TY - JOUR

T1 - Approximation complexity of tensor product-type random fields with heavy spectrum

AU - Khartov, A.A.

PY - 2013

Y1 - 2013

N2 - We consider a sequence of Gaussian tensor product-type random fields, where (λi)i ∈ ℕ̃ and ((Ψi)i ∈ ℕ̃ are all positive eigenvalues and eigenfunctions of the covariance operator of the process X1, (ξk)k ∈ ℕ̃ are standard Gaussian random variables, and ℕ̃ is a subset of positive integers. For each d ∈ ℕ, the sample paths of Xd almost surely belong to L2([0, 1]d) with norm ∥·∥2,d. The tuples, are the eigenpairs of the covariance operator of Xd. We approximate the random fields Xd, d ∈ ℕ̃, by the finite sums Xd (n) corresponding to the n maximal eigenvalues λk, k ∈ ℕ̃d. We investigate the logarithmic asymptotics of the average approximation complexity, and the probabilistic approximation complexity, as the parametric dimension d → ∞ the error threshold e{open} ∈ (0, 1) is fixed, and the confidence level δ = δ(d, e{open}) is allowed to approach zero. Supplementing recent results of M.A. Lifshits and E.V. Tulyakova, we consider the case where the sequence (λi)i ∈ ℕ̃ decreases regularly and sufficiently slowly to z

AB - We consider a sequence of Gaussian tensor product-type random fields, where (λi)i ∈ ℕ̃ and ((Ψi)i ∈ ℕ̃ are all positive eigenvalues and eigenfunctions of the covariance operator of the process X1, (ξk)k ∈ ℕ̃ are standard Gaussian random variables, and ℕ̃ is a subset of positive integers. For each d ∈ ℕ, the sample paths of Xd almost surely belong to L2([0, 1]d) with norm ∥·∥2,d. The tuples, are the eigenpairs of the covariance operator of Xd. We approximate the random fields Xd, d ∈ ℕ̃, by the finite sums Xd (n) corresponding to the n maximal eigenvalues λk, k ∈ ℕ̃d. We investigate the logarithmic asymptotics of the average approximation complexity, and the probabilistic approximation complexity, as the parametric dimension d → ∞ the error threshold e{open} ∈ (0, 1) is fixed, and the confidence level δ = δ(d, e{open}) is allowed to approach zero. Supplementing recent results of M.A. Lifshits and E.V. Tulyakova, we consider the case where the sequence (λi)i ∈ ℕ̃ decreases regularly and sufficiently slowly to z

U2 - 10.3103/S1063454113020040

DO - 10.3103/S1063454113020040

M3 - Article

SP - 98

EP - 101

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 2

ER -