35 research outputs found
Hot new directions for quasi-Monte Carlo research in step with applications
This article provides an overview of some interfaces between the theory of
quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC
theoretical settings: first order QMC methods in the unit cube and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
under appropriate conditions on the function spaces. Another important feature
is that good parameters for these QMC methods can be obtained by fast efficient
algorithms even when is large. We outline three different applications and
explain how they can tap into the different QMC theory. We also discuss three
cost saving strategies that can be combined with QMC in these applications.
Many of these recent QMC theory and methods are developed not in isolation, but
in close connection with applications
A Tool for Custom Construction of QMC and RQMC Point Sets
We present LatNet Builder, a software tool to find good parameters for lattice rules, polynomial lattice rules, and digital nets in base 2, for quasi-Monte Carlo (QMC) and randomized quasi-Monte Carlo (RQMC) sampling over the s-dimensional unit hypercube. The selection criteria are figures of merit that give different weights to different subsets of coordinates. They are upper bounds on the worst-case error (for QMC) or variance (for RQMC) for integrands rescaled to have a norm of at most one in certain Hilbert spaces of functions. We summarize what are the various Hilbert spaces, discrepancies, types of weights, figures of merit, types of constructions, and search methods supported by LatNet Builder. We briefly discuss its organization and we provide simple illustrations of what it can do.NSERC Discovery Grant, IVADO Grant, Corps des Mines Stipend, ERDF, ESF, EXP. 2019/0043
On a projection-corrected component-by-component construction
The component-by-component construction is the standard method of finding
good lattice rules or polynomial lattice rules for numerical integration.
Several authors have reported that in numerical experiments the generating
vector sometimes has repeated components. We study a variation of the classical
component-by-component algorithm for the construction of lattice or polynomial
lattice point sets where the components are forced to differ from each other.
This avoids the problem of having projections where all quadrature points lie
on the main diagonal. Since the previous results on the worst-case error do not
apply to this modified algorithm, we prove such an error bound here. We also
discuss further restrictions on the choice of components in the
component-by-component algorithm