Lattice Rules: Numerical Integration, Approximation, and Discrepancy (Springer Series in Computational Mathematics, 58)

Preface
Contents
List of Symbols
Chapter 1 Introduction
	1.1 Monte Carlo and Quasi-Monte Carlo Integration
	1.2 Lattice Rules
	1.3 The Structure of Lattice Rules
	1.4 Lattice Rules for Numerical Integration—the Classical Theory
	1.5 QMC Integration in Reproducing Kernel Hilbert Spaces
	1.6 Discrepancy and Koksma–Hlawka Type Inequalities
	1.7 The Curse of Dimensionality
	1.8 Further Quality Criteria for Lattice Rules
	Notes and Remarks
Chapter 2 Integration of Smooth Periodic Functions
	2.1 Korobov Spaces
	2.2 Integration in Korobov Spaces
	2.3 Error Bounds for the Unweighted Case
	2.4 Weighted Korobov Spaces
	2.5 Integration in Weighted Korobov Spaces
	2.6 Tractability
	Notes and Remarks
Chapter 3 Constructions of Lattice Rules
	3.1 Exhaustive Search for Generating Vectors
	3.2 Korobov Type Generating Vectors
	3.3 Component-By-Component Constructions
	3.4 The Fast CBC Construction for Product Weights
	3.5 The Fast CBC Construction for POD Weights
	3.6 A CBC Algorithm Based on the Quality Criterion R
	Notes and Remarks
Chapter 4 Modified Construction Schemes
	4.1 The Reduced CBC Construction
	4.2 The Reduced Fast CBC Construction for Product and POD Weights
	4.3 The Successive Coordinate Search Construction
	4.4 The Reduced Fast SCS Construction
	4.5 Projection-Corrected Constructions
	4.6 The Component-By-Component Digit-By-Digit Construction
	Notes and Remarks
Chapter 5 Discrepancy of Lattice Point Sets
	5.1 Extreme Discrepancy
	5.2 CBC Construction of Low Discrepancy Lattice Point Sets
	5.3 Weighted Star-Discrepancy
	5.4 Tractability of the Weighted Star-Discrepancy
	5.5 Korobov Type Lattice Point Sets With Low Weighted Star-Discrepancy
	5.6 Isotropic Discrepancy and Lattice Point Sets on the Sphere
	Notes and Remarks
Chapter 6 Extensible Lattice Point Sets
	6.1 The Definition of Extensible Lattice Point Sets
	6.2 Existence of Extensible Lattice Point Sets With Good Properties
	6.3 Constructions of Extensible Lattice Rules—Embedded Lattice Rules
	6.4 A Sieve Principle for Constructing Embedded Lattice Rules
	6.5 The CBC Sieve Algorithm
	6.6 The Fast CBC Sieve Algorithm
	6.7 A Digit-By-Digit Construction
	Notes and Remarks
Chapter 7 Lattice Rules for Nonperiodic Integrands
	7.1 Shifted Lattice Rules and Integration in Weighted Sobolev Spaces
	7.2 Sobolev Spaces of Higher Smoothness and Cosine Spaces
	7.3 Folded Lattice Rules
	7.4 Symmetrized Lattice Rules
	Notes and Remarks
Chapter 8 Integration With Respect to Probability Measures
	8.1 Transforming the Points Versus Transforming the Integrand
	8.2 Function Space Setting
	8.3 Unanchored Spaces
	8.4 The Shift-Invariant Kernel
	8.5 Integration Error
	Notes and Remarks
Chapter 9 Integration of Analytic Functions
	9.1 General Korobov Spaces and Korobov Spaces of Analytic Functions
	9.2 Integration in Korobov Spaces of Analytic Functions
	9.3 Exponential Tractability
	Notes and Remarks
Chapter 10 Korobov’s p-Sets
	10.1 The Construction of Korobov’s p-Sets
	10.2 The Weighted Star-Discrepancy of the p-Sets
	10.3 Integration of Hölder Continuous Fourier Series
	Notes and Remarks
Chapter 11 Lattice Rules in the Randomized Setting
	11.1 The Randomized Algorithm for Korobov Spaces
	11.2 Randomized Folded Lattice Rules
	11.3 A Brief Discussion of Tractability
	Notes and Remarks
Chapter 12 Stability of Lattice Rules
	12.1 A Stability Result
	12.2 The CBC Algorithm With Respect to More Than One Criterion
	12.3 Random Weights
	Notes and Remarks
Chapter 13 L2-Approximation Using Lattice Rules
	13.1 L2-Approximation of Functions in Korobov Spaces
	13.2 Lower Error Bounds for L2-Approximation in Korobov Spaces Using Lattice-Based Algorithms
	13.3 Tractability of L2-Approximation Using Lattice Rules
	13.4 Adaptions for General Weights
	Notes and Remarks
Chapter 14 L∞-Approximation Using Lattice Rules
	14.1 L∞-Approximation of Functions in Korobov Spaces
	14.2 L∞-Approximation of Functions in Korobov Spaces Using Splines
	14.3 Tractability of L∞-Approximation Using Lattice Rules and Splines
	Notes and Remarks
Chapter 15 Multiple Rank-1 Lattice Point Sets
	15.1 Multiple Rank-1 Lattice Point Sets for Approximation in Korobov Spaces
	15.2 Error Analysis
	15.3 Comparison to Previous Results and Tractability
	Notes and Remarks
Chapter 16 Fast QMC Matrix-Vector Multiplication
	16.1 The General Idea
	16.2 Fast QMC Matrix-Vector Multiplication for Lattice Point Sets
	16.3 Fast QMC Matrix-Vector Multiplication for a Special Case of Korobov’s p-Sets
	16.4 Applications
	16.5 Numerical Experiments
Appendix A Partial Differential Equations With Random Coefficients
	A.1 Uniform Random Coefficients
	A.2 Log-Normal Random Coefficients
Appendix B Numerical Experiments for Lattice Rule Construction Algorithms
	B.1 Numerical Results for the CBC Construction
	B.2 Numerical Results for Alternative Constructions
References
Index