David Munger

Random Numbers for Parallel Computers: Requirements and Methods, With Emphasis on GPUs

We examine the requirements and the available methods and software to provide (or imitate) uniform random numbers in parallel computing environments. In this context, for the great majority of applications, independent streams of random numbers are required, each being computed on a single processing element at a time. Sometimes, thousands or even millions of such streams are needed. We explain how they can be produced and managed. We devote particular attention to multiple streams for GPU devices.

On Figures of Merit for Randomly-Shifted Lattice Rules

Randomized quasi-Monte Carlo (RQMC) can be seen as a variance reduction method that provides an unbiased estimator of the integral of a function f over the s-dimensional unit hypercube, with smaller variance than standard Monte Carlo (MC) under certain conditions on f and on the RQMC point set. When f is smooth enough, the variance converges faster, asymptotically, as a function of the number of points n, than the MC rate of \(\mathcal{O}(1/n)\). The RQMC point sets are typically constructed to minimize a given parameterized measure of discrepancy between their empirical distribution and the uniform distribution. These parameters can give different weights to the different subsets of coordinates (or lower-dimensional projections) of the points, for example. The ideal parameter values (to minimize the variance) depend very much on the integrand f and their choice (or estimation) is far from obvious in practice. In this paper, we survey this question for randomly-shifted lattice rules, an important class of RQMC point sets, and we explore the practical issues that arise when we want to use the theory to construct lattices for applications. We discuss various ways of selecting figures of merit and for estimating their ideal parameters (including the weights), we examine how they can be implemented in practice, and we compare their performance on examples inspired from real-life problems. In particular, we look at how much improvement (variance reduction) can be obtained, on some examples, by constructing the points based on function-specific figures of merit compared with more traditional general-purpose lattice-rule constructions.

On the distribution of integration error by randomly-shifted lattice rules

A lattice rule with a randomly-shifted lattice estimates a math- ematical expectation, written as an integral over the s-dimensional unit hy- percube, by the average of n evaluations of the integrand, at the n points of the shifted lattice that lie inside the unit hypercube. This average provides an unbiased estimator of the integral and, under appropriate smoothness conditions on the integrand, it has been shown to converge faster as a func- tion of n than the average at n independent random points (the standard Monte Carlo estimator). In this paper, we study the behavior of the esti- mation error as a function of the random shift, as well as its distribution for a random shift, under various settings. While it is well known that the Monte Carlo estimator obeys a central limit theorem when n ! 1, the ran- domized lattice rule does not, due to the strong dependence between the function evaluations. We show that for the simple case of one-dimensional integrands, the limiting error distribution is uniform over a bounded in- terval if the integrand is non-periodic, and has a square root form over a bounded interval if the integrand is periodic. We find that in higher dimen- sions, there is little hope to precisely characterize the limiting distribution in a useful way for computing confidence intervals in the general case. We nevertheless examine how this error behaves as a function of the random shift from different perspectives and on various examples. We also point out a situation where a classical central-limit theorem holds when the dimen- sion goes to infinity, we provide guidelines on when the error distribution should not be too far from normal, and we examine how far from normal is the error distribution in examples inspired from real-life applications.

Algorithm 958: Lattice Builder: A General Software Tool for Constructing Rank-1 Lattice Rules

We introduce a new software tool and library named Lattice Builder, written in C++, that implements a variety of construction algorithms for good rank-1 lattice rules. It supports exhaustive and random searches, as well as component-by-component (CBC) and random CBC constructions, for any number of points, and for various measures of (non)uniformity of the points. The measures currently implemented are all shift-invariant and represent the worst-case integration error for certain classes of integrands. They include, for example, the weighted Pα square discrepancy, the Rα criterion, and figures of merit based on the spectral test, with projection-dependent weights. Each of these measures can be computed as a finite sum. For the Pα and Rα criteria, efficient specializations of the CBC algorithm are provided for projection-dependent, order-dependent, and product weights. For numbers of points that are integer powers of a prime base, the construction of embedded rank-1 lattice rules is supported through any of these algorithms, and through a fast CBC algorithm, with a variety of possibilities for the normalization of the merit values of individual embedded levels and for their combination into a single merit value. The library is extensible, thanks to the decomposition of the algorithms into decoupled components, which makes it easy to implement new types of weights, new search domains, new figures of merit, and so on.

A level set approach to simulate magnetohydrodynamic-instabilities in aluminum reduction cells

Magnetohydrodynamic instabilities at the metal-bath interface in aluminum reduction cells is an important and not fully understood topic. To simulate the two-fluid three-dimensional unstationary flow subject to a background magnetic field, a level set approach is proposed. It features a formulation in terms of the magnetic vector potential to avoid a numerical growth of the divergence of the magnetic field. The same exact projection scheme (with staggered grids) is used for both the velocity field and the magnetic vector potential. Test simulations show that the overall method behaves well in purely hydrodynamic as well as in fully magnetohydrodynamic regimes, in both cases with a single fluid and with two fluids. We also simulate with our technique the metal pad roll instability and trace the behavior of coupled interracial modes.

On the dynamics of 3-D single thermal plumes at various Prandtl numbers and Rayleigh numbers

Three-dimensional (3-D) numerical simulations of single turbulent thermal plumes in the Boussinesq approximation are used to understand more deeply the interaction of a plume with itself and its environment. In order to do so, we varied the Rayleigh and Prandtl numbers from Ra ∼ 105 to Ra ∼ 108 and from Pr ∼ 0.025 to Pr ∼ 70. We found that thermal dissipation takes place mostly on the border of the plume. Moreover, the rate of energy dissipation per unit mass ε T has a critical point around Pr ∼ 0.7. The reason is that at Pr greater than ∼0.7, buoyancy dominates inertia and thermal advection dominates wave formation whereas this trend is reversed at Pr less than ∼0.7. We also found that for large enough Prandtl number (Pr ∼ 70), the velocity field is mostly poloidal although this result was known for Rayleigh–Bénard convection (see Schmalzl et al. [On the validity of two-dimensional numerical approaches to time-dependent thermal convection. Europhys. Lett. 2004, 67, 390--396]). On the other hand, at small Prandtl numbers, the plume has a large helicity at large scale and a non-negligible toroidal part. Finally, as observed recently in details in weakly compressible turbulent thermal plume at Pr = 0.7 (see Plourde et al. [Direct numerical simulations of a rapidly expanding thermal plume: structure and entrainment interaction. J. Fluid Mech. 2008, 604, 99--123]), we also noticed a two-time cycle in which there is entrainment of some of the external fluid to the plume, this process being most pronounced at the base of the plume. We explain this as a consequence of calculated Richardson number being unity at Pr = 0.7 when buoyancy balance inertia.

Constructing adapted lattice rules using problem-dependent criteria

We describe a new software tool named Lattice Builder, designed to construct lattice point sets for quasi-Monte Carlo integration via randomly-shifted lattice rules. This tool permits one to search for good lattice parameters in terms of various uniformity criteria, for an arbitrary number of points and arbitrary dimension. It also constructs lattices that are extensible in the number of points and in the dimension. A numerical illustration is given.