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arXiv:1407.1615 · De Gruyter eBooks

Tractability of multivariate analytic problems

Peter Kritzer, Friedrich Pillichshammer, Henryk Woźniakowski

Published Jul 25, 2014• 25 citations
arXivHTMLPDF

Abstract

In the theory of tractability of multivariate problems one usually studies problems with finite smoothness. Then we want to know which $s$-variate problems can be approximated to within $\varepsilon$ by using, say, polynomially many in $s$ and $\varepsilon^{-1}$ function values or arbitrary linear functionals. There is a recent stream of work for multivariate analytic problems for which we want to answer the usual tractability questions with $\varepsilon^{-1}$ replaced by $1+\log \varepsilon^{-1}$. In this vein of research, multivariate integration and approximation have been studied over Korobov spaces with exponentially fast decaying Fourier coefficients. This is work of J. Dick, G. Larcher, and the authors. There is a natural need to analyze more general analytic problems defined over more general spaces and obtain tractability results in terms of $s$ and $1+\log \varepsilon^{-1}$. The goal of this paper is to survey the existing results, present some new results, and propose further questions for the study of tractability of multivariate analytic questions.

DOI
10.1515/9783110317930.147

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A note about EC-(s,t)(s,t)(s,t)-weak tractability of multivariate approximation with analytic Korobov kernels
This note is devoted to discussing multivariate approximation of continuous functions on [0,1]d[0,1]^d[0,1]d with analytic Korobov kernels in the worst and average case settings. We only consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We study EC-(s,t)(s, t)(s,t)-weak tractability under the absolute or normalized error criterion, and obtain necessary and sufficient conditions for 0<min⁡(s,t)<10<\min(s,t)<10<min(s,t)<1 and max⁡(s,t)≤1\max(s,t)\le 1max(s,t)≤1 in the worst case setting and for s,t>0s,t>0s,t>0 in the average case setting.
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In this paper we study multivariate integration for a weighted Korobov space for which the Fourier coefficients of the functions decay exponentially fast. This implies that the functions of this space are infinitely times differentiable. Weights of the Korobov space monitor the influence of each variable and each group of variables. We show that there are numerical integration rules which achieve an exponential convergence of the worst-case integration error. We also investigate the dependence of the worst-case error on the number of variables s s , and show various tractability results under certain conditions on weights of the Korobov space. Tractability means that the dependence on s s is never exponential, and sometimes the dependence on s s is polynomial or there is no dependence on s s at all.
We study d-variate approximation for a weighted unanchored Sobolev space having smoothness m ≥ 1. Folk wisdom would lead us to believe that this problem should become easier as its smoothness increases. This is true if we are only concerned with asymptotic analysis: the nth minimal error is of order n^-(m-δ) for any δ > 0. However, it is unclear how long we need to wait before this asymptotic behavior kicks in. How does this waiting period depend on d and m? We prove that no matter how the weights are chosen, the waiting period is at least m^d, even if the error demand ε is arbitrarily close to 1. Hence, for m ≥ 2, this waiting period is exponential in d, so that the problem suffers from the curse of dimensionality and is intractable. In other words, the fact that the asymptotic behavior improves with m is irrelevant when d is large. So, we will be unable to vanquish the curse of dimensionality unless m = 1 , i.e., unless the smoothness is minimal. We then show that our problem can be tractable if m = 1. That is, we can find an ε-approximation using polynomially-many (in d and ε^-1) information operations, even if only function values are permitted. When m = 1, it is even possible for the problem to be strongly tractable, i.e., we can find an ε-approximation using polynomially-many (in ε^-1) information operations, independent of d. These positive results hold when the weights of the Sobolev space decay sufficiently quickly or are bounded finite-order weights, i.e., the d-variate functions we wish to approximate can be decomposed as sums of functions depending on at most ω variables, where ω is independent of d.
This paper studies the multivariate approximation of functions in weighted Korobov spaces using multiple rank-1 lattice rules. It has been shown by Kämmerer and Volkmer (2019) that algorithms based on multiple rank-1 lattices achieve the optimal convergence rate for the L∞L_{\infty}L∞​ error in Wiener-type spaces, up to logarithmic factors. While this result was translated to weighted Korobov spaces in the recent monograph by Dick, Kritzer, and Pillichshammer (2022), the analysis requires the smoothness parameter ααα to be greater than 111 and is restricted to product weights. In this paper, we extend this result for multiple rank-1 lattice-based algorithms to the case where 1/2<α≤11/2<α\le 11/2<α≤1 and for general weights, covering a broader range of periodic functions with low smoothness and general relative importance of variables. We also provide a summability condition on the weights to ensure strong polynomial tractability for any α>1/2α>1/2α>1/2. Furthermore, by incorporating random shifts into multiple rank-1 lattice-based algorithms, we prove that the resulting randomized algorithm achieves a nearly optimal convergence rate in terms of the worst-case root mean squared L2L_2L2​ error, while retaining the same tractability property.