Abstract
We introduce a new configuration of node sets: crosslet grids for high-dimensional numerical integration, and develop symmetric quadrature rules on the unit cube of the d-dimensional Euclidean space based on these node sets. Our algorithms give the same order of accuracy as those established on full grids, but require much fewer nodes, and therefore encounter far less computational complexity in execution. Theoretical analysis and numerical simulations show that quadrature rules based on crosslet grids are effective when applied to integrands that have localized nonsmoothness. The research work here reveals a close connection between quadrature rules and quasi-interpolation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bourgain, J., Lindenstrauss, J.: Distribution of points on spheres and approximation by zonotopes. Israel J. Math. 64, 25–31 (1988)
Bourgain, J., Lindenstrauss, J., Milman, V.: Approximation of zonoids by zonotopes. Acta Math. 162, 73–141 (1989)
Bourgain, J., Lindenstrauss, J.: Approximating the ball by a Minkowski sum of segments with equal length. Discret. Comput. Geom. 9, 131–144 (1993)
Bungartz, H.J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)
Cheney, E.W., Kincaid, D.: Numerical Mathematics and Computing, 7th edn. Brooks/Cole Publishing Company, USA (2013)
Dick, J., Kuo, F.Y., Sloan, I.H.: High-dimensional integration: the quasi-Monte Carlo way. Acta Numer. 22, 133–288 (2013)
Gao, W., Fasshauer, G., Sun, X., Zhou, X.: Optimality and regularization properties of quasi-interpolation: deterministic and stochastic approaches. SIAM J. Numer. Anal. 58(4), 2059–2078 (2020)
Gao, W., Sun, X., Wu, Z., Zhou, X.: Multivariate monte carlo approximation based on scattered data. SIAM J. Sci. Comput. 42, 2262–2280 (2020)
Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algorithm. 18, 209–232 (1998)
Jeong, B., Kersey, S., Yoon, J.: Approximation of multivariate functions on sparse grids by quasi-interpolation based on radial basis functions, to appear in SIAM J. Sci. Comput.
Keshavarzzadeh, V., Kirby, R.M., Narayan, A.: Numerical integration in multiple dimensions with designed quadrature. SIAM J. Sci. Comput. 40, 2033–2061 (2018)
Marchi, S.D., Wendland, H.: On the convergence of the rescaled localized radial basis function method. Appl. Math. Lett. 99, 105996 (2019)
Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 148(5), 1042–1045 (1963)
Usta, F., Levesley, J.: Mutlilevel quasi-interpolation on a sparse grid with the Gaussian. Numer. Algorithm. 77, 793–808 (2018)
Wagner, G.: On a new method for constructing good point sets on spheres. Discrete Comput. Geom. 9, 111–129 (1993)
Williams, D.M., Frontin, C.V., Miller, E.A., Darmofal, D.L.: A family of symmetric, optimized quadrature rules for pentatopes. Comput. Math. Appl. 80, 1405–1420 (2020)
Acknowledgements
An anonymous reviewer has pointed out some typos and inconsistencies of notations in the initial version of the article, to whom we are grateful.
Funding
The first and the third authors are supported by the National Natural Science Foundation of China (No. 12001487) and the Characteristic & Preponderant Discipline of Key Construction Universities in Zhejiang Province (Zhejiang Gongshang University- Statistics).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gao, Q., Sun, X. & Zhang, S. Multivariate quadrature rules on crosslet sparse grids. Numer Algor 90, 951–962 (2022). https://doi.org/10.1007/s11075-021-01217-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-021-01217-3