In this dissertation, we develop multiscale finite element methods and uncertainty quantification technique for Richards' equation, a mathematical model to describe fluid flow in unsaturated porous media. Both coarse-level and fine-level numerical computation techniques are presented. To develop an accurate coarse-scale numerical method, we need to construct an effective multiscale map that is able to capture the multiscale features of the large-scale solution without resolving the small scale details. With a careful choice of the coarse spaces for multiscale finite element methods, we can significantly reduce errors. We introduce several methods to construct coarse spaces for multiscale finite element methods. A coarse space based on local spectral problems is also presented. The construction of coarse spaces begins with an initial choice of multiscale basis functions supported in coarse regions. These basis functions are complemented using weighted local spectral eigenfunctions. These newly constructed basis functions can capture the small scale features of the solution within a coarse-grid block and give us an accurate coarse-scale solution. However, it is expensive to compute the local basis functions for each parameter value for a nonlinear equation. To overcome this difficulty, local reduced basis method is discussed, which provides smaller dimension spaces with which to compute the basis functions. Robust solution techniques for Richards' equation at a fine scale are discussed. We construct iterative solvers for Richards' equation, whose number of iterations is independent of the contrast. We employ two-level domain decomposition pre-conditioners to solve linear systems arising in approximation of problems with high contrast. We show that, by using the local spectral coarse space for the preconditioners, the number of iterations for these solvers is independent of the physical properties of the media. Several numerical experiments are given to support the theoretical results. Last, we present numerical methods for uncertainty quantification applications for Richards' equation. Numerical methods combined with stochastic solution techniques are proposed to sample conductivities of porous media given in integrated data. Our proposed algorithm is based on upscaling techniques and the Markov chain Monte Carlo method. Sampling results are presented to prove the efficiency and accuracy of our algorithm.

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The availability of high performance computing clusters has allowed scientists and engineers to study more challenging problems. However, new algorithms need to be developed to take advantage of the new computer architecture (in particular, distributed memory clusters). Since the solution of linear systems still demands most of the computational effort in many problems (such as the approximation of partial differential equation models) iterative methods and, in particular, efficient preconditioners need to be developed. In this study, we consider application of incomplete LU (ILU) preconditioners for finite element models to partial differential equations. Since finite elements lead to large, sparse systems, reordering the node numbers can have a substantial influence on the effectiveness of these preconditioners. We study two implementations of the ILU preconditioner: a stucturebased method and a threshold-based method. The main emphasis of the thesis is to test a variety of breadth-first ordering strategies on the convergence properties of the preconditioned systems. These include conventional Cuthill-McKee (CM) and Reverse Cuthill-McKee (RCM) orderings as well as strategies related to the physical distance between nodes and post-processing methods based on relative sizes of associated matrix entries. Although the success of these methods were problem dependent, a number of tendencies emerged from which we could make recommendations. Finally, we perform a preliminary study of the multi-processor case and observe the importance of partitioning quality and the parallel ILU reordering strategy. / Master of Science

We report a Hessian-implicit optimization method to quickly solve the charge optimization problem over protein molecules: given a ligand and its complex with a receptor, determine the ligand charge distribution that minimizes the electrostatic free energy of binding. The new optimization couples boundary element method (BEM) and primal-dual interior point method (PDIPM); initial results suggest that the method scales much better than the previous methods. The quadratic objective function is the electrostatic free energy of binding where the Hessian matrix serves as an operator that maps the charge to the potential. The unknowns are the charge values at the charge points, and they are limited by equality and inequality constraints that model physical considerations, i.e. conservation of charge. In the previous approaches, finite-difference method is used to model the Hessian matrix, which requires significant computational effort to remove grid-based inaccuracies. In the novel approach, BEM is used instead, with precorrected FFT (pFFT) acceleration to compute the potential induced by the charges. This part will be explained in detail by Shihhsien Kuo in another talk. Even though the Hessian matrix can be calculated an order faster than the previous approaches, still it is quite expensive to find it explicitly. Instead, the KKT condition is solved by a PDIPM, and a Krylov based iterative solver is used to find the Newton direction at each step. Hence, only Hessian times a vector is necessary, which can be evaluated quickly using pFFT. The new method with proper preconditioning solves a 500 variable problem nearly 10 times faster than the techniques that must find a Hessian matrix explicitly. Furthermore, the algorithm scales nicely due to the robustness in number of IPM iterations to the size of the problem. The significant reduction in cost allows the analysis of much larger molecular system than those could be solved in a reasonable time using the previous methods. / Singapore-MIT Alliance (SMA)

