Angles between subspaces and their tangents

Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced via their cosines. The cosines and sines of PABS are commonly defined using the singular value decomposition. We utilize the same idea for the tangents, i.e., explicitly construct matrices, such that their singular values are equal to the tangents of PABS, using several approaches: orthonormal and non-orthonormal bases for subspaces, as well as projectors. Such a construction has applications, e.g., in analysis of convergence of subspace iterations for eigenvalue problems.Comment: 15 pages, 1 figure, 2 tables. Accepted to Journal of Numerical Mathematic

Establishing global error bounds for model reduction in combustion

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 223-239).In addition to theory and experiment, simulation of reacting flows has become important in policymaking, industry, and combustion science. However, simulations of reacting flows can be extremely computationally demanding due to the wide range of length scales involved in turbulence, the wide range of time scales involved in chemical reactions, and the large number of species in detailed chemical reaction mechanisms in combustion. To compensate for limited available computational resources, reduced chemistry is used. However, the accuracy of these reduced chemistry models is usually unknown, which is of great concern in applications; if the accuracy of a simplified model is unknown, it is risky to rely on the results of that model for critical decision-making. To address this issue, this thesis derives bounds on the global error in reduced chemistry models. First, it is shown that many model reduction methods in combustion are based on projection; all of these methods can be described using the same equation. After that, methods from the numerical solution of ODEs are used to derive separate a priori bounds on the global error in the solutions of reduced chemistry models for both projection-based reduced chemistry models and non-projection-based reduced chemistry models. The distinguishing feature between the two sets of bounds is that bounds on projection-based reduced chemistry models are stronger than those on non-projection-based reduced chemistry models. In both cases, the bounds are tight, but tend to drastically overestimate the error in the reduced chemistry. The a priori bounds on the global error in the solutions of reduced chemistry models demonstrate that if the error in the time derivatives of the state variables in the reduced model is controlled, then the error in the reduced model solution is also controlled; this thesis proves that result for the first time. Source code is included for all results presented. After presenting these results, the development of more accurate global error information is discussed. Using the error bounds above, in concert with more accurate global error information, it should be possible to assess better the accuracy and reliability of reduced chemistry models in applications.by Geoffrey Malcolm Oxberry.Ph.D

Analysis of the Self Projected Matching Pursuit Algorithm

The convergence and numerical analysis of a low memory implementation of the Orthogonal Matching Pursuit greedy strategy, which is termed Self Projected Matching Pursuit, is presented. This approach renders an iterative way of solving the least squares problem with much less storage requirement than direct linear algebra techniques. Hence, it is appropriate for solving large linear systems. The analysis highlights its suitability within the class of well posed problems

Efficient projection space updates for the approximation of iterative solutions to linear systems with successive right hand sides

Accurate initial guesses to the solution can dramatically speed convergence of iterative solvers. In the case of successive right hand sides, it has been shown that accurate initial solutions may be obtained by projecting the newest right hand side vector onto a column space of recent prior solutions. We propose a technique to efficiently update the column space of prior solutions. We find this technique can modestly improve solver performance, though its potential is likely limited by the problem step size and the accuracy of the solver