Technical Reports from UMIACS
Permanent URI for this collectionhttp://hdl.handle.net/1903/7
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Item Exploiting Structure of Symmetric or Triangular Matrices on a GPU(2008-01) Jung, Jin Hyuk; O'Leary, Dianne P.Matrix computations are expensive, and GPUs have the potential to deliver results at reduced cost by exploiting parallel computation. We focus on dense matrices of the form A D2 A^T, where A is an m x n matrix (m less than or equal to n) and D is an n x n diagonal matrix. Many important numerical problems require solving linear systems of equations involving matrices of this form. These problems include normal equations approaches to solving linear least squares and weighted linear least squares problems, and interior point algorithms for linear and nonlinear programming problems. We develop in this work efficient GPU algorithms for forming and factoring A D2 A^T by exploiting the triangular rastorization capabilities of the GPU.Item A polynomial-time interior-point method for conic optimization, with inexact barrier evaluations(2008-04) Schurr, Simon P.; O'Leary, Dianne P.; Tits, Andre L.We consider a primal-dual short-step interior-point method for conic convex optimization problems for which exact evaluation of the gradient and Hessian of the primal and dual barrier functions is either impossible or prohibitively expensive. As our main contribution, we show that if approximate gradients and Hessians of the primal barrier function can be computed, and the relative errors in such quantities are not too large, then the method has polynomial worst-case iteration complexity. (In particular, polynomial iteration complexity ensues when the gradient and Hessian are evaluated exactly.) In addition, the algorithm requires no evaluation---or even approximate evaluation---of quantities related to the barrier function for the dual cone, even for problems in which the underlying cone is not self-dual.Item Modified Cholesky Algorithms: A Catalog with New Approaches(2006-08-08) Fang, Haw-ren; O'Leary, Dianne P.Given an $n \times n$ symmetric possibly indefinite matrix $A$, a modified Cholesky algorithm computes a factorization of the positive definite matrix $A+E$, where $E$ is a correction matrix. Since the factorization is often used to compute a Newton-like downhill search direction for an optimization problem, the goals are to compute the modification without much additional cost and to keep $A+E$ well-conditioned and close to $A$. Gill, Murray and Wright introduced a stable algorithm, with a bound of $\|E\|_2=O(n2)$. An algorithm of Schnabel and Eskow further guarantees $\|E\|_2=O(n)$. We present variants that also ensure $\|E\|_2=O(n)$. Mor\'{e} and Sorensen and Cheng and Higham used the block $LBL^T$ factorization with blocks of order $1$ or $2$. Algorithms in this class have a worst-case cost $O(n3)$ higher than the standard Cholesky factorization, We present a new approach using an $LTL^T$ factorization, with $T$ tridiagonal, that guarantees a modification cost of at most $O(n2)$.Item Efficient Iterative Algorithms for the Stochastic Finite Element Method with Application to Acoustic Scattering(2002-12-19) Elman, Howard; Ernst, Oliver G.; O'Leary, Dianne P.; Stewart, MichaelIn this study, we describe the algebraic computations required to implement the stochastic finite element method for solving problems in which uncertainty is restricted to right hand side data coming from forcing functions or boundary conditions. We show that the solution can be represented in a compact outer product form which leads to efficiencies in both work and storage, and we demonstrate that block iterative methods for algebraic systems with multiple right hand sides can be used to advantage to compute this solution. We also show how to generate a variety of statistical quantities from the computed solution. Finally, we examine the behavior of these statistical quantities in one setting derived from a model of acoustic scattering. UMIACS-TR-2002-102Item Stagnation of GMRES(2001-11-12) Zavorin, Ilya; O'Leary, Dianne P.; Elman, HowardWe study problems for which the iterative method \gmr for solving linear systems of equations makes no progress in its initial iterations. Our tool for analysis is a nonlinear system of equations, the stagnation system, that characterizes this behavior. For problems of dimension 2 we can solve this system explicitly, determining that every choice of eigenvalues leads to a stagnating problem for eigenvector matrices that are sufficiently poorly conditioned. We partially extend this result to higher dimensions for a class of eigenvector matrices called extreme. We give necessary and sufficient conditions for stagnation of systems involving unitary matrices, and show that if a normal matrix stagnates then so does