Deterministic and stochastic error bounds in numerical analysis
type: Item
instance of: scholarly article
title: Deterministic and stochastic error bounds in numerical analysis
zbMATH Open document ID: 0656.65047
author: Erich Novak
published in: Lecture Notes in Mathematics
publication date: 1993-06-05T00:00:00Z
review text: In this book different deterministic and stochastic error bounds of numerical analysis are investigated. The author analyses approximation (App), integration or quadrature (Int) and optimization problems (with two different aspects: only sup f is requested (the \(Opt^*\) problem) or it seeks an \(x\in X\) such that sup f-f(x) is small (the Opt problem)). Different problems connected with the approximate solution of differential and integral equations can be formulated as particular cases. In the first chapter maximal errors of deterministic methods are discussed. These error bounds were compared with different stochastic error bounds for Monte Carlo methods (chapter 2). The maximal error is a very pessimistic estimate for the true error of a given function. Hence it seems to be reasonable to consider additionally the average error (chapter 3). All these error bounds are also deduced for some special problems: the linear case or for different classes of functions (Lipschitz, Sobolev or Hölder classes). In the appendix the author deals with different aspects concerning the existence and uniqueness of optimal algorithms. The book contains a lot of new results obtained by the author. Besides all this a unified and comparative study of deterministic and stochastic error bounds is presented.
Mathematics Subject Classification ID: 65Gxx
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 65Dxx
Mathematics Subject Classification ID: 65Kxx
zbMATH DE Number: 193590
zbMATH Keywords: deterministic error bounds
zbMATH Keywords: stochastic error bounds
zbMATH Keywords: approximation
zbMATH Keywords: quadrature
zbMATH Keywords: optimization
zbMATH Keywords: maximal errors
zbMATH Keywords: Monte Carlo methods
zbMATH Keywords: average error
zbMATH Keywords: optimal algorithms
reviewed by: V.S.Stefănescu
MaRDI profile type: Publication
DOI: 10.1007/BFB0079792
description: scientific article; zbMATH DE number 193590