The curse of dimensionality for numerical integration on general domains

Hinrichs, Aicke; Prochno, Joscha; Ullrich, Mario

doi:10.1016/j.jco.2018.08.003

Cited by 34 publications

(21 citation statements)

References 24 publications

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“…. Since the solution of the response surface function easily falls into a "curse of dimensionality" when there are too many structural parameters and the parameters may not be linearly independent, the principal component analysis was used to reduce the number of variables [36,54]. According to the probability distribution parameters listed in calculations.…”

Section: Principal Component Analysis Resultsmentioning

confidence: 99%

Embankment Seismic Fragility Assessment under the Near‐Fault Pulse‐like Ground Motions by Applying the Response Surface Method

Che

Yin

Zhou

et al. 2021

Shock and Vibration

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Uncertainties of the ground motions and structural parameters are the main factors that limit the accuracy of embankment seismic fragility assessment. In response to the uncertainties of the ground motions, artificial synthesizing method of the near-fault pulse-like ground motions was proposed, and 15 ground motions with the rupture fault distances ranging from 1 to 15 km were synthesized by taking the Chi-Chi earthquake in Taiwan, China, as an example. The Xi’an-Baoji expressway K1125 + 470 embankment was taken as the research object, and a total of 12 structural parameters were selected as the design variables, namely, the elastic modulus, bulk modulus, shear modulus, density, cohesion force, and internal friction angle of the embankment fill and soil foundation, respectively. In response to the uncertainties of these parameters, 3 principal components with large impacts on the embankment seismic fragility were extracted based on the principal component analysis. Mapping relationships among the principal components and embankment seismic damages were analyzed using the uniform design response surface method, and the seismic fragility assessment was carried out and the fragility curves were plotted. The research results are consistent with the actual embankment seismic damage conditions of the Chi-Chi earthquake, indicating that the proposed method is scientific and reasonable. It also shows that it would obviously overestimate the seismic performance in the embankment seismic fragility assessment without considering the uncertainties of the ground motions and structural parameters.

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Section: Principal Component Analysis Resultsmentioning

confidence: 99%

Embankment Seismic Fragility Assessment under the Near‐Fault Pulse‐like Ground Motions by Applying the Response Surface Method

Che

Yin

Zhou

et al. 2021

Shock and Vibration

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“…for some a > 0 and all d ∈ N. The upper bounds, however, heavily exploit the geometry of the unit cube. We remark that the curse of dimensionality for the integration problem on general domains is studied in the recent paper [4].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Uniform recovery of high-dimensional Cr-functions

Krieg

2019

Journal of Complexity

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We consider functions on the d-dimensional unit cube whose partial derivatives up to order r are bounded by one. It is known that the minimal number of function values that is needed to approximate the integral of such functions up to the error ε is of order (d/ε) d/r . Among other things, we show that the minimal number of function values that is needed to approximate such functions in the uniform norm is of order (d r/2 /ε) d/r whenever r is even.

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“…3) It is known that the integration problem for C 2 (D d ) and certain D d suffers from the curse of dimensionality. This is true if D d has a small radius, see [17] for the best known results. For example, the curse is known if D d is a ℓ d p ball and p ≥ 2.…”

Section: Some Comments Are In Ordermentioning

confidence: 93%

Algorithms and complexity for functions on general domains

Novak

2020

Journal of Complexity

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Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the approximation or integration of functions defined on D d ⊂ R d and only assume that D d is a bounded Lipschitz domain. Some results are even more general. We study three different concepts to measure the complexity: order of convergence, asymptotic constant, and explicit uniform bounds, i.e., bounds that hold for all n (number of pieces of information) and all (normalized) domains.It is known for many problems that the order of convergence of optimal algorithms does not depend on the domain D d ⊂ R d . We present examples for which the following statements are true:1. Also the asymptotic constant does not depend on the shape of D d or the imposed boundary values, it only depends on the volume of the domain.2. There are explicit and uniform lower (or upper, respectively) bounds for the error that are only slightly smaller (or larger, respectively) than the asymptotic error bound. *Much is known about the order of convergence of the numbers e n , in particular for simple domains D d , such as the cube or the torus. General bounded Lipschitz domains are studied in [7,10,11,12,26,27,29,38,39,40,42,43]. In many cases, the optimal order of the e n is of the form e n ≍ n −α (log n) β , where α and β do not depend on the domain D d . It is interesting to know also the exact asymptotic constant C := lim n→∞ e n n α (log n) −β , if it exists. The value of C is known only in rare cases (unless d = 1, we do not discuss the univariate case in detail), and usually only for very special domains,

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The curse of dimensionality for numerical integration on general domains

Cited by 34 publications

References 24 publications

Embankment Seismic Fragility Assessment under the Near‐Fault Pulse‐like Ground Motions by Applying the Response Surface Method

Embankment Seismic Fragility Assessment under the Near‐Fault Pulse‐like Ground Motions by Applying the Response Surface Method

Uniform recovery of high-dimensional Cr-functions

Algorithms and complexity for functions on general domains

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