Financial engineering problems are of great importance in the academic community and BlackScholes equation is a revolutionary concept in the modern financial theory. Financial instruments such as stocks and derivatives can be evaluated using this model. Option evaluation, is extremely important to trade in the stocks. The numerical solutions of the Black-Scholes equation are used to simulate these options. In this thesis, the explicit and the implicit Euler methods are used for the approximation of Black-scholes partial differential equation and a second order finite difference scheme is used for the spatial derivatives. These temporal and spatial discretizations are used to gain an insight about the stability properties of the explicit and the implicit methods in general. The numerical results show that the explicit methods have some constraints on the stability, whereas, the implicit Euler method is unconditionally stable. It is also demostrated that both the explicit and the implicit Euler methods are only first order convergent in time and this implies too small step-sizes to achieve a good accuracy.

The thesis is dedicated to the study and numerical analysis of the non- linear flows in the porous media, using general Forchheimer models. In the numerical analysis, the local discontinuous Galerkin method is chosen. The first part of the paper is dedicated to the derivation of the studied equations based on the physical motivation and summarizing the theory needed for the further analysis. Core of the thesis consists of the introduction of the chosen discretization method and the derivation of the main stability and a priory error estimates, optimal for the linear ansatz functions. At the end we present a couple of numerical experiments to verify the results. 1

<div>In this work, we introduce a framework for constructing light-weight neural network approximations to the solution operators for partial differential equations (PDEs). Using a data-driven offline training procedure, the resulting operator network models are able to effectively reduce the computational demands of traditional numerical methods into a single forward-pass of a neural network. Importantly, the network models can be calibrated to specific distributions of input data in order to reflect properties of real-world data encountered in practice. The networks thus provide specialized solvers tailored to specific use-cases, and while being more restrictive in scope when compared to more generally-applicable numerical methods (e.g. procedures valid for entire function spaces), the operator networks are capable of producing approximations significantly faster as a result of their specialization.</div><div><br></div><div>In addition, the network architectures are designed to place pointwise posterior distributions over the observed solutions; this setup facilitates simultaneous training and uncertainty quantification for the network solutions, allowing the models to provide pointwise uncertainties along with their predictions. An analysis of the predictive uncertainties is presented with experimental evidence establishing the validity of the uncertainty quantification schema for a collection of linear and nonlinear PDE systems. The reliability of the uncertainty estimates is also validated in the context of both in-distribution and out-of-distribution test data.</div><div><br></div><div>The proposed neural network training procedure is assessed using a novel convolutional encoder-decoder model, ConvPDE-UQ, in addition to an existing fully-connected approach, DeepONet. The convolutional framework is shown to provide accurate approximations to PDE solutions on varying domains, but is restricted by assumptions of uniform observation data and homogeneous boundary conditions. The fully-connected DeepONet framework provides a method for handling unstructured observation data and is also shown to provide accurate approximations for PDE systems with inhomogeneous boundary conditions; however, the resulting networks are constrained to a fixed domain due to the unstructured nature of the observation data which they accommodate. These two approaches thus provide complementary frameworks for constructing PDE-based operator networks which facilitate the real-time approximation of solutions to PDE systems for a broad range of target applications.</div>

<p>In this work, we mainly focus on the topic related to dimension reduction, operator learning and uncertainty quantification for problems of differential equations. The supervised machine learning methods introduced here belong to a newly booming field compared to traditional numerical methods. The building blocks for our works are mainly Gaussian process and neural network. </p> <p><br></p> <p>The first work focuses on supervised dimension reduction problems. A new framework based on rotated multi-fidelity Gaussian process regression is introduced. It can effectively solve high-dimensional problems while the data are insufficient for traditional methods. Moreover, an accurate surrogate Gaussian process model of original problem can be formulated. The second one we would like to introduce is a physics-assisted Gaussian process framework with active learning for forward and inverse problems of partial differential equations(PDEs). In this work, Gaussian process regression model is incorporated with given physical information to find solutions or discover unknown coefficients of given PDEs. Three different models are introduce and their performance are compared and discussed. Lastly, we propose attention based MultiAuto-DeepONet for operator learning of stochastic problems. The target of this work is to solve operator learning problems related to time-dependent stochastic differential equations(SDEs). The work is built on MultiAuto-DeepONet and attention mechanism is applied to improve the model performance in specific type of problems. Three different types of attention mechanism are presented and compared. Numerical experiments are provided to illustrate the effectiveness of our proposed models.</p>

