Liberating the dimension

  • Authors:

  • Frances Y. Kuo;Ian H. Sloan;Grzegorz W. Wasilkowski;Henryk Woniakowski

  • Affiliations:

  • School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia;School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia;Department of Computer Science, University of Kentucky, Lexington, KY 40506, USA;Department of Computer Science, Columbia University, New York, NY 10027, USA and Institute of Applied Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland

  • Venue:
  • Journal of Complexity
  • Year:
  • 2010

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Abstract

Many recent papers considered the problem of multivariate integration, and studied the tractability of the problem in the worst case setting as the dimensionality d increases. The typical question is: can we find an algorithm for which the error is bounded polynomially in d, or even independently of d? And the general answer is: yes, if we have a suitably weighted function space. Since there are important problems with infinitely many variables, here we take one step further: we consider the integration problem with infinitely many variables-thus liberating the dimension-and we seek algorithms with small error and minimal cost. In particular, we assume that the cost for evaluating a function depends on the number of active variables. The choice of the cost function plays a crucial role in the infinite dimensional setting. We present a number of lower and upper estimates of the minimal cost for product and finite-order weights. In some cases, the bounds are sharp.