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Please use this identifier to cite or link to this item: http://hdl.handle.net/11375/22733
Title: Rigorous defect control and the numerical solution of ordinary differential equations
Authors: Ernsthausen, John+
Advisor: Nedialkov, Nedialko
Department: Computational Engineering and Science
Keywords: Reliable computing;Taylor models;Automated stepsize control;Ordinary Differential Equation;Taylor series;Sollya;Rigorous Polynomial Approximation;Continuous output;Backward error analysis
Publication Date: Oct-2017
Abstract: Modern numerical ordinary differential equation initial-value problem (ODE-IVP) solvers compute a piecewise polynomial approximate solution to the mathematical problem. Evaluating the mathematical problem at this approximate solution defines the defect. Corless and Corliss proposed rigorous defect control of numerical ODE-IVP. This thesis automates rigorous defect control for explicit, first-order, nonlinear ODE-IVP. Defect control is residual-based backward error analysis for ODE, a special case of Wilkinson's backward error analysis. This thesis describes a complete software implementation of the Corless and Corliss algorithm and extensive numerical studies. Basic time-stepping software is adapted to defect control and implemented. Advances in software developed for validated computing applications and advances in programming languages supporting operator overloading enable the computation of a tight rigorous enclosure of the defect evaluated at the approximate solution with Taylor models. Rigorously bounding a norm of the defect, the Corless and Corliss algorithm controls to mathematical certainty the norm of the defect to be less than a user specified tolerance over the integration interval. The validated computing software used in this thesis happens to compute a rigorous supremum norm. The defect of an approximate solution to the mathematical problem is associated with a new problem, the perturbed reference problem. This approximate solution is often the product of a numerical procedure. Nonetheless, it solves exactly the new problem including all errors. Defect control accepts the approximate solution whenever the sup-norm of the defect is less than a user specified tolerance. A user must be satisfied that the new problem is an acceptable model.
URI: http://hdl.handle.net/11375/22733
Appears in Collections:Open Access Dissertations and Theses

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