Optimal Point Sets for Quasi-Monte Carlo Integration of Bivariate Periodic Functions with Bounded Mixed Derivatives

Abstract

We investigate quasi-Monte Carlo (QMC) integration of bivariate periodic functions with dominating mixed smoothness of order one. While there exist several QMC constructions which asymptotically yield the optimal rate of convergence ofit is yet unknown which point set is optimal in the sense that it is a global minimizer of the worst case integration error. We will present a computerassisted proof by exhaustion that the Fibonacci lattice is the unique minimizer of the QMC worst case error in periodic H 1 mix for small Fibonacci numbers N . Moreover, we investigate the situation for point sets whose cardinality N is not a Fibonacci number. It turns out that for N = 1, 2, 3, 5, 7, 8, 12, 13 the optimal point sets are integration lattices.

The Role of Frolov's Cubature Formula for Functions with Bounded Mixed Derivative

“…This is supported by the fact that for d = 2 the corresponding lower bound was proven for the Fibonacci cubature formula, see [47,Theorem 2.5], which is conjectured to be optimal, cf. [21].…”

“…The by now classical research topic of numerically integrating d-variate functions with mixed smoothness properties goes back to the work of Korobov [25], Hlawka [22], and Bakhvalov [2] in the 1960s and was continued later by numerous authors including Frolov [12], Temlyakov [43,45,46,49], Dubinin [7,8], Skriganov [39], Triebel [53], Hinrichs et al [16,17,21], Hinrichs, Novak, M. Ullrich, Woźniakowski [18,19], Hinrichs, Novak, M. Ullrich [20], Dũng, T. Ullrich [9], T. Ullrich [56], Dick and Pillichshammer [6], and Markhasin [28,29,30] to mention just a few. In contrast to the quadrature of univariate functions, where equidistant point grids lead to optimal formulas, the multivariate problem is much more involved.…”

Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions

“…This can also be observed in Figures 2 and 4 where the Fibonacci lattice yields the same (optimal) rate of convergence as the order 2 digital nets, although it seems to have a significantly smaller constant. For small Fibonacci numbers, it is even known that the Fibonacci lattice is the globally optimal point set [26]. In summary, we can see that the order 2 net, the Fibonacci lattice and the sparse grid are able to benefit from the higher order smoothness, while the Halton sequence does not improve over N −1 (log N ) (d−1)/2 .…”

6. Fibonacci lattices have minimal dispersion on the two-dimensional torus

“…For the classical L 2 -discrepancy this was first proved by V. Sós and S. K. Zaremba in [9]. For the periodic L 2 -discrepancy it is even conjectured that the Fibonacci lattice is globally optimal among all point sets with the same number of points, see [3]. This is proved in [3] for n = F m ≤ 13.…”