Digital Nets and Sequences

Josef Dick is a lecturer in the School of Mathematics and Statistics at the University of New South Wales, Australia. Friedrich Pillichshammer is a Professor in the Institute for Financial Mathematics at the University of Linz, Austria.

"It will give readers the confidence that their estimates of variance are tractable, and they can therefore use quasi-Monte Carlo (QMC) integration to do the software engineering tradeoff analysis that is critical to professional software project management and architecture. This textbook--and believe me, it is a textbook--will lead students to a deep understanding of the potential errors that can be expected." Larry Bernstein, Computing Reviews

• Preface; Notation; 1. Introduction; 2. Quasi–Monte Carlo integration, discrepancy and reproducing kernel Hilbert spaces; 3. Geometric discrepancy; 4. Nets and sequences; 5. Discrepancy estimates and average type results; 6. Connections to other discrete objects; 7. Duality Theory; 8. Special constructions of digital nets and sequences; 9. Propagation rules for digital nets; 10. Polynomial lattice point sets; 11. Cyclic digital nets and hyperplane nets; 12. Multivariate integration in weighted Sobolev spaces; 13. Randomisation of digital nets; 14. The decay of the Walsh coefficients of smooth functions; 15. Arbitrarily high order of convergence of the worst-case error; 16. Explicit constructions of point sets with best possible order of L2-discrepancy; Appendix A. Walsh functions; Appendix B. Algebraic function fields; References; Index.