Integration in Hermite spaces of analytic functions

Irrgeher, Christian; Kritzer, Peter; Leobacher, Gunther; Pillichshammer, Friedrich

doi:10.1016/j.jco.2014.08.004

Cited by 46 publications

(60 citation statements)

References 29 publications

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“…Besides the case of polynomially decaying coefficients, Hermite spaces with exponentially decaying coefficients were also considered. Multivariate integration for such Hermite spaces has been analyzed in [9]. It is also shown there that the elements of those function spaces are analytic.…”

Section: Definitionmentioning

confidence: 99%

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On the Optimal Order of Integration in Hermite Spaces with Finite Smoothness

Dick¹,

Irrgeher²,

Leobacher³

et al. 2018

SIAM J. Numer. Anal.

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We study the numerical approximation of integrals over R s with respect to the standard Gaussian measure for integrands which lie in certain Hermite spaces of functions. The decay rate of the associated sequence is specified by a single integer parameter which determines the smoothness classes and the inner product can be expressed via L 2 norms of the derivatives of the function.We map higher order digital nets from the unit cube to a suitable subcube of R s via a linear transformation and show that such rules achieve, apart from powers of log N , the optimal rate of convergence of the integration error.

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Section: Definitionmentioning

confidence: 99%

“…The results in [10] and [9] make heavy use of the facts that Hermite spaces are reproducing kernel Hilbert spaces with canonical kernel…”

Section: Definitionmentioning

confidence: 99%

On the Optimal Order of Integration in Hermite Spaces with Finite Smoothness

Dick¹,

Irrgeher²,

Leobacher³

et al. 2018

SIAM J. Numer. Anal.

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“…The error of A 0,s is called the initial (worst-case) error and is given bySince we will study a class of weighted reproducing kernel Hilbert spaces with exponentially decaying weights, which will be introduced in Section 1.2, we are concerned with spaces H s of smooth functions. We remark that reproducing kernel Hilbert spaces of a similar flavor were previously considered in [2,3,5,6,7,8,9]. In this case it is natural to expect that, by using suitable algorithms, we should be able to obtain errors that converge to zero very quickly as n increases, namely exponentially fast.…”

mentioning

confidence: 91%

“…However, it is not clear how long we have to wait to see this nice asymptotic behavior especially for large s. This, of course, depends on how C(s), M(s) and p(s) depend on s, and this is the subject of tractability. Thus, we intend to study how the information complexity depends on log ε −1 and s by considering the following tractability notions, which were already considered in [2,3,5,6,8]. The nomenclature was introduced in [9].…”

Section: Introductionmentioning

confidence: 99%

Integration and approximation in cosine spaces of smooth functions

Irrgeher

Kritzer

Pillichshammer

2018

Mathematics and Computers in Simulation

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We study multivariate integration and approximation for functions belonging to a weighted reproducing kernel Hilbert space based on half-period cosine functions in the worst-case setting. The weights in the norm of the function space depend on two sequences of real numbers and decay exponentially. As a consequence the functions are infinitely often differentiable, and therefore it is natural to expect exponential convergence of the worst-case error. We give conditions on the weight sequences under which we have exponential convergence for the integration as well as the approximation problem. Furthermore, we investigate the dependence of the errors on the dimension by considering various notions of tractability. We prove sufficient and necessary conditions to achieve these tractability notions.Keywords: numerical integration, function approximation, cosine space, worst-case error, exponential convergence, tractability 2010 MSC: 41A63, 41A25, 65C05, 65D30, 65Y20Without loss of generality, see, e.g., [15] or [12, Section 4], we approximate S s by a linear algorithm A n,s using n information evaluations which are given by linear functionals from the class Λ ∈ {Λ all , Λ std }. More precisely, we approximate S s by algorithms of the formObviously, for multivariate integration only the class Λ std makes sense. Furthermore, we remark that in this paper we consider only function spaces for which Λ std ⊂ Λ all .We measure the error of an algorithm A n,s in terms of the worst-case error, which is defined aswhere · Hs , · Gs denote the norms in H s and G s , respectively. The nth minimal (worstcase) error is given bywhere the infimum is taken over all admissible algorithms A n,s . When we want to emphasize that the nth minimal error is taken with respect to algorithms using information from the class Λ ∈ {Λ all , Λ std }, we write e(n, S s ; Λ). For n = 0, we consider algorithms that do not use information evaluations and therefore we use A 0,s ≡ 0. The error of A 0,s is called the initial (worst-case) error and is given bySince we will study a class of weighted reproducing kernel Hilbert spaces with exponentially decaying weights, which will be introduced in Section 1.2, we are concerned with spaces H s of smooth functions. We remark that reproducing kernel Hilbert spaces of a similar flavor were previously considered in [2,3,5,6,7,8,9]. In this case it is natural to expect that, by using suitable algorithms, we should be able to obtain errors that converge to zero very quickly as n increases, namely exponentially fast. By exponential convergence (EXP) for the worst-case error we mean that there exist a number q ∈ (0, 1) and functions p, C, M : N → (0, ∞) such that e(n, S s ) ≤ C(s) q (n/M (s)) p (s) for all s, n ∈ N.(1)If the function p in (1) can be taken as a constant function, i.e., p(s) = p > 0 for all s ∈ N, we say that we achieve uniform exponential convergence (UEXP) for e(n, S s ). Furthermore, we denote by p * (s) and p * the largest possible rates p(s) and p such that EXP and UEXP holds, respectively. When stu...

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“…Numerical integration of infinitely many times differentiable functions in certain function spaces has recently been considered in [8,11,15,16]. However, the results on higher order digital nets in [2,3] do not improve if one assumes that the integrand is infinitely many times differentiable.…”

Section: Introductionmentioning

confidence: 99%

Construction of interlaced polynomial lattice rules for infinitely differentiable functions

et al. 2017

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We study multivariate integration over the s-dimensional unit cube in a weighted space of infinitely differentiable functions. It is known from a recent result by Suzuki that there exists a good quasi-Monte Carlo (QMC) rule which achieves a super-polynomial convergence of the worst-case error in this function space, and moreover, that this convergence behavior is independent of the dimension under a certain condition on the weights.In this paper we provide a constructive approach to finding a good QMC rule achieving such a dimension-independent super-polynomial convergence of the worst-case error. Specifically, we prove that interlaced polynomial lattice rules, with an interlacing factor chosen properly depending on the number of points N and the weights, can be constructed using a fast component-by-component algorithm in at most O(sN (log N )2 ) arithmetic operations to achieve a dimension-independent super-polynomial convergence. The key idea for the proof of the worst-case error bound is to use a variant of Jensen's inequality with a purposely-designed concave function.

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Integration in Hermite spaces of analytic functions

Cited by 46 publications

References 29 publications

On the Optimal Order of Integration in Hermite Spaces with Finite Smoothness

On the Optimal Order of Integration in Hermite Spaces with Finite Smoothness

Integration and approximation in cosine spaces of smooth functions

Construction of interlaced polynomial lattice rules for infinitely differentiable functions

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