Digital nets and sequences : discrepancy and quasi-Monte Carlo integration / Josef Dick, Friedrich Pillichshammer.

Format:

Book

Author/Creator:

Dick, J. (Josef), author.

Pillichshammer, Friedrich, author.

Language:

English

Subjects (All):

Monte Carlo method.

Nets (Mathematics).

Sequences (Mathematics).

Numerical integration.

Digital filters (Mathematics).

Physical Description:

1 online resource (xvii, 600 pages) : digital, PDF file(s).

Other Title:

Digital Nets & Sequences

Place of Publication:

Cambridge : Cambridge University Press, 2010.

Language Note:

English

Summary:

Indispensable for students, invaluable for researchers, this comprehensive treatment of contemporary quasi-Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch with detailed explanations of the basic concepts and then advances to current methods used in research. As deterministic versions of the Monte Carlo method, quasi-Monte Carlo rules have increased in popularity, with many fruitful applications in mathematical practice. These rules require nodes with good uniform distribution properties, and digital nets and sequences in the sense of Niederreiter are known to be excellent candidates. Besides the classical theory, the book contains chapters on reproducing kernel Hilbert spaces and weighted integration, duality theory for digital nets, polynomial lattice rules, the newest constructions by Niederreiter and Xing and many more. The authors present an accessible introduction to the subject based mainly on material taught in undergraduate courses with numerous examples, exercises and illustrations.

Contents:

Machine generated contents note: 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.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

ISBN:

1-107-20419-4

0-511-76118-X

1-282-77141-8

9786612771415

0-511-90118-6

0-511-79882-2

0-511-90197-6

0-511-79742-7

0-511-90039-2

OCLC:

663861925