Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Iowa, 2007. / Supervisor: Shaoping Xiao.. Includes bibliographical references (leaves 174-182).

Made available in DSpace on 2014-10-09T12:32:01Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T13:56:54Z (GMT). No. of bitstreams: 1 12899.pdf: 8071135 bytes, checksum: c51cd9839b73c11d92a566a85488515f (MD5) / Tese (Doutoramento) / IPEN/T / University of Michigan

Probabilistic ideas and tools have recently begun to permeate into several fields where they had traditionally not played a major role, including fields such as numerical linear algebra and optimization. One of the key ways in which these ideas influence these fields is via the development and analysis of randomized algorithms for solving standard and new problems of these fields. Such methods are typically easier to analyze, and often lead to faster and/or more scalable and versatile methods in practice. This thesis explores the design and analysis of new randomized iterative methods for solving linear systems and inverting matrices. The methods are based on a novel sketch-and-project framework. By sketching we mean, to start with a difficult problem and then randomly generate a simple problem that contains all the solutions of the original problem. After sketching the problem, we calculate the next iterate by projecting our current iterate onto the solution space of the sketched problem. The starting point for this thesis is the development of an archetype randomized method for solving linear systems. Our method has six different but equivalent interpretations: sketch-and-project, constrain-and-approximate, random intersect, random linear solve, random update and random fixed point. By varying its two parameters – a positive definite matrix (defining geometry), and a random matrix (sampled in an i.i.d. fashion in each iteration) – we recover a comprehensive array of well known algorithms as special cases, including the randomized Kaczmarz method, randomized Newton method, randomized coordinate descent method and random Gaussian pursuit. We also naturally obtain variants of all these methods using blocks and importance sampling. However, our method allows for a much wider selection of these two parameters, which leads to a number of new specific methods. We prove exponential convergence of the expected norm of the error in a single theorem, from which existing complexity results for known variants can be obtained. However, we also give an exact formula for the evolution of the expected iterates, which allows us to give lower bounds on the convergence rate. We then extend our problem to that of finding the projection of given vector onto the solution space of a linear system. For this we develop a new randomized iterative algorithm: stochastic dual ascent (SDA). The method is dual in nature, and iteratively solves the dual of the projection problem. The dual problem is a non-strongly concave quadratic maximization problem without constraints. In each iteration of SDA, a dual variable is updated by a carefully chosen point in a subspace spanned by the columns of a random matrix drawn independently from a fixed distribution. The distribution plays the role of a parameter of the method. Our complexity results hold for a wide family of distributions of random matrices, which opens the possibility to fine-tune the stochasticity of the method to particular applications. We prove that primal iterates associated with the dual process converge to the projection exponentially fast in expectation, and give a formula and an insightful lower bound for the convergence rate. We also prove that the same rate applies to dual function values, primal function values and the duality gap. Unlike traditional iterative methods, SDA converges under virtually no additional assumptions on the system (e.g., rank, diagonal dominance) beyond consistency. In fact, our lower bound improves as the rank of the system matrix drops. By mapping our dual algorithm to a primal process, we uncover that the SDA method is the dual method with respect to the sketch-and-project method from the previous chapter. Thus our new more general convergence results for SDA carry over to the sketch-and-project method and all its specializations (randomized Kaczmarz, randomized coordinate descent ... etc.). When our method specializes to a known algorithm, we either recover the best known rates, or improve upon them. Finally, we show that the framework can be applied to the distributed average consensus problem to obtain an array of new algorithms. The randomized gossip algorithm arises as a special case. In the final chapter, we extend our method for solving linear system to inverting matrices, and develop a family of methods with specialized variants that maintain symmetry or positive definiteness of the iterates. All the methods in the family converge globally and exponentially, with explicit rates. In special cases, we obtain stochastic block variants of several quasi-Newton updates, including bad Broyden (BB), good Broyden (GB), Powell-symmetric-Broyden (PSB), Davidon-Fletcher-Powell (DFP) and Broyden-Fletcher-Goldfarb-Shanno (BFGS). Ours are the first stochastic versions of these updates shown to converge to an inverse of a fixed matrix. Through a dual viewpoint we uncover a fundamental link between quasi-Newton updates and approximate inverse preconditioning. Further, we develop an adaptive variant of the randomized block BFGS (AdaRBFGS), where we modify the distribution underlying the stochasticity of the method throughout the iterative process to achieve faster convergence. By inverting several matrices from varied applications, we demonstrate that AdaRBFGS is highly competitive when compared to the well established Newton-Schulz and approximate preconditioning methods. In particular, on large-scale problems our method outperforms the standard methods by orders of magnitude. The development of efficient methods for estimating the inverse of very large matrices is a much needed tool for preconditioning and variable metric methods in the big data era.

