Friedrich Pillichshammer

This textbook offers a comprehensive introduction to quasi-Monte Carlo methods and several of their applications. Throughout, the authors use modern concepts and notations to provide an overview of how the theory behind quasi-Monte Carlo methods developed. While the main focus of this text is on the theory, it also explores several applications with a particular emphasis on financial problems. This second edition contains substantial revisions and additions, including several new sections that more thoroughly cover weighted problems. New sections include coverage of the weighted Koksma-Hlawka inequality, weighted discrepancy of lattice point sets and tractability properties, polynomial lattice point sets, and more. In addition, the authors have corrected minor errors from the first edition and updated the bibliography and "Further reading" sections. Introduction to Quasi-Monte Carlo Integration and Applications is suitable for advanced undergraduate students in mathematics and computer science. Readers should possess a basic knowledge of algebra, calculus, linear algebra, and probability theory. It may also be used for self-study or as a reference for researchers interested in the area.

Lattice rules are a powerful and popular form of quasi-Monte Carlo rules based on multidimensional integration lattices. This book provides a comprehensive treatment of the subject with detailed explanations of the basic concepts and the current methods used in research. This comprises, for example, error analysis in reproducing kernel Hilbert spaces, fast component-by-component constructions, the curse of dimensionality and tractability, weighted integration and approximation problems, and applications of lattice rules.

Lattice rules are a powerful and popular form of quasi-Monte Carlo rules based on multidimensional integration lattices. This book provides a comprehensive treatment of the subject with detailed explanations of the basic concepts and the current methods used in research. This comprises, for example, error analysis in reproducing kernel Hilbert spaces, fast component-by-component constructions, the curse of dimensionality and tractability, weighted integration and approximation problems, and applications of lattice rules.

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.