We consider high order discontinuous-Galerkin finite element methods for partial differential equations, with a focus on the neutron transport equation. We begin by examining a method for preprocessing block-sparse matrices, of the type that arise from discontinuous-Galerkin methods, prior to factorisation by a multifrontal solver. Numerical experiments on large two and three dimensional matrices show that this pre-processing method achieves a significant reduction in fill-in, when compared to methods that fail to exploit block structures. A discontinuous-Galerkin finite element method for the neutron transport equation is derived that employs high order finite elements in both space and angle. Parallel Krylov subspace based solvers are considered for both source problems and $k_{eff}$-eigenvalue problems. An a-posteriori error estimator is derived and implemented as part of an h-adaptive mesh refinement algorithm for neutron transport $k_{eff}$-eigenvalue problems. This algorithm employs a projection-based error splitting in order to balance the computational requirements between the spatial and angular parts of the computational domain. An hp-adaptive algorithm is presented and results are collected that demonstrate greatly improved efficiency compared to the h-adaptive algorithm, both in terms of reduced computational expense and enhanced accuracy. Computed eigenvalues and effectivities are presented for a variety of challenging industrial benchmarks. Accurate error estimation (with effectivities of 1) is demonstrated for a collection of problems with inhomogeneous, irregularly shaped spatial domains as well as multiple energy groups. Numerical results are presented showing that the hp-refinement algorithm can achieve exponential convergence with respect to the number of degrees of freedom in the finite element space

The paper describes a storage scheme for sparse symmetric or nonsymmetric matrices which has been developed and used for many years at the Technical University of Chemnitz. An overview of existing library subroutines using such matrices is included.

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The first part of this thesis presents a convenient and flexible calibration method to estimate the relative rotation and translation among multiple cameras. A simple planar pattern is used for accurate calibration and is not required to be simultaneously observed by all cameras. Thus the method is especially suitable for widely spaced camera array. In order to fairly evaluate the calibration results for different camera setups, a novel accuracy metric is introduced based on the deflection angles of projection rays, which is insensitive to a number of setup factors. / The objective of this thesis is to develop a multiview system that can synthesize photorealistic novel views of the scene captured by sparse cameras distributed in a wide area. The system cost is largely reduced due to the small number of required cameras, and the image capture is greatly facilitated because the cameras are allowed to be widely spaced and flexibly placed. The key techniques to achieve this goal are investigated in this thesis. / Cui, Chunhui. / "November 2010." / Adviser: Ngan King Ngi. / Source: Dissertation Abstracts International, Volume: 73-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 140-155). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

The numerical solution of Stein (aka discrete Lyapunov) equations is the primary step in Newton's method for the solution of discrete-time algebraic Riccati equations (DARE). Here we present a low-rank Smith method as well as a low-rank alternating-direction-implicit-iteration to compute low-rank approximations to solutions of Stein equations arising in this context. Numerical results are given to verify the efficiency and accuracy of the proposed algorithms.

No description available.

Dissertation ueber die effektive Vorkonditionierung linearer Gleichungssysteme resultierend aus der Diskretisierung eines elliptischen Randwertproblems 2. Ordnung mittels der Methode der Finiten Elementen. Als Vorkonditionierer werden multi-level artige Vorkonditionierer (BPX, Multi-grid, Wavelets) benutzt.

Large sparse matrices arise in many applications, especially in the major problems of Cryptography of factoring integers and computing discrete logarithms. We focus attention on such matrices called sieve matrices generated after the sieving stage of the algorithms for integer factoring. We need to solve large sparse system of equations Bx = 0, with sieve matrices B arising in this context. The traditional Gaussian elimination, with a cubic run time, is not efficient for handling such matrices. Better algorithms for such input matrices are the quadratic runtime algorithms based on Block Lanczos(BL) or Wiedemann techniques. Of these two, BL is even better for large integer factoring algorithms. We carry out an efficient implementation of the Block Lanczos algorithm for finding the vectors in the null space of the the sieve matrix. We report our test results using our implementation for matrices of sizes up to 106. We plan to use this implementation in our ongoing projects on factoring the large RSA challenge integers of sizes 640 bits(called RSA-640) and beyond. So it is useful to exploit possible parallelism. We propose a scheme for parallelizing certain steps of the Block Lanczos method, taking advantage of structural properties of the sieve matrix. The sizes of matrices arising in integer factoring context are quite large. Hence we also discuss some techniques that are used to reduce the size of the sieve matrix. We also consider the last stage of the NFS Algorithm for finding square roots of large algebraic numbers and outline a sketch of our algorithm.

