This is the first book devoted to lattice methods, a recently developed way of calculating multiple integrals in many variables. Multiple integrals of this kind arise in fields such as quantum physics and chemistry, statistical mechanics, Bayesian statistics and many others. Lattice methods are an effective tool when the number of integrals are large.
The book begins with a review of existing methods before presenting lattice theory in a thorough, self-contained manner, with numerous illustrations and examples. Group and number theory are included, but the treatment is such that no prior knowledge is needed.
Not only the theory but the practical implementation of lattice methods is covered. An algorithm is presented alongside tables not available elsewhere, which together allow the practical evaluation of multiple integrals in many variables. Most importantly, the algorithm produces an error estimate in a very efficient manner. The book also provides a fast track for readers wanting to move rapidly to using lattice methods in practical calculations. It concludes with extensive numerical tests which compare lattice methods with other methods, such as the Monte Carlo.
`This nice book gives a thorough coverage of the numerical integration of high dimensional functions by means of lattice rules.....................I found the book very readable, and busy readers are well advised by the authors how to spend best their time with the book'
Monatshefte fur Mathematik Vol. 124 1997
1: Lattice rules
2: Lattice rules as multiple sums
3: Rank-1 rules - the method of good lattice points
4: Lattice rules of higher rank - a first look
5: Maximal rank lattice rules
6: Intermediate rank lattice rules
7: Lattice rules for nonperiodic integrands
8: Lattice rules - other topics
9: Practical implementation of lattice rules
10: Comparisons with other methods