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9780198535898

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

by
  • ISBN13:

    9780198535898

  • ISBN10:

    0198535899

  • Format: Hardcover
  • Copyright: 1991-11-07
  • Publisher: Oxford University Press

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Summary

This book is concerned with a central question in numerical analysis: theapproximate solution of differential or integral equations by algorithms usingincomplete 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 differentialoperator. The function f may not be given exactly - we might only know its valueat a finite number of points in the domain. Consequently the best that can behoped for is to solve the equation to within a given accuracy at minimal cost orcomplexity.The author develops the theory of the complexity of the solutions todifferential and integral equations and discusses the relationship between theworst-case setting and other (sometimes more tractable) related settings such asthe average case, probabilistic, asymptotic, and randomized settings.Furthermore, he studies to what extent standard algorithms (such as finiteelement methods for elliptic problems) are optimal.This approach is discussed in depth in the context of two-point boundary valueproblems, linear elliptic partial differential equations, integral equations,ordinary differential equations, and ill-posed problems. As a result, thisvolume should appeal to mathematicians and numerical analysts working on theapproximate solution of differential and integral equations as well as tocomplexity theorists addressing related questions in this area.

Table of Contents

Introduction
Example: A Two-Point Boundary Value Problem
General Information: The Worst Case Setting
General Results
Elliptic Partial Differential Equations in the Worst Case Setting
Other Problems in the Worst Case Setting
The Average Case Setting
Complexity in the Asymptotic and Randomized Settings
Table of Contents provided by Publisher. All Rights Reserved.

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