This thesis presents the mathematical analysis of a continuous-times system for sparse signal recovery. Sparse recovery arises in Compressed Sensing (CS), where signals of large dimension must be recovered from a small number of linear measurements, and can be accomplished by solving a complex optimization program. While many solvers have been proposed and analyzed to solve such programs in digital, their high complexity currently prevents their use in real-time applications. On the contrary, a continuous-time neural network implemented in analog VLSI could lead to significant gains in both time and power consumption. The contributions of this thesis are threefold. First, convergence results for neural networks that solve a large class of nonsmooth optimization programs are presented. These results extend previous analysis by allowing the interconnection matrix to be singular and the activation function to have many constant regions and grow unbounded. The exponential convergence rate of the networks is demonstrated and an analytic expression for the convergence speed is given. Second, these results are specialized to the L1-minimization problem, which is the most famous approach to solving the sparse recovery problem. The analysis relies on standard techniques in CS and proves that the network takes an efficient path toward the solution for parameters that match results obtained for digital solvers. Third, the convergence rate and accuracy of both the continuous-time system and its discrete-time equivalent are derived in the case where the underlying sparse signal is time-varying and the measurements are streaming. Such a study is of great interest for practical applications that need to operate in real-time, when the data are streaming at high rates or the computational resources are limited. As a conclusion, while existing analysis was concentrated on discrete-time algorithms for the recovery of static signals, this thesis provides convergence rate and accuracy results for the recovery of static signals using a continuous-time solver, and for the recovery of time-varying signals with both a discrete-time and a continuous-time solver.

Cette thèse traite des méthodes d’éléments finis Galerkin discontinus d’ordre élevé pour la résolution d’équations aux dérivées partielles, avec un intérêt particulier pour l’équation de transport des neutrons. Nous nous intéressons tout d’abord à une méthode de pré-traitement de matrices creuses par blocs, qu’on retrouve dans les méthodes Galerkin discontinues, avant factorisation par un solveur multifrontal. Des expériences numériques conduites sur de grandes matrices bi- et tri-dimensionnelles montrent que cette méthode de pré-traitement permet une réduction significative du ’fill-in’, par rapport aux méthodes n’exploitant pas la structure par blocs. Ensuite, nous proposons une méthode d’éléments finis Galerkin discontinus, employant des éléments d’ordre élevé en espace comme en angle, pour résoudre l’équation de transport des neutrons. Nous considérons des solveurs parallèles basés sur les sous-espaces de Krylov à la fois pour des problèmes ’source’ et des problèmes aux valeur propre multiplicatif. Dans cet algorithme, l’erreur est décomposée par projection(s) afin d’équilibrer les contraintes numériques entre les parties spatiales et angulaires du domaine de calcul. Enfin, un algorithme HP-adaptatif est présenté ; les résultats obtenus démontrent une nette supériorité par rapport aux algorithmes h-adaptatifs, à la fois en terme de réduction de coût de calcul et d’amélioration de la précision. Les valeurs propres et effectivités sont présentées pour un panel de cas test industriels. Une estimation précise de l’erreur (avec effectivité de 1) est atteinte pour un ensemble de problèmes aux domaines inhomogènes et de formes irrégulières ainsi que des groupes d’énergie multiples. Nous montrons numériquement que l’algorithme HP-adaptatif atteint une convergence exponentielle par rapport au nombre de degrés de liberté de l’espace éléments finis. / We consider high order discontinuous-Galerkin finite element methods for partial differential equations, with a focus on the neutron transport equation. We begin by examining a method for preprocessing block-sparse matrices, of the type that arise from discontinuous-Galerkin methods, prior to factorisation by a multifrontal solver. Numerical experiments on large two and three dimensional matrices show that this pre-processing method achieves a significant reduction in fill-in, when compared to methods that fail to exploit block structures. A discontinuous-Galerkin finite element method for the neutron transport equation is derived that employs high order finite elements in both space and angle. Parallel Krylov subspace based solvers are considered for both source problems and $k_{eff}$-eigenvalue problems. An a-posteriori error estimator is derived and implemented as part of an h-adaptive mesh refinement algorithm for neutron transport $k_{eff}$-eigenvalue problems. This algorithm employs a projection-based error splitting in order to balance the computational requirements between the spatial and angular parts of the computational domain. An hp-adaptive algorithm is presented and results are collected that demonstrate greatly improved efficiency compared to the h-adaptive algorithm, both in terms of reduced computational expense and enhanced accuracy. Computed eigenvalues and effectivities are presented for a variety of challenging industrial benchmarks. Accurate error estimation (with effectivities of 1) is demonstrated for a collection of problems with inhomogeneous, irregularly shaped spatial domains as well as multiple energy groups. Numerical results are presented showing that the hp-refinement algorithm can achieve exponential convergence with respect to the number of degrees of freedom in the finite element space