Pairwise sampling for the nonlinear interpolation of functions of very many variables

Abstract

The difficulty is first pointed out concerning the nonlinear interpolation of functions defined in a space of very many dimensions. There is a method using a sampling technique that works well fork≈10–20 (k=number of dimensions). The sampling errors, however, increase in powers ofk, so that fork greater than the above-quoted values the computation is no more feasible. This is due to the subtraction between two large sums of about the same magnitude, each of which suffers stochastic fluctuations accompanying samplings. To avoid this unfavorable effect, a pairwise sampling technique is considered where one draws two samples at a time when required, one from each of the two sums of terms. By this new avenue of approach, the probabilistic interpretation becomes much more straighforward than hitherto conceived and the reduction of standard errors is also remarkable especially for the cases of very many dimensions.