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

Abstract

We consider a sequence of Gaussian tensor product-type random fields

, where and are all positive eigenvalues and eigenfunctions of the covariance operator of the process X ₁, are standard Gaussian random variables, and is a subset of positive integers. For each d ∈ ℕ, the sample paths of X _d almost surely belong to L ₂([0, 1]^d) with norm ∥·∥_2,d . The tuples , are the eigenpairs of the covariance operator of X _d . We approximate the random fields X _d , d ∈ , by the finite sums X ⁽ⁿ⁾ _d corresponding to the n maximal eigenvalues λ _k , .

We investigate the logarithmic asymptotics of the average approximation complexity

$n_d^{pr} (\varepsilon ,\delta ): = \min \left\{ {n \in \mathbb{N}:\mathbb{P}(\left\| {X_d - X_d^{(n)} } \right\|_{2,d}^2 > \varepsilon ^2 \mathbb{E}\left\| {X_d } \right\|_{2,d}^2 ) \leqslant \delta } \right\},$

and the probabilistic approximation complexity

$n_d^{avg} (\varepsilon ): = \min \left\{ {n \in \mathbb{N}:\mathbb{E}\left\| {X_d - X_d^{(n)} } \right\|_{2,d}^2 \leqslant \varepsilon ^2 \mathbb{E}\left\| {X_d } \right\|_{2,d}^2 } \right\}$

, as the parametric dimension d → ∞ the error threshold ɛ ∈ (0, 1) is fixed, and the confidence level δ = δ(d, ɛ) is allowed to approach zero. Supplementing recent results of M.A. Lifshits and E.V. Tulyakova, we consider the case where the sequence decreases regularly and sufficiently slowly to zero, which has not been previously studied.