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.

• 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.

"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