The inverse problem in finance consists of determining the unknown parameters of the pricing equation from the values quoted from the market. We formulate the inverse problem as a minimization problem for an appropriate cost function to minimize the difference between the solution of the model and the market observations. Efficient gradient based optimization requires accurate gradient estimation of the cost function. In this thesis we highlight the adjoint method for computing gradients of the cost function in the context of gradient based optimization and show its importance. We derive the continuous adjoint equations with appropriate boundary conditions for three main option pricing models: the Black-Scholes model, the Heston's model and the jump diffusion model, for European type options. These adjoint equations can be used to compute the gradient of the cost function accurately for parameter estimation problems. The adjoint method allows efficient evaluation of the gradient of a cost function F(σ) with respect to parameters σ where F depends on σ indirectly, via an intermediate variable. Compared to the finite difference method and the sensitivity equation method, the adjoint equation method is very efficient in computing the gradient of the cost function. The sensitivity equations method requires solving a PDE corresponding to each parameter in the model to estimate the gradient of the cost function. The adjoint method requires solving a single adjoint equation once. Hence, for a large number of parameters in the model, the adjoint equation method is very efficient. Due to its nature, the adjoint equation has to be solved backward in time. The adjoint equation derived from the jump diffusion model is harder to solve due to its non local integral term. But algorithms that can be used to solve the Partial Integro-Differential Equation (PIDE) derived from jump diffusion model can be modified to solve the adjoint equation derived from the PIDE. / Ph. D.

Continuous adjoint methods and optimal control theory are applied to a pressure-matching inverse design problem of quasi 1-D nozzle flows. Pontryagin’s Minimum Principle is used to derive the adjoint system and the reduced gradient of the cost functional. The properties of adjoint variables at the sonic throat and the shock location are studied, revealing a logarithmic singularity at the sonic throat and continuity at the shock location. A numerical method, based on the Steger-Warming flux-vector-splitting scheme, is proposed to solve the adjoint equations. This scheme can finely resolve the singularity at the sonic throat. A non-uniform grid, with points clustered near the throat region, can resolve it even better. The analytical solutions to the adjoint equations are also constructed via Green’s function approach for the purpose of comparing the numerical results. The pressure-matching inverse design is then conducted for a nozzle parameterized by a single geometric parameter. In the second part, the adjoint methods are applied to the problem of minimizing drag coefficient, at fixed lift coefficient, for 2-D transonic airfoil flows. Reduced gradients of several functionals are derived through application of a Lagrange Multiplier Theorem. The adjoint system is carefully studied including the adjoint characteristic boundary conditions at the far-field boundary. A super-reduced design formulation is also explored by treating the angle of attack as an additional state; super-reduced gradients can be constructed either by solving adjoint equations with non-local boundary conditions or by a direct Lagrange multiplier method. In this way, the constrained optimization reduces to an unconstrained design problem. Numerical methods based on Jameson’s finite volume scheme are employed to solve the adjoint equations. The same grid system generated from an efficient hyperbolic grid generator are adopted in both the Euler flow solver and the adjoint solver. Several computational tests on transonic airfoil design are presented to show the reliability and efficiency of adjoint methods in calculating the reduced (super-reduced) gradients. / Ph. D.

The Community Multiscale Air Quality (CMAQ) system is the Environmental Protection Agency's main modeling tool for atmospheric pollution studies. CMAQ-ADJ, the adjoint model of CMAQ, offers new analysis capabilities such as receptor-oriented sensitivity analysis and chemical data assimilation. This thesis presents the construction, validation, and properties of new adjoint modules in CMAQ, and illustrates their use in sensitivity analyses and data assimilation experiments. The new module of discrete adjoint of advection is implemented with the aid of automatic differentiation tool (TAMC) and is fully validated by comparing the adjoint sensitivities with finite difference values. In addition, adjoint sensitivity with respect to boundary conditions and boundary condition scaling factors are developed and validated in CMAQ. To investigate numerically the impact of the continuous and discrete advection adjoints on data assimilation, various four dimensional variational (4D-Var) data assimilation experiments are carried out with the 1D advection PDE, and with CMAQ advection using synthetic and real observation data. The results show that optimization procedure gives better estimates of the reference initial condition and converges faster when using gradients computed by the continuous adjoint approach. This counter-intuitive result is explained using the nonlinearity properties of the piecewise parabolic method (the numerical discretization of advection in CMAQ). Data assimilation experiments are carried out using real observation data. The simulation domain encompasses Texas and the simulation period is August 30 to September 1, 2006. Data assimilation is used to improve both initial and boundary conditions. These experiments further validate the tools developed in this thesis. / Master of Science

Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method. This dissertation develops a complete framework for fully discrete adjoint sensitivity analysis and inverse problem solutions, in the context of time dependent, adaptive mesh, and adaptive step models. The discrete framework addresses all the necessary ingredients of a state–of–the–art adaptive inverse solution algorithm: adaptive mesh and time step refinement, solution grid transfer operators, a priori and a posteriori error analysis and estimation, and discrete adjoints for sensitivity analysis of flux–limited numerical algorithms. / Ph. D.

Sensitivity analysis quantifies the effect that of perturbations of the model inputs have on the model's outputs. Some of the key insights gained using sensitivity analysis are to understand the robustness of the model with respect to perturbations, and to select the most important parameters for the model. MATLODE is a tool for sensitivity analysis of models described by ordinary differential equations (ODEs). MATLODE implements two distinct approaches for sensitivity analysis: direct (via the tangent linear model) and adjoint. Within each approach, four families of numerical methods are implemented, namely explicit Runge-Kutta, implicit Runge-Kutta, Rosenbrock, and single diagonally implicit Runge-Kutta. Each approach and family has its own strengths and weaknesses when applied to real world problems. MATLODE has a multitude of options that allows users to find the best approach for a wide range of initial value problems. In spite of the great importance of sensitivity analysis for models governed by differential equations, until this work there was no MATLAB ordinary differential equation sensitivity analysis toolbox publicly available. The two most popular sensitivity analysis packages, CVODES [8] and FATODE [10], are geared toward the high performance modeling space; however, no native MATLAB toolbox was available. MATLODE fills this need and offers sensitivity analysis capabilities in MATLAB, one of the most popular programming languages within scientific communities such as chemistry, biology, ecology, and oceanogra- phy. We expect that MATLODE will prove to be a useful tool for these communities to help facilitate their research and fill the gap between theory and practice. / Master of Science

Le travail mené dans cette thèse porte sur l'étude "à posteriori" de l'assimilation variationnelle de données. Il s'agit d'une démarche de faisabilité pour la mise au point des outils permettant de faire une analyse diagnostique (qualitative et quantitative) du processus d'assimilation variationnelle, notamment en ce qui concerne l'influence du bruit des observations sur le processus d'assimilation ainsi que sa propagation sur les champs reconstitués (nous sommes alors amenés à faire une étude de sensibilité), et l'influence de la configuration spatio-temporelle des observations sur le processus d'assimilation. L'application usuelle des équations adjointes pour l'analyse de sensibilité est revisée, car dans le contexte de l'assimilation variationnelle, nous avons montré par un exemple simple qu'il faut s'y prendre différemment. Nous proposons alors une méthode pour mener correctement cette analyse de sensibilité. Cette méthode est basée sur l'utilisation des équations adjointes au second ordre, obtenues en prenant l'adjoint du système d'optimalité. La sensibilité en est déduite par inversion du Hessien de la fonction coût via la minimisation d'une fonctionnelle quadratique. L'application est faite sur un modèle de circulation générale océanique de type quasi-géostrophique, et nous faisons aussi l'étude de l'existence et l'unicité de la solution de l'équation adjointe au second ordre du modèle considéré, pour justifier l'utilisation du Hessien et l'applicabilité de notre méthode. Nous étudions aussi l'influence de la configuration spatio-temporelle des observations sur le processus d'assimilation au travers du Hessien (à l'optimum) dont les éléments propres varient lorsqu'on fait varier la configuration. Enfin, nous étudions la prédicibilité du système d'optimalité.

