1,242 research outputs found
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
Dependence on the Dimension for Complexity of Approximation of Random Fields
We consider an \eps-approximation by n-term partial sums of the
Karhunen-Lo\`eve expansion to d-parametric random fields of tensor product-type
in the average case setting. We investigate the behavior, as d tends to
infinity, of the information complexity n(\eps,d) of approximation with error
not exceeding a given level \eps. It was recently shown by M.A. Lifshits and
E.V. Tulyakova that for this problem one observes the curse of dimensionality
(intractability) phenomenon. The aim of this paper is to give the exact
asymptotic expression for the information complexity n(\eps,d).Comment: 18 pages. The published in Theory Probab. Appl. (2010) extended
English translation of the original paper "Zavisimost slozhnosti approximacii
sluchajnyh polej ot rasmernosti", submitted on 15.01.2007 and published in
Theor. Veroyatnost. i Primenen. 54:2, 256-27
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