Este trabalho apresenta a aplicação de técnicas de processamento paralelo na resolução de equações algébricas oriundas do Método de Elementos Finitos aplicado ao Eletromagnetismo, nos regimes estático e harmônico. As técnicas de programação paralelas utilizadas foram OpenMP, CUDA e GPUDirect, sendo esta última para as plataformas do tipo Multi-GPU. Os métodos iterativos abordados incluem aqueles do subespaço Krylov: Gradientes Conjugados, Gradientes Biconjugados, Conjugado Residual, Gradientes Biconjugados Estabilizados, Gradientes Conjugados para equações normais (CGNE e CGNR) e Gradientes Conjugados ao Quadrado. Todas as implementações fizeram uso das bibliotecas CUSP, CUSPARSE e CUBLAS. Para problemas estáticos, os seguintes pré-condicionadores foram adotados, todos eles com implementações paralelizadas e executadas na GPU: Decomposições Incompletas LU e de Cholesky, Multigrid Algébrico, Diagonal e Inversa Aproximada. Para os problemas harmônicos, apenas os dois primeiros pré-condicionadores foram utilizados, porém na sua versão sequencial, com execução na CPU, resultando em uma implementação híbrida CPU-GPU. As ferramentas computacionais desenvolvidas foram testadas na simulação de problemas de aterramento elétrico. No caso do regime harmônico, em que o fenômeno é regido pela Equação de Onda completa com perdas e não homogênea, a formulação adotada foi aquela em dois potenciais, A-V aresta-nodal. Em todas as situações, os aplicativos desenvolvidos para GPU apresentaram speedups apreciáveis, demonstrando a potencialidade dessa tecnologia para a simulação de problemas de larga escala na Engenharia Elétrica, com excelente relação custo-benefício. / This work presents the use of parallel processing techniques in Graphics Processing Units (GPU) for the solution of algebraic equations arising from the Finite Element modeling of electromagnetic phenomena, both in steadystate and time-harmonic regime. The techniques used were parallel programming OpenMP, CUDA and GPUDirect, the latter for those platforms of type Multi-GPU. The iterative methods discussed include those of the Krylov subspace: Conjugate Gradients, Bi-conjugate Gradients, Conjugate Residual, Bi-conjugate Gradients Stabilized, Conjugate Gradients for Normal Equations (CGNE and CGNR) and Conjugate Gradients Squared. All implementations have made use of CUSP, CUSPARSE and CUBLAS libraries. For the static problems, the following pre-conditioners were adopted, all with parallelized implementations and executed on the GPU: Incomplete decompositions, both LU and Cholesky, Algebraic Multigrid, Diagonal and Approximate Inverse. For the time-harmonic varying problems, only the first two pre-conditioners were used, but in their sequential version and running in the CPU, which yielded a hybrid CPU-GPU implementation. The developed computational tools were tested in the simulation of electrical grounding systems. In the case of the harmonic regime, in which the phenomenon is governed by the driven, lossy wave equation, the formulation adopted was that in two potential, the ungauged edge A-V formulation. In all cases, the developed GPU-based tools showed considerable speedups, showing that this is a promising technology for the simulation of large-scale Electrical Engineering problems, with excellent cost-benefit.

