The Computational Complexity of Differential and Integral Equations: An Information-Based Approach (Oxford Mathematical Monographs) - Hardcover

Werschulz, Arthur G.

Hardcover

ISBN 10: 0198535899 ISBN 13: 9780198535898

Publisher: OUP Oxford, 1991

Synopsis

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.

About this title

Review

'This book ... is a most welcome addition to the theoretical computer science and numerical analysis literature. Though it is intended as a summary of current research, it is of the quality that would make it an excellent textbook on the subject for advanced numerical analysis and computer science courses .. it reads easily and lucidly.'R.S. Andersen

'An excellent and accessible introduction to the complexity of basic arithmetic operations ... it adds an interesting new dimension to the study of numerical methods for the solution of PDEs.'Notices of the A.M.S.

Synopsis

This study is concerned with a central question in numerical analysis: how efficient can algorithms be made given only incomplete information about a differential or integral equation? Typically this question might arise when an equation is of the form Lu=f, where f is some function defined on a domain and L is a differential operator. We may not be given f exactly, merely 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 solution to differential and integral equations and discusses the relationship between the worst-case setting and two (sometimes more tractable) related problems: the average-case setting and the probalistic setting. He addresses the computation of the complexity of algorithms and also determines optimal algorithms (in the sense of having minimal cost). These methods are discussed in the context of two-point boundary value problems and for linear elliptic partial differential equations.

As a result, this volume should benefit 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.

Edition details

• Publisher: OUP Oxford
• Publication date: 1991
• Language: English
• ISBN 10: 0198535899
• ISBN 13: 9780198535898
• Binding: Hardcover
• Number of pages: 342