Adaptive Quasi-Monte Carlo Methods for Cubature

Fred J. Hickernell (), Lluís Antoni Jiménez Rugama () and Da Li ()
Additional contact information
Fred J. Hickernell: Illinois Institute of Technology
Lluís Antoni Jiménez Rugama: Illinois Institute of Technology
Da Li: Illinois Institute of Technology

A chapter in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, 2018, pp 597-619 from Springer

Abstract: Abstract High dimensional integrals can be approximated well by quasi-Monte Carlo methods. However, determining the number of function values needed to obtain the desired accuracy is difficult without some upper bound on an appropriate semi-norm of the integrand. This challenge has motivated our recent development of theoretically justified, adaptive cubatures based on digital sequences and lattice nodeset sequences. Our adaptive cubatures are based on error bounds that depend on the discrete Fourier transforms of the integrands. These cubatures are guaranteed for integrands belonging to cones of functions whose true Fourier coefficients decay steadily, a notion that is made mathematically precise. Here we describe these new cubature rules and extend them in two directions. First, we generalize the error criterion to allow both absolute and relative error tolerances. We also demonstrate how to estimate a function of several integrals to a given tolerance. This situation arises in the computation of Sobol’ indices. Second, we describe how to use control variates in adaptive quasi-Monte cubature while appropriately estimating the control variate coefficient.

Date: 2018
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-72456-0_27

Ordering information: This item can be ordered from
http://www.springer.com/9783319724560

DOI: 10.1007/978-3-319-72456-0_27

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().