Title page for ETD etd-050406-103442

Document Type dissertation

Author Name Simonis, Joseph P

URN etd-050406-103442

Title Inexact Newton Methods Applied to Under-Determined Systems

Degree PhD

Department Mathematical Sciences

Advisors
Homer Walker, Advisor
Suzanne L. Weekes, Committee Member
Arthur Heinricher, Committee Member
Joseph D. Fehribach, Committee Member
John Shadid, Committee Member

Keywords
Periodic Solutions
Under-Determined Systems
Continuation
Nonlinear Eigenvalue
Inexact Newton Methods
Newton's Method
Trust Region Methods

Date of Presentation/Defense 2006-04-25

Availability unrestricted

Abstract
Consider an under-determined system of nonlinear equations F(x)=0, F:R^m→R^n, where F is continuously differentiable and m > n. This system appears in a variety of applications, including parameter–dependent systems, dynamical systems with periodic solutions, and nonlinear eigenvalue problems. Robust, efficient numerical methods are often required for the solution of this system.

Newton’s method is an iterative scheme for solving the nonlinear system of equations F(x)=0, F:R^n→R^n. Simple to implement and theoretically sound, it is not, however, often practical in its pure form. Inexact Newton methods and globalized
inexact Newton methods are computationally efficient variations of Newton’s method commonly used on large-scale problems. Frequently, these variations are more robust than Newton’s method. Trust region methods, thought of here as globalized exact Newton methods, are not as computationally efficient in the large–scale case, yet notably more robust than Newton’s method in practice.

The normal flow method is a generalization of Newton’s method for solving the system F:R^m→R^n, m > n. Easy to implement, this method has a simple and useful local convergence theory; however, in its pure form, it is not well suited for solving large-scale problems. This dissertation presents new methods that improve the efficiency and robustness of the normal flow method in the large–scale case. These are developed in direct analogy with inexact–Newton, globalized inexact–Newton, and trust–region methods, with particular consideration of the associated convergence theory. Included are selected problems of interest simulated in MATLAB.

Files
simonis.pdf

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Document Type	dissertation
Author Name	Simonis, Joseph P
URN	etd-050406-103442
Title	Inexact Newton Methods Applied to Under-Determined Systems
Degree	PhD
Department	Mathematical Sciences
Advisors	Homer Walker, Advisor Suzanne L. Weekes, Committee Member Arthur Heinricher, Committee Member Joseph D. Fehribach, Committee Member John Shadid, Committee Member
Keywords	Periodic Solutions Under-Determined Systems Continuation Nonlinear Eigenvalue Inexact Newton Methods Newton's Method Trust Region Methods
Date of Presentation/Defense	2006-04-25
Availability	unrestricted
Abstract Consider an under-determined system of nonlinear equations F(x)=0, F:R^m→R^n, where F is continuously differentiable and m > n. This system appears in a variety of applications, including parameter–dependent systems, dynamical systems with periodic solutions, and nonlinear eigenvalue problems. Robust, efficient numerical methods are often required for the solution of this system. Newton’s method is an iterative scheme for solving the nonlinear system of equations F(x)=0, F:R^n→R^n. Simple to implement and theoretically sound, it is not, however, often practical in its pure form. Inexact Newton methods and globalized inexact Newton methods are computationally efficient variations of Newton’s method commonly used on large-scale problems. Frequently, these variations are more robust than Newton’s method. Trust region methods, thought of here as globalized exact Newton methods, are not as computationally efficient in the large–scale case, yet notably more robust than Newton’s method in practice. The normal flow method is a generalization of Newton’s method for solving the system F:R^m→R^n, m > n. Easy to implement, this method has a simple and useful local convergence theory; however, in its pure form, it is not well suited for solving large-scale problems. This dissertation presents new methods that improve the efficiency and robustness of the normal flow method in the large–scale case. These are developed in direct analogy with inexact–Newton, globalized inexact–Newton, and trust–region methods, with particular consideration of the associated convergence theory. Included are selected problems of interest simulated in MATLAB.
Files	simonis.pdf