More often than not, we cannot directly measure many phenomena that are crucial to us. However, we usually have access to certain partial observations on the phenomena of interest as well as a mathematical model of them. The filtering problem seeks estimation of the phenomena given all the accumulated partial information. In this thesis, we study several topics concerning the numerical approximation of the filtering problem. First, we study the continuous-time filtering problem. Given high-frequency ob- servations in discrete-time, we perform double discretization of the non-linear filter to allow for filter estimation with particle filter. By using the multilevel strategy, given any ε > 0, our algorithm achieve an MSE level of O(ε2) with a cost of O(ε−3), while the particle filter requires a cost of O(ε−4). Second, we propose a de-bias scheme for the particle filter under the partially observed diffusion model. The novel scheme is free of innate particle filter bias and discretization bias, through a double randomization method of [14]. Our estimator is perfectly parallel and achieves a similar cost reduction to the multilevel particle filter. Third, we look at a high-dimensional linear Gaussian state-space model in con- tinuous time. We propose a novel multilevel estimator which requires a cost of O(ε−2 log(ε)2) compared to ensemble Kalman-Bucy filters (EnKBFs) which requiresO(ε−3) for an MSE target of O(ε2). Simulation results verify our theory for models of di- mension ∼ 106. Lastly, we consider the model estimation through learning an unknown parameter that characterizes the partially observed diffusions. We propose algorithms to provide unbiased estimates of the Hessian and the inverse Hessian, which allows second-order optimization parameter learning for the model.

This work proposes the computational modeling of two-dimensional media mechanical behavior with a continuous approach, related to the Finite Element Method (FEM) associated with Cohesive Fracture Mechanics (CFM) models, and a discrete approach, using the Discrete Element Method (DEM). The FEM consists in a numerical tool widely used to achieve approximate solutions of Continuum Mechanics problems, involving physical and geometrical nonlinearities phenomena with quasi-static or dynamic behaviors, having already established practical applications in many areas of science and industry. On the other hand, DEM has more recent development and has been increasingly used to model discrete nature problems involving contact, impact and fragmentation phenomena and flow of particulate systems. Focused on analysis of problems with interactions between these natures, a FEMDEM coupling code is developed to solve the problem by a sub-region scheme where the FEM is used on modeling of nucleation process and crack propagation in continuous media, and DEM is employed to model granular media, whether due its nature or its conception, in a transient behavior. The possibility of opening and propagation of cracks is considered by using CFM models, intrinsically incorporated into the FEM formulation through interfaces inserted into the inner edges of the finite element mesh. Illustrative examples are presented and discussed in order to validate the proposed formulation and implementation. / FUNDEPES - Fundação Universitária de Desenvolvimento de extensão e Pesquisa / PRH-ANP - Programa de Recursos Humanos da Agência Nacional do Petróleo / Este trabalho propõe a modelagem computacional do comportamento mecânico bidimensional de meios com abordagens contínua, relacionada ao Método dos Elementos Finitos (MEF) associado a modelos da Mecânica da Fratura Coesiva (MFC) e discreta, através do Método dos Elementos Discretos (MED). O MEF consiste em uma ferramenta numérica bastante utilizada na determinação de soluções aproximadas para problemas da Mecânica do Contínuo, envolvendo fenômenos com não linearidades físicas e geométricas associadas e com comportamento quase-estático ou dinâmico, possuindo aplicações práticas já consagradas em diversas áreas do campo científico e industrial. Por outro lado, o MED tem desenvolvimento mais recente e vem sendo cada vez mais utilizado no tratamento de problemas de natureza discreta envolvendo fenômenos de contato, impacto, fragmentação e fluxo de sistemas particulados. Com foco na análise de problemas que envolvem interações entre tais naturezas, implementa-se uma estratégia de acoplamento MEF-MED para solução do problema em subregiões, onde o MEF é utilizado na modelagem de processos de nucleação e propagação de fraturas em meios contínuos e o MED é empregado na modelagem de meios granulares por natureza, ou assim concebidos, em comportamento transiente. A possibilidade de abertura e propagação de fraturas é considerada através da utilização de modelos da MFC, incorporados intrinsecamente na formulação do MEF através de interfaces inseridas nas arestas internas da malha de elementos finitos. Exemplos ilustrativos são apresentados e discutidos visando-se validar a formulação e a implementação propostas.