources, un cheminement en sens inverse de la pharmacognosie, ont été développées. Les données nécessaires à ces études n'ont jamais été aussi nombreuses et hétérogènes. Molécules, activités, cibles, organismes sources et leurs usages traditionnels, sont autant d'informations essentielles à la pharmacognosie. Le but de cette thèse a été de développer une plateforme informatique, regroupant toutes ces informations diverses au sein d'une vaste base de données, accompagnée d'une interface web dédiée et d'outils de chémoinformatique et de fouille de données. Parallèlement, un enrichissement constant de cette base a été réalisé. Cette plateforme permet de rechercher et de consulter les informations, mais également de les croiser afin d'en faire émerger de nouvelles connaissances. Réellement optimisée pour la pharmacognosie, elle simplifie et améliore l'étude des produits naturels en permettant une approche sur de multiples niveaux (ethnopharmacologie, botanique, chimie), offrant de nouvelles possibilités d'interaction entre les données de ces différents domaines comme l'organism hopping (sauts entre organismes), une approche originale de comparaison des organismes selon leurs profils chimiques. / Pharmacognosy, a multidisciplinary science which studies natural products, has greatly evolved in parallel with other disciplines particularly analytical chemistry and information technologies. New complementary approaches like reverse pharmacognosy, from molecules to source organisms (in contrary of pharmacognosy), have been developed. Data necessary for these studies are overwhelming and heterogeneous. Molecules, activities, targets, organisms, traditional uses, are some of essential information for pharmacognosy. The main goal of this thesis was to develop an information platform, including a large database, to collect the diverse data, along with a specific web interface and also mining and cheminformatics tools. A constant enrichment of this database has also been realized. This platform allows to query and browse data, but also to cross them in order to discover new knowledge. Truely optimised for pharmacognosy, it makes easier and enhances the study of natural products, due to a multi-level approach (ethnopharmacology, botany, chemistry). It offers new possibilities of interaction between data from different domains, such as organism hopping, a new concept of organism comparisons according to their chemical profiles.

Dans le problème inverse d’électrocardiographie, le cible est faire la reconstruction de l’activité électrophysiologique dans le cœur sans mesurer directement dans sa surface (sans interventions avec cathéter). Il est important remarque que en l’actualité la solution numérique du problème inverse est résolu avec le modèle quasi-statique. Ce modèle ne considère pas la dynamique du cœur et peut produire des erreurs dans la reconstruction de la solution sur la surface du cœur. Dans cette thèse, différents méthodologies était investigue pour résoudre le problème inverse d’électrocardiographie comme intelligence artificielle, et modèles dynamiques limites. Aussi, les effets de différents opérateurs en utilisant méthodes d’éléments de frontière , et méthodes d’élément finis était investigue. / In the inverse problem of electrocardiography, the target is to make the reconstruction of electrophysiological activity in the heart without measuring directly in its surface (without interventions with catheter). It is important to note that the current numerical solution of the inverse problem is solved with the quasi-static model. This model does not consider the dynamics of the heart and can cause errors in the reconstruction of the solution on the surface of the heart. This thesis investigates different methodologies was to solve the inverse problem of electrocardiography as artificial intelligence and dynamic models limits. Also, the effects of different operators using boundary element methods, finite element methods, and was investigates.

Des simulateurs numériques sont couramment utilisés pour la prédiction et l'optimisation de l'exploitation d'aquifères et pour la détermination de paramètres physiques (e.g perméabilité) par calcul inverse. L'équation de Richards, décrit l'écoulement d'un fluide dans un milieu poreux non saturé. C'est une équation aux dérivées partielles non linéaires, dont la résolution numérique en grande dimension 3D est très coûteuse et en particulier pour du calcul inverse.Dans ce travail, une méthode de réduction de modèle (ROM) est proposée par une décomposition orthogonale propre (POD) afin de réduire significativement le temps de calcul, tout maîtrisant la précision. Une stratégie de cette méthode est de remplacer localement dans l'algorithme d'optimisation, le modèle fin par un modèle réduit type POD. La méthode de Petroc-Galerkin POD est d'abord appliquée à l'équation de Richards et testée sur différents cas, puis adaptée en linéarisant les termes non linéaires. Cette adaptation ne fait pas appel à une technique d'interpolation et réduit le temps de calcul d'un facteur [10;100]. Bien qu'elle ajoute de la complexité du ROM, cette méthode évite d'avoir à ajuster les paramètres du noyau, comme c'est le cas dans les méthodes du POD par interpolation. Une exploration des propriétés d'interpolation et d'extrapolation inhérentes aux méthodes intrusives est ensuite faite. Des qualités d'extrapolation intéressantes permettent de développer une méthode d'optimisation nécessitant de petits plans d'expériences (DOE). La méthode d'optimisation recrée localement des modèles précis sur l'espace des paramètres, en utilisant une classification à vecteurs de support non linéaire pour délimiter la zone où le modèle est suffisamment précis, la région de confiance. Les méthodes sont appliquées sur un cas d'école en milieu non saturé régit par l'équation de Richards, ainsi que sur un aquifère situé dans le "Table Mountain Group" près de la ville du Cap en Afrique du Sud. / The Richards equation describes the movement of an unsaturated fluid through a porous media, and is characterised as a non-linear partial differential equation. The equation is subject to a number of parameters and is typically computationnaly expensive to solve. To determine the parameters in the Richards equation, inverse modelling studies often need to be undertaken. As a solution to overcome the computational expense incurred in inverse modelling, the use of Proper Orthogonal Decomposition (POD) as a Reduced Order Modelling (ROM) method is proposed in this thesis to speed-up individual simulations. The Petrov-Galerkin POD approach is initially applied to the Richards equation and tested on different case studies. However, due to the non-linear nature of the Richards equation the method does not result in significant speed up times. Subsquently, the Petrov-Galerkin method is adapted by linearising the nonlinear terms in the equation, resulting in speed-up times in the range of [10,100]., The adaptation, notably, does not use any interpolation techniques, favouring an intrusive, but physics-based, approach. While the use of intrusive POD approaches add to the complexity of the ROM, it avoids the problem of finding kernel parameters typically present in interpolative POD approaches. Furthermore, the interpolative and possible extrapolation properties inherent to intrusive PODROM's are explored. The good extrapolation propertie, within predetermined bounds, of intrusive POD's allows for the development of an optimisation approach requiring a very small Design of Experiments (DOE). The optimisation method creates locally accurate models within the parameters space usign Support Vector Classification. The limits of the locally accurate model are called the confidence region. The methods are demonstrated on a hypothetical unsaturated case study requiring the Richards equation, and on true case study in the Table Mountain Group near Cape Town, South Africa.

