Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Efficient algorithms for computing the L2-discrepancy
Mathematics of Computation
Computing discrepancies of Smolyak quadrature rules
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
Cubature formulas, discrepancy, and nonlinear approximation
Journal of Complexity
On obtaining higher order convergence for smooth periodic functions
Journal of Complexity
On lower bounds for the L2-discrepancy
Journal of Complexity
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We study the Fibonacci sets from the point of view of their quality with respect to discrepancy and numerical integration. Let {b"n}"n"="0^~ be the sequence of Fibonacci numbers. The b"n-point Fibonacci set F"n@?[0,1]^2 is defined as F"n:={(@m/b"n,{@mb"n"-"1/b"n})}"@m"="1^b^"^n, where {x} is the fractional part of a number x@?R. It is known that cubature formulas based on the Fibonacci set F"n give optimal rate of error of numerical integration for certain classes of functions with mixed smoothness. We give a Fourier analytic proof of the fact that the symmetrized Fibonacci set F"n^'=F"n@?{(p"1,1-p"2):(p"1,p"2)@?F"n} has asymptotically minimal L"2 discrepancy. This approach also yields an exact formula for this quantity, which allows us to evaluate the constant in the discrepancy estimates. Numerical computations indicate that these sets have the smallest currently known L"2 discrepancy among two-dimensional point sets. We also introduce quarteredL"p discrepancy, which is a modification of the L"p discrepancy symmetrized with respect to the center of the unit square. We prove that the Fibonacci set F"n has minimal in the sense of order quartered L"p discrepancy for all p@?(1,~). This in turn implies that certain two-fold symmetrizations of the Fibonacci set F"n are optimal with respect to the standard L"p discrepancy.