Average-Case Analysis of Numerical Problems

ISBN-13: 9783540674498 / Angielski / Miękka / 2000 / 252 str.

Klaus Ritter

Opis

The average-case analysis of numerical problems is the counterpart of the more traditional worst-case approach. The analysis of average error and cost leads to new insight on numerical problems as well as to new algorithms. The book provides a survey of results that were mainly obtained during the last 10 years and also contains new results. The problems under consideration include approximation/optimal recovery and numerical integration of univariate and multivariate functions as well as zero-finding and global optimization. Background material, e.g. on reproducing kernel Hilbert spaces and random fields, is provided.

Recenzje

"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." (Karel Najzar, zbMATH 0949.65146, 2022)

Spis treści

TABLE OF CONTENTS I Introduction Overview Historical and Bibliographical Remarks II Linear Problems: Definitions and a Classical Example Measures on Function Spaces Linear Problems Integration and L 2-Approximation on the Wiener Space III Second-Order Results for Linear Problems Reproducing Kernel Hilbert Spaces Linear Problems with Values in a Hilbert Space Splines and Their Optimality Properties Kriging IV Integration and Approximation of Univariate Functions Sacks-Ylvisaker Regularity Conditions Integration under Sacks-Ylvisaker Conditions Approximation under Sacks-Ylvisaker Conditions Unknown Mean and Covariance Kernel Further Smoothness Conditions and Results Local Smoothness and Best Regular Sequences V Linear Problems for Univariate Functions with Noisy Data Optimality of Smoothing Spline Algorithms Integration and Approximation of Univariate Functions Differentiation of Univariate Functions VI Integration and Approximation of Multivariate Functions Isotropic Smoothness Tensor Product Problems Tractability or Curse of Dimensionality? VII Nonlinear Methods for Linear Problems Nonlinear Algorithms, Adaptive Information, and Varying Cardinality Linear Problems with Gaussian Measures Unknown Local Smoothness Integration of Monotone Functions Computational Complexity of Continuous Problems VIII Nonlinear Problems Zero Finding Global Optimization Bibliography Author Index Subject Index Index of Notation