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

Abstract

In this paper, we study the complexity of information of approximation problem on the multivariate Sobolev space with bounded mixed derivative MW r p,α (T d ), 1 < p < ∞, in the norm of Lq(T d ), 1 < q < ∞, by adaptive Monte Carlo methods. Applying the discretization technique and some properties of pseudo-s-scale, we determine the exact asymptotic orders of this problem. Keywords: adaptive Monte Carlo method, Sobolev space with bounded mixed derivative, asymptotic order MSC(2000): 41A46, 41A63, 65C05, 65D99

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

“…Our results for L ∞ -approximation fit to the general picture for L q -approximation of the classes W r p (T d ), see Fang and Duan [4,5] for 1 < q < ∞, and also similar results for non-periodic isotropic spaces due to Heinrich [7]. We have different (open) regions within the (p, q)-domain:…”

“…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.…”

“…Linear methods are always nonadaptive, hence varying cardinality means n = n(ω). In this case, the error e ran,lin (n) for algorithms with varying cardinality, and e ran,lin (n) for fixed cardinality, are closely related, For the nonlinear setting, the lower bounds in Fang and Duan [4] and the present paper rely on a technique due to Heinrich [7], which is based on norm expectations for Gaussian measures. These lower bounds hold for adaptive algorithms as well.…”

“…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].…”

Breaking the curse for uniform approximation in Hilbert spaces via Monte Carlo methods

“…Heinrich 1992 [9] continued this work, providing nonlinear Monte Carlo approximation methods for certain parameter settings of d-variate isotropic spaces. Similar results for periodic spaces of dominating mixed smoothness by Fang and Duan [7] still rely on Mathé's sequence space result, but for some cases that have been left open there, modified techniques needed to be applied, see [4]. To summarize, for L q -approximation of functions from a Sobolev space with integrability index p, randomization turns out to speed up the convergence if max{2, p} < q.…”