Cubature  formulas for symmetric measures  in higher dimensions with few points

Hinrichs, Aicke; Novak, Erich

doi:10.1090/s0025-5718-07-01974-6

Cited by 13 publications

(8 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the number of the points is much greater than the theoretical lower bound. Very recently, starting from Smolyak method, Hinrichs and Novak [9] presented a kind of formula with n 2 + 7n + 1 points for fifth degree case and (n 3 + 21n 2 + 20n + 3)/3 points for seventh degree case, which is very close to the theoretical lower bound. Before them, except Lu's formula [11], the best upper bounds were of the form "≈ 2n 2 " and "≈ 4n 3 /3" respectively, where "≈" denotes the strong equivalence of sequences.…”

Section: Introductionmentioning

confidence: 54%

Constructing cubature formulae of degree 5 with few points

Meng

Luo

2013

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

This paper will devote to construct a family of fifth degree cubature formulae for n-cube with symmetric measure and n-dimensional spherically symmetrical region. The formula for n-cube contains at most n 2 + 5n + 3 points and for n-dimensional spherically symmetrical region contains only n 2 + 3n + 3 points. Moreover, the numbers can be reduced to n 2 + 3n + 1 and n 2 + n + 1 if n = 7 respectively, the latter of which is minimal.

show abstract

Section: Introductionmentioning

confidence: 54%

Constructing cubature formulae of degree 5 with few points

Meng

Luo

2013

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…, where the norm is a specific L 2 -norm or L ∞ -norm and Θ is the convergence rate, determined by m, d, and certain smoothness conditions, refer to [Gar12] for details. More importantly, if we define N min (2m + 1, d) as the minimal number of nodes needed by Q (m,d) (f ) admitting a total degree 2m + 1 of exactness, our interpolatory rule reduces N min (2m + 1, d) when compared to the standard SGQ due to the fully symmetric property, see Theorem 1 in [HN07].…”

Section: Deterministic Rules For Kernel Approximationmentioning

confidence: 99%

Towards a Unified Quadrature Framework for Large-Scale Kernel Machines

Liu

Huang

Chen

et al. 2022

IEEE Trans. Pattern Anal. Mach. Intell.

View full text Add to dashboard Cite

In this paper, we develop a quadrature framework for large-scale kernel machines via a numerical multiple integration representation. Leveraging the fact that the integration domain and measure of typical kernels, e.g., Gaussian kernels, arc-cosine kernels, are fully symmetric, we introduce a deterministic fully symmetric interpolatory rule to efficiently compute its quadrature nodes and associated weights to approximate such typical kernels. This interpolatory rule is able to reduce the number of needed nodes while retaining a high approximation accuracy. Further, we randomize the above deterministic rule such that the proposed stochastic version can generate dimension-adaptive feature mappings for kernel approximation. Our stochastic rule has the nice statistical properties of unbiasedness and variance reduction with fast convergence rate. In addition, we elucidate the relationship between our deterministic/stochastic interpolatory rules and current quadrature rules for kernel approximation, including the sparse grids quadrature and stochastic spherical-radial rule, thereby unifying these methods under our framework. Experimental results on several benchmark datasets show that our fully symmetric interpolatory rule compares favorably with other representative random features based methods

show abstract

“…They are embedded and result in better stability and lower node count than their GH counterparts. More e cient rules have been developed for l = 5 and l = 7 for both general ⇢(s) [22,14] and ⇢(s) > 0 [33,18]. The GK rules for l > 13 are most e cient, in the sense that they require the fewest nodes for a given degree of accuracy.…”

Section: Formulation Of Approximate Sbvp Approximation Of the Paramementioning

confidence: 99%

Stochastic Collocation Methods for Nonlinear Parabolic Equations with Random Coefficients

Barajas-Solano¹,

Tartakovsky²

2016

SIAM/ASA J. Uncertainty Quantification

View full text Add to dashboard Cite

We evaluate the performance of global stochastic collocation methods for solving nonlinear parabolic and elliptic problems (e.g., transient and steady nonlinear di↵usion) with random coe cients. The robustness of these and other strategies based on a spectral decomposition of stochastic state variables depends on the regularity of the system's response in outcome space. The latter is a↵ected by statistical properties of the input random fields. These include variances of the input parameters, whose e↵ect on the computational e ciency of this class of uncertainty quantification techniques has remained unexplored. Our analysis shows that if random coe cients have low variances and large correlation lengths, stochastic collocation strategies outperform Monte Carlo simulations (MCS). As variance increases, the regularity of the stochastic response decreases, which requires higherorder quadrature rules to accurately approximate the moments of interest and increases the overall computational cost above that of MCS.

show abstract

Cubature formulas for symmetric measures in higher dimensions with few points

Cited by 13 publications

References 26 publications

Constructing cubature formulae of degree 5 with few points

Constructing cubature formulae of degree 5 with few points

Towards a Unified Quadrature Framework for Large-Scale Kernel Machines

Stochastic Collocation Methods for Nonlinear Parabolic Equations with Random Coefficients

Contact Info

Product

Resources

About