Resumen: En esta tesis doctoral se presentan tres modelos matemáticos que describen el comportamiento de dos enfermedades parasitarias con vector de transmisión; de los cuales dos modelos están dedicados a la Toxoplasmosis donde se explora la dinámica de la enfermedad a nivel de la población humana y de gatos domésticos. Los gatos juegan un papel de agentes infecciosos del Toxoplasma gondii. La dinámica cualitativa del modelo es determinada por el umbral básico de reproducción, R0. Si el parámetro R0 < 1, entonces la solución converge al punto de equilibrio libre de la enfermedad. Por otro lado, si R0 > 1, la convergencia es al punto de equilibrio endémico. Las simulaciones numéricas ilustran diferentes dinámicas en función del parámetro umbral R0 y muestra la importancia de este parámetro en el sector salud. Finalmente la Babesiosis bovina se modela a partir de cinco ecuaciones diferenciales ordinarias, que permiten explicar la influencia de los parámetros epidemiológicos en la evolución de la enfermedad. Los estados estacionarios del sistema y el número básico de reproducción R0 son determinados. La existencia del punto endémico y libre de enfermedad se expone, puntos que dependen del R0, ratificando la importancia del parámetro umbral en la salud publica. Objetivo: Construir modelos matemáticos epidemiológicos aplicados a enfermedades parasitarias (Toxoplasmosis y Babesiosis) con vector de transmisión. Metodología: Para la construcción de los modelos matemáticos epidemiológicos es necesario representar la enfermedad a partir de modelos de flujo, permitiendo ver la dinámica de la población entre los diferentes estadíos de la enfermedad, dichos movimientos son analizados a partir de sistemas dinámicos, análisis matemático y métodos numéricos; con estas herramientas es posible hacer un estudio detallado del modelo, permitiendo calcular parámetros umbrales que dominan la dinámica de la enfermedad y a su vez simular escenarios reales e hipotéticos. / Aranda Lozano, DF. (2011). Modeling of parasitic diseases with vector of transmission: toxoplasmosis and babesiosis bovine [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/11539

Oceanic waves registered by satellite observations often have curvilinear fronts and propagate over various currents. In this thesis, we study long linear and weakly-nonlinear ring waves in a stratified fluid in the presence of a depth-dependent horizontal shear flow. It is shown that despite the clashing geometries of the waves and the shear flow, there exists a linear modal decomposition, which can be used to describe distortion of the wavefronts of surface and internal waves, and systematically derive a 2+1-dimensional cylindrical Korteweg-de Vries (cKdV)-type equation for the amplitudes of the waves. The general theory is applied to the case of the waves in a two-layer fluid with a piecewise-constant shear flow, with an emphasis on the effect of the shear flow on the geometry of the wavefronts. The distortion of the wavefronts is described by the singular solution (envelope of the general solution) of the nonlinear first order differential equation, constituting generalisation of the dispersion relation in this curvilinear geometry. There exists a striking difference in the shape of the wavefronts: the wavefront of the surface wave is elongated in the shear flow direction while the wavefront of the interfacial wave is squeezed in this direction. We solve the derived 2+1-dimensional cKdV-type equation numerically using a finite-difference scheme. The effects of nonlinearity and dispersion are studied by considering numerical results for surface and interfacial ring waves generated from a localised source with and without shear flow and the 2D dam break problem. In these examples, the linear and nonlinear surface waves are faster than interfacial waves, the wave height decreases faster at the surface, the shear flow leads to the wave height decreasing slower downstream and faster upstream, and the effect becomes more prominent as the shear flow strengthens.