Cette thèse traite des interactions et résonances dans les systèmes quantiques et se subdivise en trois sous-thématiques. Dans les premiers chapitres, nous proposons, dans le cadre de la limite locale, une méthode systématique de construction d'un hamiltonien vibrationnel mis sous forme normale pour des systèmes moléculaires à n degrés de liberté fortement excités, à partir des générateurs d'une algèbre de Lie, l'algèbre des polynômes invariants construite en mécanique classique à partir du noyau de l'opérateur adjoint adH0 . Puis, nous exposons les méthodes de construction en l'absence et en présence d'une résonance p : q. Une application à la molécule triatomique non linéaire ClOH est ensuite envisagée.D'autre part, nous réalisons, à l'aide de l'algorithme LTPA, la normalisation des molécules triatomiques linéaires AB2 et nous comparons, dans le cas de la molécule de CO2, nos résultats à ceux d'autres auteurs qui utilisèrent une approche différente. Par analogie avec la construction des hamiltoniens de systèmes moléculaires AB2 non linéaires, nous montrons ensuite que l'interaction de Fermi permet de décrire le passage d'un condensat de Bose-Einstein (CBE) atomique à un condensat moléculaire.Enfin, le dernier chapitre de cette thèse s'intéresse conjointement au phénomène de résonance1 : 1 entre un système et un champ extérieur et à l'équation de Heun. Nous utilisons le modèledu système quantique à deux niveaux d'énergie interagissant avec un champ extérieur à modulation de phase périodique et à pulsation de Rabi généralisée constante. Nous montrons lors de transitions non adiabatiques, que l'évolution des amplitudes de probabilité des états se déduit de l'équation de Heun générale pour une classe de solutions particulière. Nous mettons également en évidence trois comportements différents pour la fonction de décalage en fréquence : les non-croisements, les croisements et le level-glancing. Pour ces deux derniers comportements, une résonance 1 : 1 se produit périodiquement entre le système et le champ. / This thesis book is concerned with the interactions and resonances in quantum systems and is subdivided into three thematics. First, our work is aimed at constructing in the local limit a systematic method for a normalized vibrational Hamiltonian for a strongly excited n-degree-of-freedom molecular system from the generators of the Lie algebra, the algebra of the invariant polynomials built in classical mechanics from the the kernel of the adjoint operator adH0 . We present both the method of construction in case of absence and in case of a p : q resonance system with n degrees of freedom. Application to the non-linear triatomic molecule ClOH is then given.On the other hand, by using the LTPA Algorithm, we realize normalisation of linear triatomic molecules and we compare in case of the CO2 molecule our results to those of authors who used to another approaches. Then, we are dealing with the Fermi interaction in order to show analogously to the building of Hamiltonians of non-linear molecules AB2, that this interaction is able to describe the transition of a atomic Bose-Einstein condensate (BEC) to a molecular one.Finally, in the last chapter, we explore the non-adiabatic dynamics of a two-state system subject to excitation by a specific constant-amplitude periodic level-crossing model and we show that the evolution of the probability amplitudes of states is deduced from the Heun equation for a particular class of solutions. We also highlight three different behaviors for the detuning : non-crossing, crossing and level-glancing. For these two last behaviors, a 1 : 1 resonance occurs periodically between the system and the field.

