The complexity of function approximation on Sobolev spaces with bounded mixed derivative by linear Monte Carlo methods

Abstract

We study the information-based complexity of the approximation problem on the multivariate Sobolev space with bounded mixed derivative MW r p, in the norm of L q by linear Monte Carlo methods. Applying the Maiorov's discretization technique and some properties of pseudo-s-scale, we determine the exact orders of this problem for 1 < p, q < ∞.

The information-based complexity of approximation problem by adaptive Monte Carlo methods

“…Now let us look at linear versus nonlinear algorithms in the randomized setting. First, we recall the result on the linear randomized approximation numbers obtained in [24]. Theorem 3 [24] .…”

“…First, we recall the result on the linear randomized approximation numbers obtained in [24]. Theorem 3 [24] . Let 1 < p, q < ∞ and 0 < r = r 1 = · · · = r ν < r ν+1 · · · r d , 1 ν d. Then…”

Monte Carlo methods for uniform approximation on periodic Sobolev spaces with mixed smoothness

“…that have been obtained by Fang and Duan for nonlinear Monte Carlo approximation [4], and for linear methods [5], respectively. Note that the proofs for the lower bounds in the papers of Fang and Duan work for a larger range of the smoothness parameter r than the corresponding upper bounds.…”

“…One may employ Algorithm 4.3 for similar calculations on the L q -approximation, 2 < q < ∞, using the corresponding results from Lemma 4.5. That way one can reproduce the exact asymptotic order for the Monte Carlo approximation of W r 2 (T d ) ֒→ L q (T d ) that has been determined by Fang and Duan [5,Theorem 1]. The algorithm behind their estimates, however, is hidden within theory of pseudo s-scales.…”

“…Let T d := [0, 1) d be the d-dimensional torus and W r p (T d ) be the periodic Sobolev spaces of dominating mixed smoothness r on T d . In this paper we study L ∞ -approximation of the class W r p (T d ) where we supplement the results of Fang and Duan [4,5] on L q -approximation. Let us mention that the study of L ∞ -approximation of function spaces with dominating mixed smoothness is much harder compared to the case 1 < q < ∞ and different techniques are needed, see comments and open problems in [3,Section 4.6].…”