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De nombreux algorithmes de simulations informatiques permettent de reproduire des comportements physiques d'objets en trois dimensions. Des phénomènes liés à différents domaines, comme la mécanique classique (animation de tissus par exemple), la mécanique des fluides (animation de feux par exemple) ou encore la photométrie (visualisation photo-réaliste de scènes par exemple) peuvent être recréés virtuellement. Ces algorithmes complexes possèdent généralement de nombreux paramètres. Leurs réglages sont souvent délicats et peu conformes au réel. Pour utiliser ces algorithmes en simulations chirurgicales ou pour les calculs de stabilité d'un pont suspendu, il est indispensable d’avoir des paramètres exacts.Malheureusement, il est difficile, voire impossible, même pour un expert de trouver les bonnes valeurs de ces paramètres pour produire un effet donné, et tout particulièrement sur des objets réels complexes comme les organes. En effet, même si nous disposons d’un simulateur très puissant, l'adéquation de ses paramètres avec le monde réel est difficile, et dans le meilleur des cas il s’agit toujours de « tâtonner » pour obtenir le résultat voulu, et ce parfois pendant des heures. Cette thèse vise donc à déterminer directement ces paramètres depuis des séquences d'images réelles, afin de reproduire en réalité virtuelle et de la façon la plus fidèle possible le comportement observé des objets du monde réel. Nous concevons pour cela une solution matérielle et logicielle peu onéreuse en proposant un nouveau dispositif de capture de force et un algorithme d'estimation inverse basé sur plusieurs métriques d'erreur. / Many computer-driven simulations are able to reproduce three dimensional object's physical behaviours. Phenomenons related to many different domains like "classical" mechanic (cloth simulation for example), fluids mechanic (fires or smoke for example) or photometry (photo-realistic 3d scenes visualisation for example) can be recreated virtually. However, those algorithms usually need a lot of parameters. Their tuning is often complex and not realistic. To use those algorithms in surgical simulations or for bridge stability computations, it is essential to have the right parameters. Unfortunately, it is hard (sometimes impossible), even for a domain expert, to find the right values for those parameters to produce the desired effect, and especially for complex real objects like an eye or a liver for example. Indeed, even if we had a very powerfull simulator, the adequacy with the real world is far to be obvious, and in the best case we always need to grope in order to obtain the wanted result, and it can sometimes take hours. This PhD aim to find these parameters directly from real videos, in order to reproduce in virtual reality the "real" objects behaviours in the most faithful possible way. To achieve this, we propose a low cost hardware and software solution by designing a new force capture device and an inverse estimation algorithm based on some error metrics.

We are interested in solving the time-harmonic inverse acoustic scattering problem for planar sound-soft obstacles. In this work, we introduce four methods for solving inverse scattering problems. The first method is a variation of the method introduced by Johansson and Sleeman. This method solves the inverse problem when we have the far field pattern given for only one incident wave. It is an iterative method based on a pair of integral equations used to obtain the far field pattern of a known single object. The method proposed in this thesis has a better computational performance than the method of Johansson and Sleeman. The second method we present is a multi-frequency method called the recursive linearization algorithm. This method solves the inverse problem when the far field pattern is given for multiple frequencies. The idea of this method is that from an initial guess, we solve the single frequency inverse problem for the lowest frequency. We use the result obtained as the initial guess to solve the problem for the next highest frequency. We repeat this process until we use the data from all frequencies. To solve the problem at each frequency, we use the first method proposed. To improve the quality of the reconstruction of the shadowed part of the object, we solve the inverse scattering problem of reconstructing an unknown sound-soft obstacle in the presence of known scatterers. We show that depending on the position of the scatterers, we may be able to obtain very accurate reconstructions of the entire unknown object. Next, we introduce a method for solving the inverse problem of reconstructing a convex sound-soft obstacle, given measures of the far field pattern at two frequencies that are not in the resonance region of the object. This method is based on the use of an approximation formula for the far field pattern using geometric optics. We are able to prove that for the reconstruction of the circle of radius $R$ and center at the origin, the size of the interval of convergence of this method is proportional to the inverse of the wavenumber. This procedure is effective at reconstructing the illuminated part of the object; however, it requires an initial guess close to the object for frequencies out of the resonance region. Finally, we propose a globalization technique to obtain a better initial guess to solve the inverse problem at frequencies out of the resonance region. In this technique, given the far field pattern of a convex object at two frequencies out of the resonance region, we use our extrapolation operator to generate synthetic data for low frequencies. We apply the recursive linearization algorithm, using as a single frequency solver the method that is based on geometric optics. We obtain an approximation of the object that can be used as the initial guess to apply the recursive linearization algorithm using the first method introduced as the single frequency solver.

