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There is the need for applying numerical methods to problems that cannot be solved analytically and as the spatial dimension of the problem is increased the need for computational recourses increase exponentially, a phenomenon known as the “curse of dimensionality”. In the Black-Scholes-Merton framework the American option pricing problem has no closed form solution and a numerical procedure has to be employed for solving a PDE. The multi-asset American option introduces challenging computational problems, since for every added asset the dimension of the PDE is increased by one. One way to deal with the curse of dimensionality is threw parallelism. Here the finite element method-of-lines is used for pricing a multi-asset American option dependent on up to four assets in a parallel environment. The problem is also solved with the PSOR method giving a accurate benchmark used for comparison. In finance the put option is one of the most fundamental derivatives since it is basically asset-value insurance and a lot of research is done in the field of quantitative finance on accurate and fast pricing techniques for the multi-dimensional case. “What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.” Norbert Wiener “As soon as an Analytical Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will then arise – by what course of calculation can these results be arrived at by the machine in the shortest time?” Charles Babbage

Contactless electrical impedance imaging technique is based upon the measurement of secondary electromagnetic fields caused by induced currents inside the body. In this study, a circular single-coil is used as a transmitter and a receiver. The purpose of this study is twofold: (1) to solve the induced current density distribution inside the realistic head model resulting from a sinusoidal excitation, (2) to calculate the impedance change of the same coil from the induced current distribution inside the head model. The Finite Difference Method is used to solve the induced current density in the head. The realistic head model is formed by seven tissues with a 1 mm resolution. The electrical properties of the model are assigned as a function of frequency. The quasi-stationary assumptions, especially for head tissues, are explored. It is shown that, numerical solution of only the scalar potential is sufficient to obtain the induced current density in the head below 10 MHz operating frequency. This simplification not only reduce the excessive size of the solution domain, but also reduces the number of unknowns by a factor of 4. For higher frequencies (depending on the application) induction and propagation effects become important. Additionally it is observed that dynamic monitoring of hemorrhage at any frequency seems feasible. It is concluded that the methodology provides useful information about the electrical properties of the human head via contactless measurements and has a potent as a new imaging modality for different clinical applications.

An option is a contract between two parties where the holder has the option to buy or sell some underlying asset after a predefined exercise time. Options where the holder only has the right to buy or sell at the exercise time is said to be of European-style, while options that can be exercised any time before the exercise time is said to be of American-style. Asian options are options where the payoff is determined by some average value of the underlying asset, e.g., the arithmetic or the geometric average. For arithmetic Asian options, there are no closed-form pricing formulas, and one must apply numerical methods. Several methods have been proposed and tested for Asian options. For example, the Monte Carlo method isslowforEuropean-styleAsianoptionsandnotapplicableforAmerican-styleAsian options. In contrast, the finite difference method have successfully been applied to price both European- and American-style Asian options. But from a financial point of view, one is also interested in different measures of sensitivity, called the Greeks, which are hard approximate with the finite difference method. For more accurate approximations of the Greeks, researchers have turned to the finite element method with promising results for European-style Asian options. However, the finite element method has never been applied to American-style Asian options, which still lack accurate approximations of the Greeks. Here we present a study of pricing European- and American-style Asian options using the finite element method. For European-style options, we consider two different pricing PDEs. The first equation we consider is a convection-dominated problem, which we solve by applying the so-called streamline-diffusion method. The second equation comes from modelling Asian options as options on a traded account, which we solve by using the so-called cG(1)cG(1) method. For American-style options, the model based on options on a traded account is not applicable. Therefore, we must consider the first convection-dominated problem. To handle American-style options, we study two different methods, a penalty method and the projected successive over-relaxation method. For European-style Asian options, both approaches give good results, but the model based on options on a traded account show more accurate results. For American-style Asian options, the penalty method give accurate results. Meanwhile, the projected successive over-relaxation method does not converge properly for the tested parameters. Our result is a first step towards an accurate and fast method to calculate the price and the Greeks of both European- and American-style Asian options. Because good estimations of the Greeks are crucial when hedging and trading of options, we anticipate that the ideas presented in this work can lead to new ways of trading with Asian options.

Les travaux de thèse portent sur l'identification paramétrique en dynamique transitoire à partir des mesures fortement bruitées, l'un des objectifs à long terme étant de proposer une méthode d’identification peu intrusive afin de pouvoir être implémentée dans des codes de calcul éléments finis commerciaux. Dans ce travail, le concept de l'erreur en relation de comportement modifiée a été retenu pour traiter le problème d’identification des paramètres matériau. La minimisation de la fonctionnelle coût sous contraintes débouche, dans le cas de la dynamique transitoire, sur un problème dit « aux deux bouts » dans lequel il s’agit de résoudre un problème différentiel spatio-temporel avec des conditions à la fois initiales et finales en temps. Il en résulte un problème couplé entre les champs direct et adjoint dont le traitement est délicat. Dans un premier temps, des méthodes précédemment développées telles que la « méthode de Riccati » et la « méthode de tirs » ont été étudiées. Il est montré que l’identification par ces méthodes est robuste même pour des mesures fortement corrompues, mais qu’elles sont limitées par la complexité d’implémentation dans un code industriel, des problèmes de conditionnement ou de coût de calcul. Dans un second temps, une approche itérative basée sur une méthode de sur-relaxation a été développée et comparée à celles précédemment mentionnées sur des exemples académiques, validant l’intérêt de cette nouvelle approche. Enfin, des comparaisons ont été menées entre cette technique et une variante « discrétisée » de la formulation introduite par Bonnet et Aquino [Inverse Problems, vol. 31, 2015]. / This thesis deals with parameters identification in transient dynamic in case of highly noisy experimental data. One long-term goal is the derivation of a non-intrusive method dedicated to the implementation in a commercial finite element code.In this work, the modified error in the constitutive relation framework is used to treat the identification of material parameters. The minimization of the cost function under constraints leads, in the case of transient dynamics, to a « two points boundary value problem » in which the differential space-time problem involves both initial and final time conditions. This results in a problem coupling the direct and adjoint fields, whose treatment is difficult.In the first part, methods such as those based on the « Riccati equations » and the « shooting methods » have been studied. It is shown that the identification is robust even in the case of highly corrupted measures, but these methods are limited either by the implementation intrusiveness, conditioning problems or the numerical cost.In the second part, an iterative over-relaxation approach is developed and compared to the aforementioned approaches on academic problems in order to validate the interest of the method. Finally, comparisons are carried out between this approach and a « discretized » variation of the formulation introduced by Bonnet and Aquino [Inverse Problems, vol. 31, 2015].