Elsevier

Journal of Complexity

Volume 2, Issue 3, September 1986, Pages 255-269
Journal of Complexity

Probabilistic setting of information-based complexity

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Abstract

We study the probabilistic (ϵ, δ)-complexity for linear problems equipped with Gaussian measures. The probabilistic (ϵ, δ)-complexity, comprob(ϵ, δ), is understood as the minimal cost required to compute approximations with error at most ϵ on a set of measure at least 1 − δ. We find estimates of compprob(ϵ, δ) in terms of eigenvalues of the correlation operator of the Gaussian measure over elements which we want to approximate. In particular, we study the approximation and integration problems. The approximation problem is studied for functions of d variables which are continuous after r times differentiation with respect to each variable. For the Wiener measure placed on rth derivatives, the probabilistic compprob(ϵ, S) is estimated by Θ((√2 ln(1δ/ϵ)1(r+a)(ln(√2 ln(1δ)/ϵ))(d−1)(r+1)r+a), where a = 1 for the lower bound and a = 0.5 for the upper bound. The integration problem is studied for the same class of functions with d = 1. In this case, compprob(ϵ, δ) = Θ((√2 ln(1δ)/ϵ)1(r+1)).

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This research was supported in part by the National Science Foundation under Contract DCR-82-14327.