Des inclusions de petits encombrements sont sources de perturbations pour les champs électromagnétiques ambiants (ceux qui existeraient en leurs absences, par exemple). Il est facilement imaginable que la mesure de ces perturbations puisse fournir des informations permettant l'identification des inclusions, où par iden- tification l'on signifie au minimum leur localisation, mais où l'on pourrait aussi signifier quantifocation de leurs paramètres électriques, voire dans la meilleure des hypothèses, caractérisation de leurs encombrements et formes. Récemment, une théorie mathématique a été développée pour préciser de petites inclusions à partir de mesures de frontière, voir [7] et références citées. Cette thèse porte principalement sur l'identication d'inclusions homogènes (de nombre inconnu a priori) d'un milieu donné à partir de mesures d'amplitudes de diraction lors de l'éclairement approprié de ce milieu. Premièrement, nous four- nissons de nouvelles formules asymptotiques, tant robustes que précises, des champs électromagnétiques résultant du phénomène de diffraction. Ensuite, nous les ex- ploitons pour la construction d'algorithmes d'identification non itératifs pertinents. Le problème est traité en trois grandes parties, chacune étant dédiée à une géométrie spécique : 1) Le milieu d'enfouissement de la collection est homogène, l'espace libre. 2) Le milieu est constitué de deux demi-espaces séparés par une interface plane, la collection étudiée se situant dans le demi-espace inférieur et sources et capteurs se situant dans le demi-espace supérieur. 3) Le milieu est un guide d'ondes, et la collection est dans le coeur de ce guide d'ondes.

Constitutive model parameters are identified during model calibration through trial-and-error process driven to fit test data. In this research, the calibration of constitutive models is formally handled as an inverse problem. The first phase of this research explores error propagation. Data errors, experimental biases (e.g. improper boundary conditions), and model errors affect the inversion of model parameters and ensuing numerical predictions. Drained and undrained tests are simulated to study the effect of these three classes of errors. Emphasis is placed on the analysis of error surfaces computed by successive forward simulations. The second phase of this research centers on test procedures. Conventional soil tests were developed to create uniform stress and strain fields; consequently, they provide limited amount of information, the inversion is ill-posed, and results enhance uncertainty and error propagation. This research examines soil testing using new, non-conventional loading and boundary conditions to create rich, diverse, non-uniform strain and stress fields. In particular, the flexural excitation of cylindrical soil specimens is shown to provide rich data leading to a more informative test than conventional geotechnical tests. The new test is numerically optimized. Then a set of unique experimental studies is conducted.

In this thesis, direct and inverse problems concerning nodal solutions associated with the one-dimensional p-Laplacian operators are studied. We first consider the eigenvalue problem on (0, 1), −(y0(p−1))0 + (p − 1)q(x)y(p−1) = (p − 1) £fw(x)y(p−1) (0.1) Here f(p−1) := |f|p−2f = |f|p−1 sgn f. This problem, though nonlinear and degenerate, behaves very similar to the classical Sturm-Liouville problem, which is the special case p = 2. The spectrum {£fk} of the problem coupled with linear separated boundary conditions are discrete and the eigenfunction yn corresponding to£fn has exactly n−1 zeros in (0, 1). Using a Pr¡Lufer-type substitution and properties of the generalized sine function, Sp(x), we solve the reconstruction and stablity issues of the inverse nodal problems for Dirichlet boundary conditions, as well as periodic/antiperiodic boundary conditions whenever w(x) £f 1. Corresponding Ambarzumyan problems are also solved. We also study an associated boundary value problem with a nonlinear nonhomogeneous term (p−1)w(x) f(y(x)) on the right hand side of (0.1), where w is continuously differentiable and positive, q is continuously differentiable and f is positive and Lipschitz continuous on R+, and odd on R such that f0 := lim y!0+ f(y) yp−1 , f1 := lim y!1 f(y) yp−1 . are not equal. We extend Kong¡¦s results for p = 2 to general p > 1, which states that whenever an eigenvalue _n 2 (f0, f1) or (f1, f0), there exists a nodal solution un having exactly n − 1 zeros in (0, 1), for the above nonhomogeneous equation equipped with any linear separated boundary conditions. Although it is known that there are indeed some differences, Our results show that the one-dimensional p-Laplacian operator is still very similar to the Sturm-Liouville operator, in aspects involving Pr¡Lufer substitution techniques.