<p> In this thesis, both direct and inverse elastic scattering problems are considered. For a given incident wave, the direct problem is to determine the displacement of wave field from the known structure, which could be an obstacle or a surface in this thesis; The inverse problem is to determine the structure from the measurement of displacement on an artificial boundary.</p><p>In the second chapter, we consider the scattering of an elastic plane wave by a rigid obstacle, which is immersed in a homogeneous and isotropic elastic medium in two dimensions. Based on a Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the scattering problem is formulated as a boundary value problem of the elastic wave equation in a bounded domain. By developing a new duality argument, an a posteriori error estimate is derived for the discrete problem by using the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator which decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed to solve the elastic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error.<br></p><p>In chapter 3, we extend the argument developed in chapter 2 to elastic surface grating problem, where the surface is assumed to be periodic and elastic rigid; Then, we treat the obstacle scattering in three dimensional space; The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is studied and a frequency continuation method is developed for the inverse problem. Finally, in chapter 4, a rigorous mathematical model and an efficient computational method are proposed to solve the inverse elastic surface scattering problem which arises from the near-field imaging of periodic structures. The surface is assumed to be a small and smooth perturbation of an elastically rigid plane. By placing a rectangle slab of a homogeneous and isotropic elastic medium with larger mass density above the surface, more propagating wave modes can be utilized from the far-field data which contributes to the reconstruction resolution. Requiring only a single illumination, the method begins with the far-to-near field data conversion and utilized the transformed field expansion to derive an analytic solution for the direct problem, which leads to an explicit inversion formula for the inverse problem; Moreover, a nonlinear correction scheme is developed to improve the accuracy of the reconstruction; Numerical examples are presented to demonstrate the effectiveness of the proposed methods for solving the questions mentioned above.<br></p>

The thesis presents a comprehensive study of a posteriori error estimation in the adaptive solution to some classical elliptic partial differential equations. Several new error estimators are proposed for diffusion problems with discontinuous coefficients and for convection-reaction-diffusion problems with dominated convection/reaction. The robustness of the new estimators is justified theoretically. Extensive numerical results demonstrate the robustness of the new estimators for challenging problems and indicate that, compared to the well-known residual-type estimators, the new estimators are much more accurate.

No trabalho desenvolvido estudaram-se as forças que atuam em uma partícula quando esta se movimenta em escoamentos, com intuito de obter uma metodologia capaz de representar o movimento de uma partícula em um escoamento. A equação do movimento da partícula foi integrada numericamente considerando os termos de massa aparente, arrasto estacionário, arrasto não estacionário (forças de Boussinesq/Basset) e forças de sustentação; efeito Magnus e efeito Saffman. O método dos volumes finitos foi utilizado para simulação do escoamento. Na análise das forças utilizamos tanto experimentos quanto simulações numéricas (FLUENT) para avaliar e aumentar a validade dos modelos apresentados na revisão bibliográfica. O FLUENT foi validado para obtenção do coeficiente de arrasto estacionário e sustentação devido ao efeito Magnus. Palavras-chaves: Efeito Magnus, efeito Saffman, força de Bousinesq/Basset, movimento de partículas e solução numérica. / In the developed work was studied the forces which act on a particle when these is a moving inside of a flow, in order to find out a methodology which is able to represent the particle dynamics on a flow. The equation of particle motion was integrated with a numerical approach taking in account the apparent mass, static drag, dynamic drag (history term; Boussinesq/Basset force) and lift force; Magnus effect and Saffman effect. The finite volume method was used to simulate the flow. In the force analyses we used experimental and numerical simulation (FLUENT) to evaluate and extend the models shown on the review. FLUENT was validated to determine the static drag coefficient and lift coefficient due to Magnus effect.

No trabalho desenvolvido estudaram-se as forças que atuam em uma partícula quando esta se movimenta em escoamentos, com intuito de obter uma metodologia capaz de representar o movimento de uma partícula em um escoamento. A equação do movimento da partícula foi integrada numericamente considerando os termos de massa aparente, arrasto estacionário, arrasto não estacionário (forças de Boussinesq/Basset) e forças de sustentação; efeito Magnus e efeito Saffman. O método dos volumes finitos foi utilizado para simulação do escoamento. Na análise das forças utilizamos tanto experimentos quanto simulações numéricas (FLUENT) para avaliar e aumentar a validade dos modelos apresentados na revisão bibliográfica. O FLUENT foi validado para obtenção do coeficiente de arrasto estacionário e sustentação devido ao efeito Magnus. Palavras-chaves: Efeito Magnus, efeito Saffman, força de Bousinesq/Basset, movimento de partículas e solução numérica. / In the developed work was studied the forces which act on a particle when these is a moving inside of a flow, in order to find out a methodology which is able to represent the particle dynamics on a flow. The equation of particle motion was integrated with a numerical approach taking in account the apparent mass, static drag, dynamic drag (history term; Boussinesq/Basset force) and lift force; Magnus effect and Saffman effect. The finite volume method was used to simulate the flow. In the force analyses we used experimental and numerical simulation (FLUENT) to evaluate and extend the models shown on the review. FLUENT was validated to determine the static drag coefficient and lift coefficient due to Magnus effect.