OXFORD UNIVERSITY PRESS

The Computational Complexity of Differential and Integral Equations: An Information-based Approach

ISBN : 9780198535898

参考価格(税込): 
¥6,941
著者: 
A.G. Werschulz
ページ
342 ページ
フォーマット
Hardcover
サイズ
164 x 241 mm
刊行日
1991年08月
シリーズ
Oxford Mathematical Monographs
メール送信
印刷

This book is concerned with a central question in numerical analysis: the approximate solution of differential or integral equations by algorithms using incomplete information. This situation often arises for equations of the form Lu = f, where f is some function defined on a domain and L is a differential operator. The function f may not be given exactly - we might only know its value at a finite number of points in the domain. Consequently the best that can be hoped for is to solve the equation to within a given accuracy at minimal cost or complexity. The author develops the theory of the complexity of the solutions to differential and integral equations and discusses the relationship between the worst-case setting and other (sometimes more tractable) related settings such as the average case, probabilistic, asymptotic, and randomized settings. Furthermore, he studies to what extent standard algorithms (such as finite element methods for elliptic problems) are optimal. This approach is discussed in depth in the context of two-point boundary value problems, linear elliptic partial differential equations, integral equations, ordinary differential equations, and ill-posed problems. As a result, this volume should appeal to mathematicians and numerical analysts working on the approximate solution of differential and integral equations as well as to complexity theorists addressing related questions in this area.

目次: 

Introduction

EXAMPLE: A TWO-POINT BOUNDARY VALUE PROBLEM: Introduction
Error, cost, and complexity
A minimal error algorithm
Complexity bounds
Comparison with the finite element method
Standard information
Final remarks

GENERAL FORMULATION: Introduction
Problem formulation
Information
Model of computation
Algorithms, their errors, and their costs
Complexity
Randomized setting
Asymptotic setting

THE WORST CASE SETTING: GENERAL RESULTS: Introduction
Radius and diameter
Complexity
Linear problems
The residual error criterion

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS IN THE WORST CASE SETTING
Introduction
Variational elliptic boundary value problems
Problem formulation
The normed case with arbitrary linear information
The normed case with standard information
The seminormed case
Can adaption ever help?

OTHER PROBLEMS IN THE WORST CASE SETTING: Introduction
Linear elliptic systems
Fredholm problems of the second kind
Ill-posed problems
Ordinary differential equations

THE AVERAGE CASE SETTING: Introduction
Some basic measure theory
General results for the average case setting
Complexity of shift-invariant problems
Ill-posed problems
The probabilistic setting

COMPLEXITY IN THE ASYMPTOTIC AND RANDOMIZED SETTINGS: Introduction
Asymptotic setting
Randomized setting
Appendices
Bibliography.

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