The cost of deterministic, adaptive, automatic algorithms: Cones, not balls

Clancy, Nicholas; Ding, Yuhan; Hamilton, Caleb; Hickernell, Fred J.; Zhang, Yizhi

doi:10.1016/j.jco.2013.09.002

Cited by 14 publications

(16 citation statements)

References 15 publications

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“…There are also corresponding adaption-does-not-help results in other settings, see, e.g., [12, 14, 17, 18]. On the other hand, if the class is not convex and/or a different from the worst case error criterion is used to compare algorithms then adaption can significantly help, see [2] or [16]. …”

Section: Introductionmentioning

confidence: 99%

Automatic integration using asymptotically optimal adaptive Simpson quadrature

Plaskota

2014

Numer. Math.

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We present a novel theoretical approach to the analysis of adaptive quadratures and adaptive Simpson quadratures in particular which leads to the construction of a new algorithm for automatic integration. For a given function with and possible endpoint singularities the algorithm produces an approximation to within a given asymptotically as . Moreover, it is optimal among all adaptive Simpson quadratures, i.e., needs the minimal number of function evaluations to obtain an -approximation and runs in time proportional to .

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Section: Introductionmentioning

confidence: 99%

Automatic integration using asymptotically optimal adaptive Simpson quadrature

Plaskota

2014

Numer. Math.

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“…In the underlying work we assume that f is from a cone F p,q,K . The idea to take "cones, not balls" was suggested and studied by Hickernell and his colleagues, see [7] and [13]. Interesting cones could be of the form…”

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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“…Our theoretically justified Algorithms A and M build upon the ideas used to construct the adaptive algorithms in [7,9,12,13,15,16,24]. In all those cases, a cone, C, of input functions is identified for which the adaptive algorithms succeed, just as is done here.…”

Section: Related Work On Adaptive Algorithmsmentioning

confidence: 99%

Local adaption for approximation and minimization of univariate functions

Choi

Ding

Hickernell

et al. 2017

Journal of Complexity

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Most commonly used adaptive algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a user-specified absolute error tolerance for a cone, C, of non-spiky input functions in the Sobolev space W 2,∞ [a, b]. Our algorithms automatically determine where to sample the function-sampling more densely where the second derivative is larger. The computational cost of our algorithm for approximating a univariate function f on a bounded interval with L ∞ -error no greater than ε is O f 1 2 /ε as ε → 0. This is the same order as that of the best function approximation algorithm for functions in C. The computational cost of our global minimization algorithm is of the same order and the cost can be substantially less if f significantly exceeds its minimum over much of the domain. Our Guaranteed Automatic Integration Library (GAIL) contains these new algorithms. We provide numerical experiments to illustrate their superior performance.

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The cost of deterministic, adaptive, automatic algorithms: Cones, not balls

Cited by 14 publications

References 15 publications

Automatic integration using asymptotically optimal adaptive Simpson quadrature

Automatic integration using asymptotically optimal adaptive Simpson quadrature

Solvable integration problems and optimal sample size selection

Local adaption for approximation and minimization of univariate functions

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