Multilevel quasi-interpolation on a sparse grid with the Gaussian

Usta, Fuat; Levesley, Jeremy

doi:10.1007/s11075-017-0340-y

Cited by 11 publications

(6 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If we consider functions in the native space of the Gaussian, a space of infinitely differentiable functions, then the order of convergence is d − ln(d) ln (2) . This work is motivated by the desire to prove convergence of the multilevel sparse grid quasi-interpolation introduced by Levesley and Usta [5]. Multilevel sparse grid algorithms using smooth functions were introduced by the author and collaborators [4].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Convergence of Multilevel Stationary Gaussian Convolution

Hubbert

Levesley

2019

Lecture Notes in Computational Science and Engineering

Self Cite

View full text Add to dashboard Cite

In this paper we give a short note showing convergence rates for multilevel periodic approximation of smooth functions by multilevel Gaussian convolution. We will use the Gaussian scaling in the convolution at the finest level as a proxy for degrees of freedom d in the model. We will show that, for functions in the native space of the Gaussian, convergence is of the order d − ln(d) ln(2) . This paper provides a baseline for what should be expected in discrete convolution, which will be the subject of a follow up paper.

show abstract

Section: Introductionmentioning

confidence: 99%

“…Using this we replicate the steps from Proposition 1 (associated to the β > 5 2 case) to reach the required bound. For the remaining cases captured by j ≥ 2β−1 4 (or equivalently β ≤ 2j + 1 2 ) we follow Proposition 1 again and define m j to be the integer satisfying…”

mentioning

confidence: 99%

Convergence of Multilevel Stationary Gaussian Convolution

Hubbert

Levesley

2019

Lecture Notes in Computational Science and Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…Therefore, compared with other meshless techniques ( such as RBF interpolation ), this method can reduce the computation time and even approximate functions in high-dimensions. QI has been successfully applied to scattered data approximation and interpolation, numerical solution and quadrature of PDEs, see [14].…”

Section: Yong Duanmentioning

confidence: 99%

On the meshless quasi-interpolation methods for solving 2D sine-Gordon equations

Shanshan,

DUAN,

Bai

2024

Preprint

View full text Add to dashboard Cite

This paper is devoted to applying a numerical method for solution of the 2D sine-Gordon equation. The bivariate multiple quadratic quasi-interpolation ( MQQI ) method is adopted to simulate this equations, which the first order spatial derivative is approximated by MQQI, the second spatial and time derivative are approximated by forward difference. One of the merit of this scheme is its simple structure and easy implementation. In the meanwhile, we present truncation and total error of this scheme, high accuracy and efficiency of the method are verified by numerical experiments. In addition, the optimal value of parameters are investigated in this article based on Luh [10, 11].

show abstract

“…This approach is different from ours, that is based on identifying the fully symmetric sets a sparse grid is a union of, and specific to sparse grids. Other sparse grid based kernel methods appear in [17,22,15,59]. The construction of sparse grids that we present in this section is not the most general possible as we work in the fully symmetric framework.…”

Section: Sparse Gridsmentioning

confidence: 99%

Fully Symmetric Kernel Quadrature

Karvonen¹,

Särkkä²

2018

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Kernel quadratures and other kernel-based approximation methods typically suffer from prohibitive cubic time and quadratic space complexity in the number of function evaluations. The problem arises because a system of linear equations needs to be solved. In this article we show that the weights of a kernel quadrature rule can be computed efficiently and exactly for up to tens of millions of nodes if the kernel, integration domain, and measure are fully symmetric and the node set is a union of fully symmetric sets. This is based on the observations that in such a setting there are only as many distinct weights as there are fully symmetric sets and that these weights can be solved from a linear system of equations constructed out of row sums of certain submatrices of the full kernel matrix. We present several numerical examples that show feasibility, both for a large number of nodes and in high dimensions, of the developed fully symmetric kernel quadrature rules. Most prominent of the fully symmetric kernel quadrature rules we propose are those that use sparse grids.

show abstract

Multilevel quasi-interpolation on a sparse grid with the Gaussian

Cited by 11 publications

References 17 publications

Convergence of Multilevel Stationary Gaussian Convolution

Convergence of Multilevel Stationary Gaussian Convolution

On the meshless quasi-interpolation methods for solving 2D sine-Gordon equations

Fully Symmetric Kernel Quadrature

Contact Info

Product

Resources

About