Fast Component-By-Component Construction of Rank-1 Lattice Rules for (Non-)Primes (Part II)

Abstract

Part I: (this part of the talk by Ronald Cools)
We restate our previous result which showed that it is possible to
construct the generating vector of a rank-1 lattice rule in a fast way,
i.e. O(s n log(n)), with s the number of dimensions and n the number of
points assumed to be prime.  Here we explicitly use basic facts from
algebra to exploit the structure of a matrix – which introduces the
crucial cost in the construction – to get a matrix-vector
multiplication in time O(n log(n)) instead of O(n^2).  We again stress
the fact that the algorithm works for any tensor product reproducing
kernel Hilbert space.

Part II: (this part of the talk by Dirk Nuyens)
In the second part we generalize the tricks used for primes to
non-primes, by basically falling back to algebraic group theory.  In
this way it can be shown that also for a non-prime number of points,
this crucial matrix-vector multiplication can be done in time O(n
log(n)). We conclude that the construction of rank-1 lattice rules in
an arbitrary r.k.h.s. for an arbitrary amount of points can be done in a
fast way of O(s n log(n)).