Numerical performance of optimized Frolov lattices in tensor product reproducing kernel Sobolev spaces

2018/02/23 by Kacwin, Christopher, Oettershagen, Jens, Ullrich, Mario +1 · 1 citation
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1802.08666

Abstract

In this paper, we deal with several aspects of the universal Frolov cubature method, that is known to achieve optimal asymptotic convergence rates in a broad range of function spaces. Even though every admissible lattice has this favorable asymptotic behavior, there are significant differences concerning the precise numerical behavior of the worst-case error. To this end, we propose new generating polynomials that promise a significant reduction of the integration error compared to the classical polynomials. Moreover, we develop a new algorithm to enumerate the Frolov points from non-orthogonal lattices for numerical cubature in the d-dimensional unit cube [0,1]^d. Finally, we study Sobolev spaces with anisotropic mixed smoothness and compact support in [0,1]^d and derive explicit formulas for their reproducing kernels. This allows for the simulation of exact worst-case errors which numerically validate our theoretical results.

Cited by

• Discretization of Continuous Frames by Quasi-Monte Carlo Methods

Related

• On the orthogonality of the Chebyshev-Frolov lattice and applications
• On the Randomization of Frolov's Algorithm for Multivariate Integration
• Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square
• Enumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes
• Optimal Stencils in Sobolev Spaces