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Tractability of multi-parametric Euler and Wiener integrated processes

Lifshits M. A., Papageorgiou A., Woźniakowski H.

Probability and Mathematical Statistics

2012

Vol. 32, Fasc. 1

131--165

We study average case approximation of Euler and Wiener integrated processes of d variables which are almost surely rk-times continuously differentiable with respect to the k-th variable and 0 ≤ rk ≤ rk+1. Let n(ε; 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 ε. Strong polynomial tractability means that there are nonnegative numbers C and p such that [formula]. We prove that the Wiener process is much more difficult to approximate than the Euler process. Namely, strong polynomial tractability holds for the Euler case iff ...[formula]. Other types of tractability are also studied.

Curse of dimensionality in approximation of random fields

Lifshits M. A., Tulyakova V.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 1

97--112

Consider a random field of tensor product-type X(t), t∈[0,1]d, given by [formula] where (λ(i))i>0∈l2(φi)i>0 is an orthonormal system in L2 [0, 1] and (ξk)k∈Nd are non-correlated random variables with zero mean and unit variance. We investigate the quality of approximation (both in the average and in the probabilistic sense) to X by the n-term partial sums Xn minimizing the quadratic error E‖X‒Xn‖2, In the first part of the paper we consider the case of fixed dimension d. In the second part, following the suggestion of H. Woźniakowski, we consider the same problem for d→∞. We show that, for any fixed level of relative error, approximation complexity increases exponentially and we find the ex- plosion coefficient. We also show that the behavior of the probabilistic and average complexity is essentially the same in the large domain of parameters.