In this thesis we generalize the problem of phase retrieval of vector to that of multi-vector. The identification of the multi-vector is done up to some special classes of isometries in the space. We give some upper and lower estimates on the minimal number of multi-linear operators needed for the retrieval. The results are preliminary and far from sharp.

Les problèmes inverses de localisation de sources en électroencéphalographie (EEG) consistent à retrouver le lieu d'origine dans le cerveau des signaux mesurés sur le scalp. La qualité du résultat de localisation dépend des modèles géométriques et de conductivité électrique utilisés pour la résolution du problème. Parmi les tissus composant la tête, le crâne est celui dont la conductivité est la plus influente, en particulier à cause de sa faible valeur. De plus, le crâne humain est un tissu osseux comportant des parties dures et spongieuses, d'épaisseurs variables. Sa composition est très variable selon les individus, en termes de géométrie et de valeurs des conductivités, d'où la nécessité de développer des technique d'estimation de conductivités inconnues dans le crâne. Le but de cette thèse est de réduire l'incertitude sur la conductivité du crâne, pour des géométries sphériques et réalistes, en particulier en vue d’améliorer les résultats d'estimation des sources dans le problème inverse EEG. Dans le cas d'un domaine sphérique à 3 couches, l'existence, l'unicité et la stabilité de la conductivité dans la couche intermédiaire (crâne) sont discutées, et une procédure de reconstruction est proposée. Puis deux modèles plus réalistes de tête sont étudiés, l'un pour lequel le crâne est modelisé par un seul compartiment, l'autre dans lequel les parties spongieuses et dure sont distinguées. Des simulations numériques mettent en évidence le rôle de la structure interne du crâne pour la détermination de sa conductivité. / One of the major issues related to electroencephalography (EEG) is to localize where in the brain signals are generated, this is so called inverse problem of source localization. The quality of the source localization depends on the accuracy of the geometry and the electrical conductivity model used to solve the problem. Among the head tissues, the skull conductivity is the one that influences most the accuracy of the source localization, due to its low value. Moreover, the human skull is a bony tissue consisting of compact and spongy bone layers, whose thickness vary across the skull. As the skull tissue composition has strong inter-individual variability both in terms of geometry and of individual conductivity, conductivity estimation techniques are required in order to determine the unknown skull conductivity. The aim of this thesis is to reduce the uncertainty on the skull conductivity both in spherical and realistic head geometries in order to increase the quality of the inverse source localization problem. Therefore, conductivity estimation is first performed on a 3-layered spherical head model. Existence, uniqueness and stability of the conductivity in the intermediate skull layer are discussed, together with a constructive recovery scheme. Then a simulation study is performed comparing two realistic head models, a bulk model where the skull is modelled as a single compartment and a detailed one accounting for the compact and spongy bone layers, in order to determine the importance of the internal skull structure for conductivity estimation in EEG.

Fractional calculus has been introduced as an e cient tool for modeling physical phenomena, thanks to its memory and hereditary properties. For example, fractional models have been successfully used to describe anomalous di↵usion processes such as contaminant transport in soil, oil flow in porous media, and groundwater flow. These models capture important features of particle transport such as particles with velocity variations and long-rest periods. Mathematical modeling of physical phenomena requires the identification of pa- rameters and variables from available measurements. This is referred to as an inverse problem. In this work, we are interested in studying theoretically and numerically inverse problems for space Fractional Advection-Dispersion Equation (FADE), which is used to model solute transport in porous media. Identifying parameters for such an equa- tion is important to understand how chemical or biological contaminants are trans- ported throughout surface aquifer systems. For instance, an estimate of the di↵eren- tiation order in groundwater contaminant transport model can provide information about soil properties, such as the heterogeneity of the medium. Our main contribution is to propose a novel e cient algorithm based on modulat-ing functions to estimate the coe cients and the di↵erentiation order for space FADE, which can be extended to general fractional Partial Di↵erential Equation (PDE). We also show how the method can be applied to the source inverse problem. This work is divided into two parts: In part I, the proposed method is described and studied through an extensive numerical analysis. The local convergence of the proposed two-stage algorithm is proven for 1D space FADE. The properties of this method are studied along with its limitations. Then, the algorithm is generalized to the 2D FADE. In part II, we analyze direct and inverse source problems for a space FADE. The problem consists of recovering the source term using final observations. An analytic solution for the non-homogeneous case is derived and existence and uniqueness of the solution are established. In addition, the uniqueness and stability of the inverse problem is studied. Moreover, the modulating functions-based method is used to solve the problem and it is compared to a standard Tikhono-based optimization technique.

Ablative materials can sustain very high temperatures in which surface thermochemical processes are significant enough to cause surface recession. Existence of moving boundary over a wide range of temperatures, temperature-dependent thermophysical properties of ablators, and no prior knowledge about the location of the moving surface augment the difficulty for predicting the exposed heat flux at the receding surface of ablators. In this paper, the conjugate gradient method is proposed to estimate the unknown surface recession and time-varying net surface heat flux for these kinds of problems. The first order Tikhonov regularization is employed to stabilize the inverse solution. Considering the complicated phenomena that are taking place, it is shown via simulated experiment that unknown quantities can be obtained with reasonable accuracy using this method despite existing noises in the measurement data.

In this thesis, we present a sensor model comprised of a geometrically nonlinear beam coupled with a nonlinear viscoelastic Pasternak foundation via a distributed system of compliant elements. The governing equations of the system are obtained. By posing an inverse problem, the model is used to simulate the estimation of coupled substrates' material (constitutive) parameters. In the inverse problem, beam deformations are considered as measured parameters, and therefore an eventual hardware implementation would require measurements of these quantities. Different case studies are simulated to assess the robustness and applicability of this sensor model.

No description available.

This paper studies the inverse source problem for the Helmholtz equation with a point source in a two dimensional domain. Given complete boundary data and appropriate discretization Tikhonov regularization is established to be an effective method at finding the point source. Furthermore, it was found that Tikhonov regularization can locate point sources even given significant noise, as well as incomplete boundary data in complicated domains.

We proved that the topological conjugacy relation on $T_1$, a subclass of Toeplitz subshifts, is hyperfinite, extending Kaya's result that the topological conjugate relation of Toeplitz subshifts with growing blocks is hyperfinite. A close concept about the topological conjugacy is the flip conjugacy, which has been broadly studied in terms of the topological full groups. Particularly, we provided an equivalent characterization on Toeplitz subshifts with single hole structure to be flip invariant.

In this thesis, the perturbation-based decomposition technique developed by Szlavik [1] was used in an attempt to solve the inverse problem in EEG source localization. A set of dipole locations were forward modeled using a 4-layer sphere model of the head at uniformly distributed lead locations to form the vector basis necessary for the method. Both a two-dimensional and a pseudo-three-dimensional versions of the model were assessed with the two-dimensional model yielding decompositions with minimal error and the pseudo-three-dimensional version having unacceptable levels of error. The utility of interpolation as a method to reduce the number of data points to become overdefined was assessed as well. The approach was effective as long as the number of component functions did not exceed the number of data points and stayed relatively small (less than 77 component functions). This application of the method to a spatially variate system indicates its potential for other systems and with some tweaking to the least squares algorithm used, could be applied to multivariate systems.

No description available.