an entire family of nonnormal matrices with the same eigenvalues. Finally, we show that there are real matrices for which stagnation occurs for certain complex right-hand sides but not for real ones. (Also UMIACS-TR-2001-74)Item Text Summarization via Hidden Markov Models and Pivoted QR Matrix Decomposition(2001-05-10) Conroy, John; O'Leary, Dianne P.A sentence extract summary of a document is a subset of the document's sentences that contains the main ideas in the document. We present two approaches to generating such summaries. The first uses a pivoted QR decomposition of the term-sentence matrix in order to identify sentences that have ideas that are distinct from those in other sentences. The second is based on a hidden Markov model that judges the likelihood that each sentence should be contained in the summary. We compare the results of these methods with summaries generated by humans, showing that we obtain higher agreement than do earlier methods. (Cross-referenced as UMIACS-TR-2001-11)Item Image Restoration through Subimages and Confidence Images(2000-07-20) Nagy, James G.; O'Leary, Dianne P.Some very effective but expensive image reconstruction algorithms cannot be applied to large images because of their cost. In this work, we first show how to apply such algorithms to subimages, giving improved reconstruction of regions of interest. Our second contribution is to construct confidence intervals for pixel values, by generalizing a theorem of O'Leary and Rust to allow both upper and lower bounds on variables. All current algorithms for image deblurring or deconvolution output an image. This provides an estimated value for each pixel in the image. What is lacking is an estimate of the statistical confidence that we can have in those pixel values or in the features they form in the image. There are two obstacles in determining confidence intervals for pixel values: first, the process is computationally quite intensive, and second, there has been no proposal for providing the results in a visually useful way. In this work we overcome the first of those limitations and use a recently developed algorithm called {\sf Twinkle} to overcome the second. We demonstrate the usefulness of these techniques on astronomical and motion-blurred images. (Also cross-referenced as UMIACS-TR-2000-52)Item Displaying Confidence Images(2000-07-20) Nagy, James G.; O'Leary, Dianne P.Algorithms for computing images result in an estimate of an image. The image may result from deblurring a measured image, from deconvolving a set of measurements, or from computing an image by modeling physical processes such as the weather. These computations provide an estimated value for each pixel in the image. What is lacking, however, is an estimate of the statistical confidence that we can have in those pixel values or in the features they form. In this work we discuss novel ways to display confidence information, using an algorithm called {\sf Twinkle}, in order to give the viewer valuable visual insight into uncertainties. The technique is useful whether the confidence information is in the form of a confidence interval or a distribution of possible values. We demonstrate how to display confidence information in a variety of applications: weather forecasts, intensity of a star, and rating a potential tumor in a diagnostic image. (Also cross-referenced as UMIACS-TR-2000-51)Item A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations(1999-08-27) Elman, Howard C.; Ernst, Oliver G.; O'Leary, Dianne P.Standard multigrid algorithms have proven ineffective for the solution of discretizations of Helmholtz equations. In this work we modify the standard algorithm by adding GMRES iterations at coarse levels and as an outer iteration. We demonstrate the algorithm's effectiveness through theoretical analysis of a model problem and experimental results. In particular, we show that the combined use of GMRES as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as two hundred. For fixed wave numbers, it displays grid-independent convergence rates and has costs proportional to number of unknowns. Also cross-referenced as UMIACS-TR-99-36Item Symbiosis between Linear Algebra and Optimization(1999-05-28) O'Leary, Dianne P.The efficiency and effectiveness of most optimization algorithms hinges on the numerical linear algebra algorithms that they utilize. Effective linear algebra is crucial to their success, and because of this, optimization applications have motivated fundamental advances in numerical linear algebra. This essay will highlight contributions of numerical linear algebra to optimization, as well as some optimization problems encountered within linear algebra that contribute to a symbiotic relationship. Also cross-referenced as UMIACS-TR-99-30
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