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Tractability of Multivariate Problems: Volume III: Standard Information for Operators
Erich Novak, University of Jena, Germany, and Henryk Woźniakowski, Columbia University, New York, NY
A publication of the European Mathematical Society.
cover
EMS Tracts in Mathematics
2012; 604 pp; hardcover
Volume: 18
ISBN-10: 3-03719-116-3
ISBN-13: 978-3-03719-116-3
List Price: US$128
Member Price: US$102.40
Order Code: EMSTM/18
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This is the third volume of a three-volume set comprising a comprehensive study of the tractability of multivariate problems. The third volume deals with algorithms using standard information consisting of function values. Linear and selected nonlinear operators are studied.

The most important example studied in volume III is the approximation of multivariate functions. Many other linear and some nonlinear problems are closely related to the approximation of multivariate functions. While the lower bounds obtained in volume I for the class of linear information also yield lower bounds for the standard class of function values, new techniques for upper bounds are presented in volume III. One of the main issues here is to verify when the power of standard information is nearly the same as the power of linear information. In particular, for the approximation problem defined over Hilbert spaces, the power of standard and linear information is the same in the randomized and average case (with Gaussian measures) settings, whereas in the worst case setting this is not true.

The book is of interest to researchers working in computational mathematics, especially in approximation of high-dimensional problems. It may be well suited for graduate courses and seminars. The text contains 58 open problems for future research in tractability.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in computational mathematics.

Table of Contents

  • Examples of multivariate approximation
  • Randomized setting: Multivariate approximation
  • Randomized setting: Linear problems
  • Average case setting: Multivariate approximation
  • Average case setting: Linear problems
  • Worse case setting: Multivariate approximation
  • Worse case setting: Linear problems
  • Nonlinear problems
  • Power of function values for multivariate approximation
  • List of open problems
  • Errata for volumes I and II
  • Bibliography
  • Index
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