Average-case analysis of numerical problems
type: Item
instance of: scholarly article
title: Average-case analysis of numerical problems
zbMATH Open document ID: 0949.65146
author: Klaus Ritter
published in: Lecture Notes in Mathematics
publication date: 2000-07-18T00:00:00Z
review text: This research monograph is devoted to the analysis of numerical problems in an average case setting. The author studies the average error and cost and is looking for optimal methods with respect to these two quantities. A probability measure is used to specify the smothness and geometrical properties of the underlying functions. Equivalently, a random function or stochastic process is considered. The choice of the measure in the average case setting corresponds to the choice of the function class in the worst case setting. The authors studies continuous problems on infinite-dimensional function spaces, where only partial information such as a finite number of function values is available. The average case and worst case setting are among the several settings that are used in information-based complexity to analyze such problems. The first three chapters contain some background material concerning the average case setting for linear problems. Furthermore, they contain some definitions and facts about measures on function spaces and basic properties of reproducing kernel Hilbert spaces. Chapters IV--VI are concerned with minimal average errors for integration, approximation, and differentiation in the linear case. Chapter VII deals with nonlinear methods for linear problems. The author discusses integration and approximation of univariate functions with unknown local smothness. Integration of monotone functions are also discussed. The final chapter presents average case results for nonlinear problems of zero finding and global optimization. The monograph is carefully written and gives a systematic up-to-date treatment of theoretical questions concerning the average-case analysis of numerical problems. It is helpful for both theoretical research and practical applications.
reviewed by: Karel Najzar
Mathematics Subject Classification ID: 65Y20
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 65Dxx
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 65C20
Mathematics Subject Classification ID: 65K10
zbMATH DE Number: 1478502
zbMATH Keywords: linear methods
zbMATH Keywords: linear problems
zbMATH Keywords: nonlinear problems
zbMATH Keywords: numerical approximation
zbMATH Keywords: numerical integration
zbMATH Keywords: smoothig splines
zbMATH Keywords: average case setting
zbMATH Keywords: minimal average errors
zbMATH Keywords: multivariate case
zbMATH Keywords: probability measure
zbMATH Keywords: covariance kernel
zbMATH Keywords: Gaussian measures
zbMATH Keywords: computational case
zbMATH Keywords: Wiener measure
zbMATH Keywords: monograph
zbMATH Keywords: worst case setting
zbMATH Keywords: information-based complexity
zbMATH Keywords: global optimization
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0103934
OpenAlex ID: W1598589417
DOI: 10.1007/BFB0103934
description: scientific article; zbMATH DE number 1478502