Design optimization is a method that can be used to automate the design process to obtain better results. When applied to aeroelastic structures, design optimization often leads to the creation of highly flexible aeroelastic structures. There are, however, a number of conventional design procedures that must be modified when dealing with highly flexible aeroelastic structures. First, the deformed geometry must be the baseline for weight, structural, and stability analyses. Second, potential couplings between aeroelasticity and rigid-body dynamics must be considered. Third, dynamic analyses must be modified to handle large nonlinear displacements. These modifications to the conventional design process significantly increase the difficulty of developing an optimization framework appropriate for highly flexible aeroelastic structures. As a result, when designing these structures, often either gradient-free optimization is performed (which limits the optimization to relatively few design variables) or optimization is simply omitted from the design process. Both options significantly decrease the design exploration capabilities of a designer compared to a scenario in which gradient-based optimization is used. This dissertation therefore presents various contributions that allow gradient-based optimization to be more easily used to optimize highly flexible aeroelastic structures. One of our primary motivations for developing these capabilities is to accurately capture the design constraints of solar-regenerative high-altitude long-endurance (SR-HALE) aircraft. In this dissertation, we therefore present a SR-HALE aircraft optimization framework which accounts for the peculiarities of structurally flexible aircraft while remaining suitable for use with gradient-based optimization. These aircraft tend to be extremely large and light, which often leads to significant amounts of structural flexibility. Using this optimization framework, we design an aircraft that is capable of flying year-round at \SI{35}{\degree} latitude at \SI{18}{\kilo\meter} above sea level. We subject this aircraft to a number of constraints including energy capture, energy storage, material failure, local buckling, stall, static stability, and dynamic stability constraints. Critically, these constraints were designed to accurately model the actual design requirements of SR-HALE aircraft, rather than to provide a rough approximation of them. To demonstrate the design exploration capabilities of this framework, we also performed several parameters sweeps to determine optimal design sensitivities to altitude, latitude, battery specific energy, solar efficiency, avionics and payload power requirements, and minimum design velocity. Through this optimization framework, we demonstrate both the potential of gradient-based optimization applied to highly flexible aeroelastic structures and the challenges associated with it. One challenge associated with the gradient-based optimization of highly flexible aeroelastic structures, is the ability to accurately, efficiently, and reliably model the large deflections of these structures in gradient-based optimization frameworks. To enable large-scale optimization involving structural models with large deflections to be performed more easily, we present a finite-element implementation of geometrically exact beam theory which is designed specifically for gradient-based optimization. A key feature of this implementation of geometrically exact beam theory is its compatibility with forward and reverse-mode automatic differentiation, which allows accurate design sensitivities to be obtained with minimal development effort. Another key feature is its native support for unsteady adjoint sensitivity analysis, which allows design sensitivities to be obtained efficiently from time-marching simulations. Other features are also presented that build upon previous implementations of geometrically exact beam theory, including a singularity-free rotation parameterization based on Wiener-Milenkovi\'c parameters, an implementation of stiffness-proportional structural damping using a discretized form of the compatibility equations, and a reformulation of the equations of motion for geometrically exact beam theory from a fully implicit index-1 differential algebraic equation to a semi-explicit index-1 differential algebraic equation. Several examples are presented which verify the utility and validity of each of these features. Another challenge associated with the gradient-based optimization of highly flexible aeroelastic structures is the ability to reliably track and constrain individual dynamic stability modes across the design iterations of an optimization framework. To facilitate the development of mode-specific dynamic stability constraints in gradient-based optimization frameworks we develop a mode tracking method that uses an adaptive step size in order to maintain an arbitrarily high degree of confidence in mode correlations. This mode tracking method is then applied to track the modes of a linear two-dimensional aeroelastic system and a nonlinear three-dimensional aeroelastic system as velocity is increased. When used in a gradient-based optimization framework, this mode tracking method has the potential to allow continuous dynamic stability constraints to be constructed without constraint aggregation. It also has the potential to allow the stability and shape of specific modes to be constrained independently. Finally, to facilitate the development and use of highly flexible aeroelastic systems for use in gradient-based optimization frameworks, we introduce a general methodology for coupling aerodynamic and structural models together to form modular monolithic aeroelastic systems. We also propose efficient methods for computing the Jacobians of these coupled systems without significantly increasing the amount of time necessary to construct these systems. For completeness we also discuss how to ensure that the resulting system of equations constitutes a set of first-order index-1 differential algebraic equations. We then derive direct and adjoint sensitivities for these systems which are compatible with automatic differentiation so that derivatives for gradient-based optimization can be obtained with minimal development effort.

