Searcharxiv⌕ Search

arXiv subjects

H. Woźniakowski

Publications and source records attributed to H. Woźniakowski.

2 recordsLinked to original sources
arXiv2011

Tractability of Multi-Parametric Euler and Wiener Integrated Processes

We study average case approximation of Euler and Wiener integrated processes of d variables which are almost surely r_k-times continuously differentiable with respect to the k-th variable. Let n(h,d) denote the minimal number of continuous linear functionals which is needed to find an algorithm that uses n such functionals and whose average case error improves the average case error of the zero algorithm by a factor h. Strong polynomial tractability means that there are nonnegative numbers C and p such that n(h,d)< C h^{-p} for all d and 0 1/(2\ln 3), whereas it holds for the Wiener case iff liminf r_k/k^s > 0 for some s>1/2. Other types of tractability are also studied.

math.PR↗
arXiv2011

Average Case Tractability of Non-homogeneous Tensor Product Problems

We study d-variate approximation problems in the average case setting with respect to a zero-mean Gaussian measure. Our interest is focused on measures having a structure of non-homogeneous linear tensor product, where covariance kernel is a product of univariate kernels. 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 n(h,d) of such evaluations for error in the d-variate case to be at most h. The growth of n(h,d) as a function of h^{-1} and d depends on the eigenvalues of the covariance operator 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 univariate kernels. We illustrate our results by considering approximation problems related to the product of Korobov kernels characterized by a weights g_k and smoothnesses r_k. We assume that weights are non-increasing and smoothness parameters are non-decreasing. Furthermore they may be related, for instance g_k=g(r_k) for some non-increasing function g. In particular, we show that approximation problem is strongly polynomially tractable, i.e., n(h,d)\le C h^{-p} for all d and 0 1. For other types of tractability we also show necessary and sufficient conditions in terms of the sequences g_k and r_k.

math.PR↗