Tractability of Multi-Parametric Euler and Wiener Integrated Processes

TL;DR

This paper investigates the computational complexity of approximating multi-parametric Euler and Wiener integrated processes, revealing that Euler processes are easier to approximate than Wiener processes under certain smoothness conditions.

Contribution

It establishes precise conditions under which strong polynomial tractability holds for both Euler and Wiener processes, highlighting their differing complexities.

Findings

Strong polynomial tractability for Euler processes requires liminf r_k /ln k > 1/(2ln 3).

Wiener process approximation is strongly polynomially tractable if liminf r_k /k^s > 0 for some s>1/2.

Wiener processes are significantly more difficult to approximate than Euler processes.

Abstract

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<h<1. 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 liminf r_k /ln k > 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.