2021/01/15 by A. A. Khartov, Khartov, A. A., I. A. Limar +1
Computer Science · Mathematics · Physics and Astronomy · #41A63 #41A65 #60G60 #65Y20 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Probability (math.PR) #Scientific Research and Discoveries #cs.NA #math.NA #math.PR #msc:41A63 #msc:41A65 #msc:60G60 #msc:65Y20
paper · pdf · doi:10.48550/arxiv.2101.06331
openalex publication_date 2021/01/15 · arxiv created 2021/03/02 · arxiv updated 2021/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider tensor product random fields Yd, d∈ℕ, whose covariance funtions are Gaussian kernels. The average case approximation complexity nYd(ε) is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate Yd, with relative 2-average error not exceeding a given threshold ε∈(0,1). We investigate the growth of nYd(ε) for arbitrary fixed ε∈(0,1) and d→∞. Namely, we find criteria of boundedness for nYd(ε) on d and of tending nYd(ε)→∞, d→∞, for any fixed ε∈(0,1). In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics ln nYd(ε)= ad+q(ε)bd+o(bd), d→∞, with any ε∈(0,1). Here q\colon (0,1)→ℝ is a non-decreasing function, (ad)d∈ℕ is a sequence and (bd)d∈ℕ is a positive sequence such that bd→∞, d→∞. We show that only special quantiles of self-decomposable distribution functions appear as functions q in a given asymptotics. These general results apply to nYd(ε) under particular assumptions on the length scale parameters.