High-dimensional integration on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:math>, weighted Hermite spaces, and orthogonal transforms

Irrgeher, Christian; Leobacher, Gunther

doi:10.1016/j.jco.2014.09.002

Cited by 28 publications

(44 citation statements)

References 16 publications

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“…In this paper, we continue the work on exponentially decreasing r, thereby extending the results in [12]. Polynomially decreasing r will be studied also in [4].…”

Section: Hermite Polynomials and The Hermite Spacementioning

confidence: 79%

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Integration in Hermite spaces of analytic functions

Irrgeher

Kritzer

Leobacher

et al. 2015

Journal of Complexity

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a b s t r a c tWe study integration in a class of Hilbert spaces of analytic functions defined on the R s . The functions are characterized by the property that their Hermite coefficients decay exponentially fast. We use Gauss-Hermite integration rules and show that the errors of our algorithms decay exponentially fast. Furthermore, we study tractability in terms of s and log ε −1 and give necessary and sufficient conditions under which we achieve exponential convergence with EC-weak, EC-polynomial, and EC-strong polynomial tractability.

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“…In this paper, we continue the work on exponentially decreasing r, thereby extending the results in [12]. Polynomially decreasing r will be studied also in [4].…”

Section: Hermite Polynomials and The Hermite Spacementioning

confidence: 79%

“…In that paper, the m j defined a regular grid that served as integration node set. Here, we make similar choices for the parameters m j , but they now determine the order of the Gauss-Hermite rule (12).…”

Section: Proposition 2 Let a Ns Be The S-dimensional Tensor Productmentioning

confidence: 99%

Integration in Hermite spaces of analytic functions

Irrgeher

Kritzer

Leobacher

et al. 2015

Journal of Complexity

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“…dx. Now we define the Hermite space analogous to [10], where the construction as well as basic properties are given in great detail. Definition 1.…”

Section: Hermite Spaces Of Functions Of Finite Smoothnessmentioning

confidence: 99%

“…This definition of a Hermite space is slightly more general than that given in [10]. There it was required k∈N s 0 r(k) < ∞.…”

Section: Hermite Spaces Of Functions Of Finite Smoothnessmentioning

confidence: 99%

“…The authors there consider the sequencer s,α (k) = k −α , which asymptotically is same as the choice of r s,α in this paper. But due to the stronger (and unnecessary) requirement of summability in [10] there is the restriction of α > 1, which is relaxed to α ≥ 1 here. Besides the case of polynomially decaying coefficients, Hermite spaces with exponentially decaying coefficients were also considered.…”

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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The Real Case: Towards Extending Krivine’s Approach

Oertel

2024

Lecture Notes in Mathematics

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High-dimensional integration on Rd, weighted Hermite spaces, and orthogonal transforms

Cited by 28 publications

References 16 publications

Integration in Hermite spaces of analytic functions

Integration in Hermite spaces of analytic functions

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

The Real Case: Towards Extending Krivine’s Approach

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