## Deterministic and Stochastic Error Bounds in Numerical Analysis, Issue 1349In 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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### Contents

Introduction | 1 |

Error bounds for Monte Carlo methods | 43 |

Average error bounds | 66 |

Copyright | |

2 other sections not shown

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adaption helps adaptive methods Algebraic App and Int approximation arbitrary average a posteriori average error Bakhvalov Banach space Borel Borel set Chebyshev center class F compact metric space compute consider convex and symmetric cube define deterministic methods dn(F dn(Fu edge length Edited en(F equidistribution error function estimates example existence and uniqueness finite dimensional function classes Gaussian measures given hence Hilbert space ill.posed inequality inf sup Jordan measurable Lebesgue measure Lecture Notes Lemma linear problems mapping Math maximal error measure on F methods with varying metric space Micchelli Monte Carlo methods n.widths nonadaptive methods normed space notion Novak numbers numerical analysis optimal algorithm optimal information p.center p.optimal probability measure problem Int problems App Proceedings random numbers Remark Seminar smaller stochastic error bounds Sukharev theorem Theory Tikhomirov Traub upper bound valid varying cardinality Vlll Wasilkowski Werschulz worst case error Wozniakowski 1984