In this thesis, we consider the inverse Toeplitz eigenvalue problem which recover a real symmetric Toeplitz with desired eigenvalues. First some lower dimensional cases are solved by algebraic methods. This gives more insight on the inverse problem. Next, we explore the geometric meaning of real symmetric Toeplitz matrices. For high dimensional cases, numerical are unavoidable. From our numerical experiments, Newton-like methods are very effective for this problem.

Abstract This paper proposes a novel approach to analyze the inverse problems of elastohydrodynamic lubrication (EHL). First, a finite-difference method is employed to discretize the elastic deformation and the force balance equations. The discretizing equations can be rearranged into a matrix form. The pressure distribution can be expressed in an appropriate function and then substituting it into the matrix form. The least-squares error method is adopted to find the undetermined coefficients, which generates a smooth pressure distribution based upon a small number of measuring points on the film thickness map and overcomes the problems of pressure fluctuations obtained from the traditional methods. The apparent viscosity can be solved from the Reynolds equations by using the least-squares method to predict the optimum value of the pressure-viscosity index ( ). The proposed method is applied to analyze four kinds of the inverse problems, namely, EHL of line contacts, EHL of point contacts, pure squeeze EHL motion of circular contacts, and elastohydrodynamic thin film lubrication (EHTFL). This paper discusses the effects of the implemented errors on the predicted value of apparent viscosity. The errors are implemented at the film thickness, the load, the effective elastic modulus, the viscosity at ambient pressure, and the mean velocity. Results show that the implemented errors in load and effective elastic modulus have a significant influence on the accuracy of the results, but the errors in average velocity and in the viscosity at ambient pressure do not have a significant effect. In these implemented errors, the resolution of the film thickness measurement plays the most important role in determining the accuracy of the apparent viscosity. Even when errors in the film thickness measurements are deliberately introduced, the inverse approach still provides a satisfactory value of the pressure-viscosity index. The resulting apparent viscosity errors are much smaller than those generated when using the traditional inverse method. The inverse approach can allow higher measurement error than traditional inverse method, and the allowable resolution range can be increased to about 3-10 times. Base on the viscous adsorption theory, the modified Reynolds equation is derived for EHTFL. In this theory, the film thickness between lubricated surfaces is simplified as three fixed layers across the film, and the viscosity and the density of lubricant vary with pressure in each layer. The difference between classical EHL and EHTFL is investigated to find the important parameters of EHTFL. Results show that the proposed model can reasonably calculate the film thickness and the viscosity under EHTFL. Adsorbent layer thickness and viscosity significantly influence the lubrication characteristics of the contact conjunction. The inverse approach is developed to evaluate the pressure of contact region, pressure-viscosity index ( ) of oil film, and the film thickness and viscosity of adsorbent layer under EHTFL. This paper also uses a self-development EHL tester with the optical interferometry equipment to observe the EHL film thickness map of circular contacts under the steady state and the pure squeeze motion. The inverse approach can be used to estimate the pressure distribution on a film thickness map obtained from optical EHL tester. By using this pressure distribution, the estimated pressure-viscosity index can be obtained. Result shows that the inverse approach predicts a larger value of the pressure-viscosity index than the actual value. The error between the actual and the estimated values of is less than 7 percent. When the minimum film thickness is less than 30 nm, the inverse approach based on EHTFL theory can reduce the error between the actual and the estimated values of .