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Deterministic and stochastic error bounds in numerical analysis / Erich Novak.

By: Material type: TextTextSeries: Lecture notes in mathematics (Springer-Verlag) ; 1349.Publication details: Berlin ; New York : Springer-Verlag, ©1988.Description: 1 online resource (113 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9783540459873
  • 3540459871
Subject(s): Additional physical formats: Print version:: Deterministic and stochastic error bounds in numerical analysis.DDC classification:
  • 510 s 519.4 19
LOC classification:
  • QA3 .L28 no. 1349 QA297
Other classification:
  • 31.73
  • 31.76
  • PA 29
  • *65Gxx
  • 65-02
  • 65C05
  • 65Dxx
  • 65Kxx
  • SI 850
  • 27
  • MAT 661f
Online resources:
Contents:
Introduction -- Deterministic Error Bounds -- Error Bounds for Monte Carlo Methods -- Average Error Bounds -- Appendix: Existence and Uniqueness of Optimal Algorithms -- Bibliography -- Notations -- Index.
Summary: In these notes different deterministic and stochastic error bounds of numerical analysis are investigated. For many computational problems we have only partial information (such as n function values) and consequently they can only be solved with uncertainty in the answer. Optimal methods and optimal error bounds are sought if only the type of information is indicated. First, worst case error bounds and their relation to the theory of n-widths are considered; special problems such approximation, optimization, and integration for different function classes are studied and adaptive and nonadaptive methods are compared. Deterministic (worst case) error bounds are often unrealistic and should be complemented by different average error bounds. The error of Monte Carlo methods and the average error of deterministic methods are discussed as are the conceptual difficulties of different average errors. An appendix deals with the existence and uniqueness of optimal methods. This book is an introduction to the area and also a research monograph containing new results. It is addressd to a general mathematical audience as well as specialists in the areas of numerical analysis and approximation theory (especially optimal recovery and information-based complexity).
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Includes bibliographical references (pages 100-109) and index.

Introduction -- Deterministic Error Bounds -- Error Bounds for Monte Carlo Methods -- Average Error Bounds -- Appendix: Existence and Uniqueness of Optimal Algorithms -- Bibliography -- Notations -- Index.

In these notes different deterministic and stochastic error bounds of numerical analysis are investigated. For many computational problems we have only partial information (such as n function values) and consequently they can only be solved with uncertainty in the answer. Optimal methods and optimal error bounds are sought if only the type of information is indicated. First, worst case error bounds and their relation to the theory of n-widths are considered; special problems such approximation, optimization, and integration for different function classes are studied and adaptive and nonadaptive methods are compared. Deterministic (worst case) error bounds are often unrealistic and should be complemented by different average error bounds. The error of Monte Carlo methods and the average error of deterministic methods are discussed as are the conceptual difficulties of different average errors. An appendix deals with the existence and uniqueness of optimal methods. This book is an introduction to the area and also a research monograph containing new results. It is addressd to a general mathematical audience as well as specialists in the areas of numerical analysis and approximation theory (especially optimal recovery and information-based complexity).

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