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<h<1, where C and p are independent of h and d, iff liminf |ln g_k|/ln k >1. For other types of tractability we also show necessary and sufficient conditions in terms of the sequences g_k and r_k

Uniform Weak Tractability of Multivariate Problems

In this dissertation we introduce a new notion of tractability which is called uniform weak tractability. We give necessary and sufficient conditions on uniform weak tractability of homogeneous linear tensor product problems in the worst case, average case and randomized settings. We then turn to the study of approximation problems defined over spaces of functions with varying regularity with respect to successive variables. In the worst case setting we study approximation problems defined over suitable Korobov and Sobolev spaces. In the average case setting we study approximation problems defined over the space of continuous functions C([0, 1]^d ) equipped with a zero-mean Gaussian measure whose covariance operator is given by Euler or Wiener integrated process. We establish necessary and sufficient conditions on uniform weak tractability of those problems in terms of their regularity parameters