Extensions of Atanassov’s methods for Halton sequences

Henri Faure (1) , Christiane Lemieux (2) , Xiaoheng Wang (3)

1 IML - Institut de mathématiques de Luminy
2 University of Waterloo [Waterloo]
3 Harvard University

We extend Atanassov’s methods for Halton sequences in two different directions: (1) in the direction of Niederreiter (t, s)-sequences, (2) in the direction of generating matrices for Halton sequences. It is quite remarkable that Atanassov’s method for classical Halton sequences applies almost “word for word” to (t, s)-sequences and gives an upper bound quite comparable to those of Sobol’, Faure, and Niederreiter. But Atanassov also found a way to improve further his bound for classical Halton sequences by means of a clever scrambling producing sequences which he named modified Halton sequences. We generalize his method to nonsingular lower triangular matrices in the last part of this article.

Mots clés: Halton sequences

Domaines: Théorie des nombres [math.NT]

Henri Faure, Christiane Lemieux, Xiaoheng Wang. Extensions of Atanassov’s methods for Halton sequences. 9th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, University of Warsaw, Aug 2010, Warsaw, Poland. pp.345-362, ⟨10.1007/978-3-642-27440-4_17⟩. ⟨hal-01273620⟩