Liberating the dimension
نویسندگان
چکیده
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.
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ورودعنوان ژورنال:
- J. Complexity
دوره 26 شماره
صفحات -
تاریخ انتشار 2010