Some Results on the Complexity of Numerical Integration

Novak, Erich

doi:10.1007/978-3-319-33507-0_6

Cited by 33 publications

(22 citation statements)

References 115 publications

(146 reference statements)

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“…Such adaptive methods have been known to be able to outperform non-adaptive methods in the following case: the hypothesis space is imbalanced or non-convex (see e.g. Section 1 of [41]). In the worst case error, the hypothesis space is the unit ball in the RKHS H, which is balanced and convex and so adaptation does not help.…”

Section: Discussionmentioning

confidence: 99%

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

2019

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This paper presents convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights, and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase quickly), or (ii) if the minimum distance between design points does not decrease very quickly. As a consequence of the latter result, we derive a rate of convergence for Bayesian quadrature in misspecified settings. We reveal a condition on design points to make Bayesian quadrature robust to misspecification, and show that, under this condition, it may adaptively achieve the optimal rate of convergence in the Sobolev space of a lesser order (i.e., of the unknown smoothness of a test integrand), under a slightly stronger regularity condition on the integrand. MSC 2010 subject classification: Primary: 65D30, Secondary: 65D32, 65D05, 46E35, 46E22.

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Section: Discussionmentioning

confidence: 99%

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

2019

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“…This paper is in the tradition of IBC, see [27,36]. A recent survey on the complexity of the integration problem is [28]. Let F 1 , F 2 and R be normed linear spaces.…”

Section: Discussionmentioning

confidence: 99%

Solvable integration problems and optimal sample size selection

Kunsch

Novak

Rudolf³

2019

Journal of Complexity

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We compute the integral of a function or the expectation of a random variable with minimal cost and use, for our new algorithm and for upper bounds of the complexity, i.i.d. samples. Under certain assumptions it is possible to select a sample size based on a variance estimation, or -more generally -based on an estimation of a (central absolute) p-moment. That way one can guarantee a small absolute error with high probability, the problem is thus called solvable. The expected cost of the method depends on the p-moment of the random variable, which can be arbitrarily large.In order to prove the optimality of our algorithm we also provide lower bounds. These bounds apply not only to methods based on i.i.d. samples but also to general randomized algorithms. They show that -up to constantsthe cost of the algorithm is optimal in terms of accuracy, confidence level, and norm of the particular input random variable. Since the considered classes of random variables or integrands are very large, the worst case cost would be infinite. Nevertheless one can define adaptive stopping rules such that for each input the expected cost is finite.We contrast these positive results with examples of integration problems that are not solvable.

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“…If performing the integration numerically, the corresponding complexity is mainly determined by the dimension of the domain of the integrand and the required error bound [30]. If the integration regarding input function P,Q has a simple closed-form formula, then evaluation of the integrated function can usually be done in constant time, so the corresponding complexities in Table 5 are either linear or quadratic in m. Moreover, the process of calculating these formulas can be easily processed in parallel, since all of them are simply summations of individual cases.…”

Section: Complexity Of Computationmentioning

confidence: 99%

On the distance between random events on a network

2019

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In this paper, we study several statistical properties regarding the distance between events that take place on random locations along the edges of a given network. We derive analytical expressions for the arbitrary moments of such a distance, its probability density function, its cumulative distribution function, as well as their conditional counterparts for the cases in which the position of one event is known in advance. As part of this study, we implement our developments as a callable library for the Python language, to provide potential users with a computational engine able to calculate and visualize these statistics for any given network. We test our implementation on several networks of different sizes and topologies, analyze some of interesting properties we observed in our experiments, and discuss several applications for our proposed methodology. In particular, we focus our discussion on applications aimed to help with the optimal design of emergency response systems on infrastructure networks. KEYWORDSdistance distributions, emergency response systems, infrastructure networks, random events, spatial random processes INTRODUCTION AND MOTIVATIONA wide variety of problems arising from different scientific domains involve the careful study of random events that occur on scattered locations of a given network [3,11,15,24,44]. Among the many different examples, perhaps the studies that have permeated the literature the most, are the ones that involve the analysis of events that take place on infrastructure networks [2]. Of particular importance, given the major role they play in our society, are the studies aimed to optimally design emergency response systems (ERS), like police, fire, and medical services [8,25,29,40]. In this particular context, a detailed analysis of the location of criminal acts, car accidents, and medical emergencies, among others, can be enormously beneficial to maximize the coverage and efficiency of such systems.Motivated by the fact that most ERS operations require one to rapidly dispatch specialized units in response to randomly occurring critical incidents, there has been a notable interest in designing methodological tools to help analyze several random properties regarding such events [4]. Moreover, with the recent emergence of new technologies that facilitate the collection, storage, and processing of large amounts data, the need for new developments able to estimate and characterize locations, distances, and the nature of such random events has become even more prevalent in recent years [8].One of the key components pertaining to the study of random events in networks is the statistical characterization of the distances between them. Given that the events take place on random locations across a given network, studying the random variables representing these distances can bring valuable information regarding the dynamics behind these complex systems [27].A few examples of key problems in the context of ERS, whose solution approaches can benefit from these methodologies are: (a...

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Some Results on the Complexity of Numerical Integration

Cited by 33 publications

References 115 publications

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

Solvable integration problems and optimal sample size selection

On the distance between random events on a network

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