Date of Award
Master of Science (MS)
Mathematics and Statistics
James D. Watson
In this paper we will consider the problem of selecting the best, or optimal, numerical method of solution to a given mathematical problem. The admissible numerical methods will be a clearly defined set for each problem. Obviously, in order to find the best method in this set, we must have a clear mathematical formulation of just what "best" means; this will be the intent of Theorem 0.1. Intuitively, the best method will be understood to be the one which minimizes the maximum possible error where this error will be measured in terms of the norm of a given Hilbert space.
Each succeeding chapter in this paper will be a different example in finding an optimal technique. The actual examples and the numerical methods used to solve them may have little in common; however, the techniques used to prove that a certain method is optimal will have a unifying theme - each of the proofs will revolve around the use of functional analysis techniques, particularly techniques involving Hilbert space theory.
The examples will be based on sections from Numerical Processes In Differential Equations by I. Babuska, M. Prager, and E. Vitasek (these examples may be found in [l, pp. 42-47, 72-78, 170-181]). This book is an English translation of the Czech version which was published in 1964. It is also a book in which the word "obvious" is used liberally. The intent of this paper is to make some 'of these missing "obvious" parts. of the proofs more obvious.
Schoenfeld, Kenneth D., "Functional Analysis Techniques in Numerical Analysis" (1974). All Graduate Plan B and other Reports. 1175.
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