The theme of diploma thesis concerning teaching and educational activities of a physical education teacher is very extensive. It comprises a great number of phenomena. For this reason this diploma thesis only concentrates on the teaching and educational methods that physical education teachers use in basic schools in the České Budějovice region. The diploma thesis mainly deals with quantitative illustration of observed phenomena. We applied a short-term observation method to determine the results. The research was executed in 7 basic schools and 20 teachers and 40 physical education classes were observed. The results were presented in graphs and tables.

Made available in DSpace on 2014-10-09T12:32:01Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T13:56:54Z (GMT). No. of bitstreams: 1 12899.pdf: 8071135 bytes, checksum: c51cd9839b73c11d92a566a85488515f (MD5) / Tese (Doutoramento) / IPEN/T / University of Michigan

Please read the abstract in the front section of this document / Dissertation (MSc (Food Science))--University of Pretoria, 2007. / Food Science / unrestricted

Typescript (photocopy).

Plant location as a centralized planning objective in which some agency has control over most of the system elements can be reduced, in many circumstances, to the problem of finding the medians of a weighted graph. This concept is feasible if it can be assumed that each location sought is constrained to a subset of p nodes on an n node network. This combinatorial programming problem can be formally stated as follows: if G is a weighted graph, [formula omitted] the weighted distance of node [symbol omitted] to node [symbol omitted], and Xp is any set of p nodes on G (x₁, x₂, •••,Xp), then the required set of p nodes Xp∗ on G is the p median of the graph if it satisfies the expression [formula omitted]. Although this objective can be explicitly optimized by branch bound algorithms, those developed to date can become computationally infeasible for some large scale problems. A fast method for estimating the medians of a weighted graph is given which will provide optimal or near optimal solutions on any type of network. The heuristic procedures adopted in this study can be generalized in terms of three basic steps; 1) partition the graph to obtain an initial feasible solution, 2) re-iterate over; step 1 to achieve a local minimum, and 3) perturb this convergence to test for a lower bound. The design of steps 1 and 3 are crucial to the success of the algorithmic method. Two procedures are given for the basic partitioning of the graph, one of which is a modification of a criterion originally developed by Singer (1968) . The other method introduces a node elimination recursion which appears, experimentally, to be the more efficient procedure for certain types of weighted networks. Efficient perturbation methods are developed for testing the lower bounds obtained. The basic model structure is modified by the introduction of heuristics for the constrained plant location problem under a wide variety of restrictions. Numerical procedures are suggested for restricting the search to a subset of m potential plant sites among all n nodes on the network. Heuristics are developed for forcing certain locations into solution, for placing upper bound constraints on plant sizes, and for restricting the maximum link distance over which a particular allocation might be made. Attention is given to the problem of estimating the joint minimization of plant and transportation cost functions over a network surface. For dynamic location-allocation systems an explicit dynamic programming formulation is developed for the optimal sequencing of plant locations over time subject, if necessary, to periodic variations in all cost functions and node weights. An application of the basic median algorithm to the problem of rural hospital locations in Southeast British Columbia is demonstrated, and computer codes are listed for all the specified models. / Arts, Faculty of / Geography, Department of / Graduate

No description available.