Este trabalho apresenta a aplicação de técnicas de processamento paralelo na resolução de equações algébricas oriundas do Método de Elementos Finitos aplicado ao Eletromagnetismo, nos regimes estático e harmônico. As técnicas de programação paralelas utilizadas foram OpenMP, CUDA e GPUDirect, sendo esta última para as plataformas do tipo Multi-GPU. Os métodos iterativos abordados incluem aqueles do subespaço Krylov: Gradientes Conjugados, Gradientes Biconjugados, Conjugado Residual, Gradientes Biconjugados Estabilizados, Gradientes Conjugados para equações normais (CGNE e CGNR) e Gradientes Conjugados ao Quadrado. Todas as implementações fizeram uso das bibliotecas CUSP, CUSPARSE e CUBLAS. Para problemas estáticos, os seguintes pré-condicionadores foram adotados, todos eles com implementações paralelizadas e executadas na GPU: Decomposições Incompletas LU e de Cholesky, Multigrid Algébrico, Diagonal e Inversa Aproximada. Para os problemas harmônicos, apenas os dois primeiros pré-condicionadores foram utilizados, porém na sua versão sequencial, com execução na CPU, resultando em uma implementação híbrida CPU-GPU. As ferramentas computacionais desenvolvidas foram testadas na simulação de problemas de aterramento elétrico. No caso do regime harmônico, em que o fenômeno é regido pela Equação de Onda completa com perdas e não homogênea, a formulação adotada foi aquela em dois potenciais, A-V aresta-nodal. Em todas as situações, os aplicativos desenvolvidos para GPU apresentaram speedups apreciáveis, demonstrando a potencialidade dessa tecnologia para a simulação de problemas de larga escala na Engenharia Elétrica, com excelente relação custo-benefício. / This work presents the use of parallel processing techniques in Graphics Processing Units (GPU) for the solution of algebraic equations arising from the Finite Element modeling of electromagnetic phenomena, both in steadystate and time-harmonic regime. The techniques used were parallel programming OpenMP, CUDA and GPUDirect, the latter for those platforms of type Multi-GPU. The iterative methods discussed include those of the Krylov subspace: Conjugate Gradients, Bi-conjugate Gradients, Conjugate Residual, Bi-conjugate Gradients Stabilized, Conjugate Gradients for Normal Equations (CGNE and CGNR) and Conjugate Gradients Squared. All implementations have made use of CUSP, CUSPARSE and CUBLAS libraries. For the static problems, the following pre-conditioners were adopted, all with parallelized implementations and executed on the GPU: Incomplete decompositions, both LU and Cholesky, Algebraic Multigrid, Diagonal and Approximate Inverse. For the time-harmonic varying problems, only the first two pre-conditioners were used, but in their sequential version and running in the CPU, which yielded a hybrid CPU-GPU implementation. The developed computational tools were tested in the simulation of electrical grounding systems. In the case of the harmonic regime, in which the phenomenon is governed by the driven, lossy wave equation, the formulation adopted was that in two potential, the ungauged edge A-V formulation. In all cases, the developed GPU-based tools showed considerable speedups, showing that this is a promising technology for the simulation of large-scale Electrical Engineering problems, with excellent cost-benefit.

Cette étude s’inscrit dans le domaine de la simulation électromagnétique des problèmes de grande taille tels que la diffraction d’ondes planes par de larges plateformes et le rayonnement d’antennes aéroportées. Elle consiste à développer une méthode combinant décomposition en sous-domaines et compression hiérarchique des équations intégrales de frontière. Pour cela, nous rappelons dans un premier temps les points importants de la méthode des équations intégrales de frontière et de leur compression hiérarchique par l’algorithme ACA (Adaptive Cross Approximation). Ensuite, nous présentons la formulation IE-DDM (Integral Equations – Domain Decomposition Method) obtenue à partir d’une représentation intégrale des sous-domaines. Les matrices résultant de la discrétisation de cette formulation sont stockées au format H-matrice (matricehiérarchique). Un solveur spécialement adapté à la résolution de la formulation IE-DDM et à sa représentation hiérarchique a été conçu. Cette étude met en évidence l’efficacité de la décomposition en sous-domaines en tant que préconditionneur des équations intégrales. De plus, la méthode développée est rapide pour la résolution des problèmes à incidences multiples ainsi que la résolution des problèmes basses fréquences / This thesis is about the electromagnetic simulation of large scale problems as the wave scattering from aircrafts and the airborne antennas radiation. It consists in the development of a method combining domain decomposition and hierarchical compression of the surface integral equations. First, we remind the principles of the boundary element method and the hierarchical representation of the surface integral equations with the Adaptive Cross Approximation algorithm. Then, we present the IE-DDM formulation obtained from a sub-domain integral representation. The matrices resulting of the discretization of the formulation are stored in the H-matrix format. A solver especially fitted with the hierarchical representation of the IE-DDM formulation has been developed. This study highlights the efficiency of the sub-domain decomposition as a preconditioner of the integral equations. Moreover, the method is fast for the resolution of multiple incidences and the resolution of low frequencies problems