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

Werschulz, Arthur G.

 
9780198535898: The Computational Complexity of Differential and Integral Equations: An Information-Based Approach (Oxford Mathematical Monographs)
Hardcover
ISBN 10: 0198535899 ISBN 13: 9780198535898
Publisher: Oxford University Press, 1991

Synopsis

Complexity theory has become an increasingly important theme in mathematical research. This book deals with an 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. We do not have complete information about f. For instance, we might only know its value at a finite number of points in the domain, or the values of its inner products with a finite set of known functions. Consequently the best that can be hoped for is to solve the equation to within a given accuracy at minimal cost or complexity. In this book, the theory of the complexity of the solution to differential and integral equations is developed. The relationship between the worst case setting and other (sometimes more tractable) related settings, such as the average case, probabilistic, asymptotic, and randomized settings, is also discussed. The author determines the inherent complexity of the problem and finds optimal algorithms (in the sense of having minimal cost). 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.

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About the Author

Arthur G. Werschulz, Associate Professor of Mathematics and Computer Science, Fordham University; Visiting Scholar, Columbia University.

"About this title" may belong to another edition of this title.

Publisher
Oxford University Press
Publication date
1991
Language
English
ISBN 10 
0198535899
ISBN 13 
9780198535898
Binding
Hardcover
Edition number
1
Number of pages
342