Average case tractability of non-homogeneous tensor product problems

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We study d-variate approximation problems in the average case setting with respect to a zero-mean Gaussian measure νd. Our interest is focused on measures having the structure of a non-homogeneous tensor product, where the covariance kernel of νd is a product of univariate kernels: Kd(s,t)=∏k=1dRk(sk,tk)for s,t∈[0,1]d. We consider the normalized average error of algorithms that use finitely many evaluations of arbitrary linear functionals. The information complexity is defined as the minimal number navg(ε,d) of such evaluations for error in the d-variate case to be at most ε. The growth of navg(ε,d) as a function of ε−1 and d depends on the eigenvalues of the covariance operator of νd and determines whether a problem is tractable or not. Four types of tractability are studied and for each of them we find the necessary and sufficient conditions in terms of the eigenvalues of the integral operator with kernel Rk.We illustrate our results by considering approximation problems related to the product of Korobov kernels Rk. Each Rk is characterized by a weight gk and a smoothness rk. We assume that weights are non-increasing and smoothness parameters are non-decreasing. Furthermore they may be related; for instance gk=g(rk) for some non-increasing function g. In particular, we show that the approximation problem is strongly polynomially tractable, i.e., navg(ε,d)≤Cε−p for all d∈N,ε∈(0,1], where C and p are independent of ε and d, iff lim infk→∞ln1gklnk>1. For other types of tractability we also show necessary and sufficient conditions in terms of the sequences gk and rk.