Inverse problems aim at estimating the unknown excitations or properties of a physical system based on available measurements of the system response. For example, wave tomography is used in geophysics for seismic waveform inversion; in biomedical engineering, optical tomography is used to detect breast cancer tissue; in structural engineering, inversion techniques are used for health monitoring and damage detection in structural safety evaluation. Inverse solvers depend on the type of measurement data the unknown parameters to be estimated. The work in this thesis focuses on structural parameter identification based on static and dynamic measurements. As an integral part of the formulated inverse solver, the associated forward problem is studded and deeply investigated. In reality, the data are associated with uncertainties caused by measurement devices or unfriendly environmental conditions during data acquisition. Traditional approaches use probability theory and model uncertainties as random variables. This approach has its own limitation due to a prior assumption on the probability structure of uncertainty. This is usually too optimistic or not realistic. However, in practice, it is usually difficult to reliably assess the statistical nature of uncertainties. Instead, only bounds on the uncertain variables and some partial information about their probabilities are known. The main source of uncertainty is due to the accuracy of measuring devices; these are designed to operate within specific allowable tolerances, as defined by National Institute of Standards and Technology (NIST). Tolerances are performance requirements that fix the limit of allowable error or departure from true performance or value. Thus closed intervals are the most realistic way to model uncertainty in measurements. In this work, uncertainties in measurement data are modeled as interval variables bounded by their endpoints. It is proven that interval analysis provides guaranteed enclosure of the exact solution set regardless of the underlying nature of the associated uncertainties. This work presents a solution of inverse problems under measurements uncertainty within the framework of Interval Finite Element Methods (IFEM) and adjoint-based optimization techniques. The solution consists of a two-step algorithm: first, an estimate of the parameters is obtained by means of a deterministic iterative solver. Then, the algorithm switches to a full interval solution, using the previous deterministic estimate as an initial guess. In general, the solution of an inverse problem requires iterative solutions of the forward problem. Efficient and accurate interval forward solutions in static and dynamic domains have been developed. In particular, overestimation due to interval dependency has been drastically reduced using a new decomposition of the load, stiffness, and mass matrices. Further improvements in the available interval iterative solvers have been achieved. Conjugate gradient and Newton-Raphson methods to gether with an inexact line search are used in the newly formulated optimization procedure. Moreover Tikhonov regularization is used to improve the conditioning of the ill-posed inverse problem. The developed interval solution for the inverse problem under uncertainty has been tested in a wide range of applications in static and dynamic domains. By comparing current solutions with other available methods in the literature, it is proven that the developed method provides guaranteed sharp bounds on the exact solution sets at a low computational cost. In addition, it contains those solutions provided by probabilistic approaches regardless of the used probability distributions. In conclusion, the developed method provides a powerful tool for the analysis of structural inverse problem under uncertainty.

published_or_final_version / Physics / Doctoral / Doctor of Philosophy

Most existing computational methods for inverting material properties of multilayer systems have focused primarily on elastic properties of materials or a static approach. Typically, they are based on a two-stage approach: (I) modeling structural responses with a computer program, and (II) estimating layer properties mathematically using the response outputs determined in stage I without interactions with the governing state partial-differential-equation (PDE) of stage I. This two-stage approach may not be accurate and efficient enough for inverting larger scale model parameters. The objective of this research was to develop a computational method to invert dynamic moduli of multilayer systems with applications to flexible pavements under falling weight deflectometer (FWD) tests, thereby advancing existing methods and fostering understanding of material behaviors. This research first developed a finite-element and Newton-Raphson method to invert layer elastic moduli using FWD data. The model improved the moduli seeds estimation and achieved a satisfactory accuracy based on Monte Carlo simulations, addressing the common back-calculation issue of no unique solutions. Consequently, a time-domain finite-element method was developed to simulate dynamic-viscoelastic responses of the multilayer systems under loading pulses. Simulation results demonstrated that the dynamic-viscoelastic-damping-coupled model could emulate structural responses more accurately, thereby advancing existing simulation approaches. By using the dynamic-viscoelastic-response model as one computation module, this research led to the development of a PDE-constrained Lagrangian optimization method to invert dynamic moduli and viscoelastic properties of multilayer systems. The Lagrangian function was used as an objective function, with a regularization term and governing-state PDE constraint. Both the first-order (gradient) and second-order variation (Hessian matrix) of the Lagrangian were computed to satisfy necessary and sufficient optimality conditions, and Armijo rule was modified to determine a stable step length. The developed method improved computation speed significantly, and it is superior for large-scale inverse problems. The model was implemented for evaluating flexible pavements under FWD tests and for inverting the master curve of dynamic moduli of the asphalt layer. Independent computer coding was developed for all numerical methods. The computational methods developed may also be applied to other multilayer systems, such as tissues and sandwich structures at different time and length scales. / text

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