The overall goals of this dissertation are to advance the field of chemical data assimilation, and to develop efficient computational tools that allow the atmospheric science community benefit from state of the art assimilation methodologies. Data assimilation is the procedure to combine data from observations with model predictions to obtain a more accurate representation of the state of the atmosphere. As models become more complex, determining the relationships between pollutants and their sources and sinks becomes computationally more challenging. The construction of an adjoint model ( capable of efficiently computing sensitivities of a few model outputs with respect to many input parameters ) is a difficult, labor intensive, and error prone task. This work develops adjoint systems for two of the most widely used chemical transport models: Harvard's GEOS-Chem global model and for Environmental Protection Agency's regional CMAQ regional air quality model. Both GEOS-Chem and CMAQ adjoint models are now used by the atmospheric science community to perform sensitivity analysis and data assimilation studies. Despite the continuous increase in capabilities, models remain imperfect and models alone cannot provide accurate long term forecasts. Observations of the atmospheric composition are now routinely taken from sondes, ground stations, aircraft, and satellites, etc. This work develops three and four dimensional variational data assimilation capabilities for GEOS-Chem and CMAQ which allow to estimate chemical states that best fit the observed reality. Most data assimilation systems to date use diagonal approximations of the background covariance matrix which ignore error correlations and may lead to inaccurate estimates. This dissertation develops computationally efficient representations of covariance matrices that allow to capture spatial error correlations in data assimilation. Not all observations used in data assimilation are of equal importance. Erroneous and redundant observations not only affect the quality of an estimate but also add unnecessary computational expense to the assimilation system. This work proposes techniques to quantify the information content of observations used in assimilation; information-theoretic metrics are used. The four dimensional variational approach to data assimilation provides accurate estimates but requires an adjoint construction, and uses considerable computational resources. This work studies versions of the four dimensional variational methods (Quasi 4D-Var) that use approximate gradients and are less expensive to develop and run. Variational and Kalman filter approaches are both used in data assimilation, but their relative merits and disadvantages in the context of chemical data assimilation have not been assessed. This work provides a careful comparison on a chemical assimilation problem with real data sets. The assimilation experiments performed here demonstrate for the first time the benefit of using satellite data to improve estimates of tropospheric ozone. / Ph. D.

Dans ce travail, une approche d’optimisation dynamique hybride est développée et utilisée pour simuler et optimiser les procédés d’adsorption modulée en pression (PSA). Elle est principalement basée sur la formulation hybride du modèle du procédé et sur l’utilisation de la méthode du système adjoint.Le problème de simulation qui consiste à déterminer le régime stationnaire cyclique (CSS) est formulé comme un problème d’optimisation où le critère de performance est défini par la condition de CSS, les variables de décision sont données par les valeurs initiales des variables d’état, et les contraintes par le modèle hybride du procédé avec les conditions aux limites associées. En optimisation, le vecteur des variables de décision contient, en plus des valeurs initiales de l’état, les paramètres de dimensionnement et de fonctionnement. La condition de CSS devient, dans ce cas, une contrainte à satisfaire par chaque solution optimale. Plusieurs modèles de procédés, allant du plus simple au plus compliqué, sont ´étudiés.Il s’agit notamment de procédés isothermes et non isothermes avec et sans états gelés. Les critères de performance considérés sont la pureté, la récupération et l´énergie. Les résultats obtenus aussi bien au niveau des performances des procédés considérés que de la robustesse de l’algorithme d’optimisation mis en œuvre, sont tout `a fait intéressants et montrent le grand potentiel de l’approche développée pour le dimensionnement et le fonctionnement optimaux des procédés PSA / The objective of the work was to develop a hybrid dynamic optimisation approach for simulation and optimisation of pressure swing adsorption (PSA) processes. It is mainly based on the hybrid formulation of the process model and on the use of adjoint system method. The simulation problem which consists in determining the cyclic steady state (CSS) is formulated as an optimisation problem where the CSS condition is considered as the performance index, initial values of state variables as decision variables and process model along with associated conditions as constraints. In optimisation, the decision vector consists of design and operation parameters in addition to the initials values of state variables whereas the CSS condition is considered in this case as a constraint to be satisfied for each optimal solution. Several process models with a varied degree of complexity have been studied. These models are isothermal and non isothermal with and without frozen states. The performance index considered are purity, recovery and energy. The results obtained are interesting vis-a-vis the performance of the processes considered as well as the robustness of the optimisation algorithm and show the great potential of the approach developed for the optimal design and operation of PSA processes