Dick J. Lattice Rules. Numerical Integration, Approximation,...Discrepancy 2022

Textbook in PDF format

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.
Preface
List of Symbols
Introduction
Integration of Smooth Periodic Functions
Constructions of Lattice Rules
Modified Construction Schemes
Discrepancy of Lattice Point Sets
Extensible Lattice Point Sets
Lattice Rules for Nonperiodic Integrands
Integration With Respect to Probability Measures
Integration of Analytic Functions
Korobov’s p-Sets
Lattice Rules in the Randomized Setting
Stability of Lattice Rules
L2-Approximation Using Lattice Rules
L∞-Approximation Using Lattice Rules
Multiple Rank-1 Lattice Point Sets
Fast QMC Matrix-Vector Multiplication
Appendix A Partial Differential Equations With Random Coefficients
Appendix B Numerical Experiments for Lattice Rule Construction Algorithms
References
Index