Wolfgang Ch. Schmid

MinT: A Database for Optimal Net Parameters

An overwhelming variety of different constructions for (t, m, s)-nets and (t, s)-sequences are known today. Propagation rules as well as connections to other mathematical objects make it a difficult task to determine the best net available in a given setting.

Techniques for parallel quasi-Monte Carlo integration with digital sequences and associated problems

Currently, in the context of quasi-Monte Carlo applications the most effective low-discrepancy sequences are digital (t, s)-sequences.

Calculation of the Quality Parameter of Digital Nets and Application to Their Construction

In quasi-Monte Carlo methods, point sets of low discrepancy are crucial for accurate results. A class of point sets with low theoretic upper bounds of discrepancy are the digital point sets known as digital (t, m, s)-nets which can be implemented very efficiently. The parameter t is indicative of the quality; i.e., small values of t lead to small upper bounds of the discrepancy. We introduce an effective way to establish this quality parameter t for digital nets constructed over arbitrary finite fields and give an application to the construction of digital nets of high quality.

Tractability properties of the weighted star discrepancy

Tractability properties of various notions of discrepancy have been intensively studied in the last decade. In this paper we consider the so-called weighted star discrepancy which was introduced by Sloan and Wozniakowski. We show that under a very mild condition on the weights one can obtain tractability with s-exponent zero (s is the dimension of the point set). In the case of product weights we give a condition such that the weighted star discrepancy is even strongly tractable. Furthermore, we give a lower bound for the weighted star discrepancy for a large class of weights. This bound shows that for such weights one cannot obtain strong tractability.

Projections of digital nets and sequences

Currently, in the context of quasi-Monte Carlo applications, the most effective low-discrepancy point sets and sequences are digital (t, m, s)-nets and (t, s)-sequences. In this survey we will consider ideas for the investigation of a new quality parameter reflecting projections of digital nets and sequences. Finally we will give several examples for commonly used digital point sets.

Parallel Quasi-Monte Carlo Integration Using (t, s)-Sequences

Currently, the most effective constructions of low-discrepancy point sets and sequences are based on the theory of (t, m, s)-nets and (t, s)-sequences. In this work we discuss parallelization techniques for quasi-Monte Carlo integration using (t, s)-sequences. We show that leapfrog parallelization may be very dangerous whereas block-based parallelization turns out to be robust.

Defects in parallel Monte Carlo and quasi-Monte Carlo integration using the leap-frog technique

Currently, the most efficient numerical techniques for evaluating high-dimensional integrals are based on Monte Carlo and quasi-Monte Carlo techniques. These tasks require a significant amount of computation and are therefore often executed on parallel computer systems. In order to keep the communication amount within a parallel system to a minimum, each processing element (PE) requires its own source of integration nodes. Therefore, techniques for using separately initialized and disjoint portions of a given point set on a single PE are classically employed. Using the so-called substreams may lead to dramatic errors in the results under certain circumstances. In this work, we compare the possible defects employing leaped quasi-Monte Carlo and Monte Carlo substreams. Apart from comparing the magnitude of the observed integration errors we give an overview under which circumstances (i.e. parallel programming models) such errors can occur.

MinT-Architecture and applications of the (t, m, s)-net and OOA database

Many different constructions for (t,m,s)-nets and (t,s)-sequences are known today. Propagation rules as well as connections to other mathematical objects make it difficult to determine the best net available in a given setting. The MinT database developed by the authors is one of the most elaborate solutions to this problem. In this article we discuss some aspects of the theory that makes MinT work. We also provide a synopsis of the strongest bounds and existence results known today as determined by MinT.

Particle approximation of convection-diffusion equations

We present a particle method for solving initial-value problems for convection–diffusion equations with constant diffusion coefficients. We sample N particles at locations xj(0) from the initial data. We discretize time into intervals of length Δt. We represent the solution at time tn=nΔt by N particles at locations xj(n). In each time interval the evolution of the system is obtained in three steps. In the first step the particles are transported under the action of the convective field. In the second step the particles are relabeled according to their position. In the third step the diffusive process is modeled by a random walk. We study the convergence of the scheme when quasi-random numbers are used. We compare several constructions of quasi-random point sets based on the theory of (t,s)-sequences. We show that an improvement in both magnitude of error and convergence rate can be achieved when quasi-random numbers are used in place of pseudo-random numbers.

A good permutation for one-dimensional diaphony

Abstract In this article we focus on two aspects of one-dimensional diaphony of generalised van der Corput sequences in arbitrary bases. First we give a permutation with the best distribution behaviour concerning the diaphony known so far. We improve a result of Chaix and Faure from 1993 from a value of 1.31574 . . . for a permutation in base 19 to 1.13794 . . . for our permutation in base 57. Moreover for an infinite sequence X and its symmetric version , we analyse the connection between the diaphony F(X, N) and the L 2-discrepancy using another result of Chaix and Faure. Therefore we state an idea how to get a lower bound for the diaphony of generalised van der Corput sequences in arbitrary base b.