Average case complexity of linear multivariate problems

We study the average case complexity of a linear multivariate problem (\lmp) defined on functions of $d$ variables. We consider two classes of information. The first \lstd consists of function values and the second \lall of all continuous linear functionals. Tractability of \lmp means that the average case complexity is O((1/\e)^p) with $p$ independent of $d$. We prove that tractability of an \lmp in \lstd is equivalent to tractability in \lall, although the proof is {\it not} constructive. We provide a simple condition to check tractability in \lall. We also address the optimal design problem for an \lmp by using a relation to the worst case setting. We find the order of the average case complexity and optimal sample points for multivariate function approximation. The theoretical results are illustrated for the folded Wiener sheet measure.Comment: 7 page

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

A survey of information-based complexity

AbstractWe survey some recent results in information-based complexity. We focus on the worst case setting and also indicate some average case results

A Probabilistic Analysis of Linear Operator Testing

AbstractWe test for β-conformance of an implementation linear operator A to a specification linear operator S where the operator domain and range are separable Hilbert spaces and the domain space F is equipped with a Gaussian measure μ. Given an error bound ε>0 and a tolerance parameter β∈(0, 1), we want to determine either that there is an element f in a ball Bq of radius q in domain F such that ‖Sf−Af‖>ε or that Aβ-conforms to S on a set of measure at least 1−β in the ball Bq; i.e., μq(f:‖Sf−Af‖⩽ε)⩾1−β where μq is the truncated Gaussian measure to Bq. We present a deterministic algorithm that solves this problem and uses almost a minimal number of tests where each test is an evaluation of operators S and A at an element of F. We prove that optimal tests are conducted on the eigenvectors of the covariance operator of μ. They are universal; they are independent of the operators under consideration and other problem parameters. We show that finite testing is conclusive in this probabilistic setting. In contrast, finite testing is inconclusive in the worst and average case settings; see [5, 7]. We discuss the upper and lower bounds on the minimal number of tests. For q=∞ we derive the exact bounds on the minimal number of tests, which depend on β very weakly. On the other hand, for a finite q, the bounds on the minimal number of tests depend on β more significantly. We explain our approach by an example with the Wiener measure