Computational Functional Analysis

Ramon E Moore, Michael J Cloud

• Publisher: Woodhead Publishing
• Year: 2007
• Print ISBN: 9781904275244
• eBook ISBN: 9780857099433
• Edition: 2
• Topic: Tecnologia e ingegneria
• Subtopic: Ingegneria elettronica e telecomunicazioni

This course text fills a gap for first-year graduate-level students reading applied functional analysis or advanced engineering analysis and modern control theory. Containing 100 problem-exercises, answers, and tutorial hints, the first edition is often cited as a standard reference. Making a unique contribution to numerical analysis for operator equations, it introduces interval analysis into the mainstream of computational functional analysis, and discusses the elegant techniques for reproducing Kernel Hilbert spaces. There is discussion of a successful ''hybrid'' method for difficult real-life problems, with a balance between coverage of linear and non-linear operator equations. The authors successful teaching philosophy: ''We learn by doing'' is reflected throughout the book. - Contains 100 problem-exercises, answers and tutorial hints for students reading applied functional analysis - Introduces interval analysis into the mainstream of computational functional analysis

1

Introduction

The outcome of any numerical computation will be a finite set of numbers. The numbers themselves will be finite decimal (or binary) expansions of rational numbers. Nevertheless, such a set of numbers can represent a function in many ways: as coefficients of a polynomial; as coefficients of a piecewise polynomial function (for example a spline function); as Fourier coefficients; as left and right hand endpoints of interval coefficients of an interval valued function; as coefficients of each of the components of a vector valued function; as values of a function at a finite set of argument points; etc.

The concepts and techniques of functional analysis we shall study will enable us to design and apply methods for the approximate solution of operator equations (differential equations, integral equations, and others). We shall be able to compute numerical representations of approximate solutions and numerical estimates of error. Armed with convergence theorems, we shall know that, by doing enough computing, we can obtain approximate solutions of any desired accuracy, and know when we have done so.

Since no previous knowledge of functional analysis is assumed here, a number of introductory topics will be discussed at the beginning in order to prepare for discussion of the computational methods.

The literature on functional analysis is now quite extensive, and only a small part of it is presented here — that which seems most immediately relevant to computational purposes. In this introductory study, we hope that the reader will be brought along far enough to be able to begin reading the more advanced literature and to apply the techniques to practical problems.

Some knowledge of linear algebra and differential equations will be assumed, and previous study of numerical methods and some experience in computing will help in understanding the applications to be discussed. No background in measure theory is assumed; in fact, we will make scant use of those concepts.

In the first part of the study (Chapters 1–10), we introduce a number of different kinds of topological spaces suitable for investigations of computational methods for solving linear operator equations. These include Hilbert spaces, Banach spaces, and metric spaces. Linear functionals will play an important role, especially in Hilbert spaces. In fact, these mappings are the source of the name functional analysis. We will see that the Riesz representation theorem plays an important role in computing when we operate in reproducing kernel Hilbert spaces.

The study of order relations in function spaces leads to important computing methods based on interval valued mappings. In Chapter 17, interval analysis is used to help construct bounds on the norm of an operator in connection with computationally verifying sufficient conditions for the convergence of an iterative method for solving a nonlinear operator equation.

In the second part of the study (Chapters 11–13), we turn our attention to methods for the approximate solution of linear operator equations.

In the third part of the study (Chapters 14–20), we investigate methods for the approximate solution of nonlinear operator equations.

It will be assumed, as the text proceeds, that the reader understands the content of the exercises to that point. Some hints (and often full solutions) appear from page 160 onwards; as mentioned in the Preface, however, the reader will learn best by attempting each exercise before consulting the hints.

2

Linear spaces

We begin with an introduction to some basic concepts and definitions in linear algebra. These are of fundamental importance for linear problems in functional analysis, and are also of importance for many of the methods for nonlinear problems, since these often involve solving a sequence of linear problems related to the nonlinear problem.

The main ideas are these: we can regard real valued functions, defined on a continuum of arguments, as points (or vectors) in the same way that we regard n-tuples of real numbers as points; that is, we can define addition and scalar multiplication. We can take linear combinations. We can form larger or smaller linear spaces containing or contained in them; and we can identify equivalent linear spaces, differing essentially only in notation.

Many numerical methods involve finding approximate solutions to operator equations (for example differential equations or integral equations) in the form of polynomial approximations (or other types of approximations) which can be computed in reasonably simple ways. Often the exact solution cannot be computed at all in finite real time, but can only be approximated as the limit of an infinite sequence of computations.

Thus, for numerical approximation of solutions as well as for theoretical analysis of properties of solutions, linear spaces are indispensable.

The basic properties of relations are introduced in this chapter, since they ...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright page
• Preface
• Notation
• 1: Introduction
• 2: Linear spaces
• 3: Topological spaces
• 4: Metric spaces
• 5: Normed linear spaces and Banach spaces
• 6: Inner product spaces and Hilbert spaces
• 7: Linear functionals
• 8: Types of convergence in function spaces
• 9: Reproducing kernel Hilbert spaces
• 10: Order relations in function spaces
• 11: Operators in function spaces
• 12: Completely continuous (compact) operators
• 13: Approximation methods for linear operator equations
• 14: Interval methods for operator equations
• 15: Contraction mappings and iterative methods for operator equations in fixed point form
• 16: Fréchet derivatives
• 17: Newton’s method in Banach spaces
• 18: Variants of Newton’s method
• 19: Homotopy and continuation methods
• 20: A hybrid method for a free boundary problem
• Hints for selected exercises
• Further reading
• Index