Friedrich Pillichshammer

Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces

We introduce a weighted reproducing kernel Hilbert space which is based on Walsh functions. The worst-case error for integration in this space is studied, especially with regard to (t, m, s)-nets. It is found that there exists a digital (t, m, s)-net, which achieves a strong tractability worst-case error bound under certain condition on the weights.We also investigate the worst-case error of integration in weighted Sobolev spaces. As the main tool we define a digital shift invariant kernel associated to the kernel of the weighted Sobolev space. This allows us to study the mean square worst-case error of randomly digitally shifted digital (t, m, s)- nets. As this digital shift invariant kernel is almost the same as the kernel for the Hilbert space based on Walsh functions, we can derive results for the weighted Sobolev space based on the analysis of the Walsh function space. We show that there exists a (t, m, s)-net which achieves the best possible convergence order for integration in weighted Sobolev spaces and are strongly tractable under the same condition on the weights as for lattice rules.

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights. We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules. We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

Discrepancy Theory and Quasi-Monte Carlo Integration

In this chapter we show the deep connections between discrepancy theory on the one hand and quasi-Monte Carlo integration on the other. Discrepancy theory was established as an area of research going back to the seminal paper by Weyl [117], whereas Monte Carlo (and later quasi-Monte Carlo) was invented in the 1940s by John von Neumann and Stanislaw Ulam to solve practical problems. The connection between these areas is well understood and will be presented here. We further include state of the art methods for quasi-Monte Carlo integration.

The construction of good extensible rank-1 lattices

It has been shown by Hickernell and Niederreiter that there exist generating vectors for integration lattices which yield small integration errors for n = p, p 2 ,... for all integers p ≥ 2. This paper provides algorithms for the construction of generating vectors which are finitely extensible for n = p, p 2 ,... for all integers p ≥ 2. The proofs which show that our algorithms yield good extensible rank-1 lattices are based on a sieve principle. Particularly fast algorithms are obtained by using the fast component-by-component construction of Nuyens and Cools. Analogous results are presented for generating vectors with small weighted star discrepancy.

Introduction to Quasi-Monte Carlo Integration and Applications

Preface.- Notation.- 1 Introduction.- 2 Uniform Distribution Modulo One.- 3 QMC Integration in Reproducing Kernel Hilbert Spaces.- 4 Lattice Point Sets.- 5 (t, m, s)-nets and (t, s)-Sequences.- 6 A Short Discussion of the Discrepancy Bounds.- 7 Foundations of Financial Mathematics.- 8 Monte Carlo and Quasi-Monte Carlo Simulation.- Bibliography.- Index.

Lattice rules for nonperiodic smooth integrands

The aim of this paper is to show that one can achieve convergence rates of

Distribution Properties of Generalized van der Corput-Halton Sequences and their Subsequences

N^{-\alpha + \delta }

Multivariate integration of infinitely many times differentiable functions in weighted Korobov spaces

N−α+δ for