Fivedimensional optimal lattice rules
Authors:
J. N. Lyness and Tor Sørevik
Journal:
Math. Comp. 75 (2006), 14671480
MSC (2000):
Primary 41A55, 41A63, 42A10; Secondary 65D32
Published electronically:
March 13, 2006
MathSciNet review:
2219038
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A major search program is described that has been used to determine a set of fivedimensional optimal lattice rules of enhanced trigonometric degrees up to 12. The program involved a distributed search, in which approximately 190 CPUyears were shared between more than 1,400 computers in many parts of the world.
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J. N. Lyness and R. Cools, Notes on a search for optimal lattice rules, in Cubature Formulae and Their Applications (M. V. Noskov, ed.), 259273, Krasnoyarsk STU, 2000. Also available as Argonne National Laboratory Preprint ANL/MCSP8290600.
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Sørevik, A search program for finding optimal integration
lattices, Computing 47 (1991), no. 2,
103–120 (English, with German summary). MR 1139431
(92k:65037), http://dx.doi.org/10.1007/BF02253429
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Sørevik, An algorithm for finding optimal integration
lattices of composite order, BIT 32 (1992),
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(93i:65038), http://dx.doi.org/10.1007/BF01994849
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Sørevik, Lattice rules by component
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(1993), no. 204, 799–820. MR 1185247
(94a:65011), http://dx.doi.org/10.1090/S00255718199311852476
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Sørevik, Fourdimensional lattice rules
generated by skewcirculant matrices, Math.
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(2004k:65041), http://dx.doi.org/10.1090/S0025571803015345
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 [SoMy01]
T. Sørevik and J. F. Myklebust, GRISK: An Internet based search for Koptimal lattice rules, in Proceedings of PARA2000, Lecture Notes in Computer Science 1947, 196205, Springer Verlag, 2001.
 [CoGo03]
 R. Cools and H. Govaert Five and sixdimensional lattice rules generated by structured matrices, Journal of Complexity 19 (2003), no. 6, 715729. MR 2039626 (2005a:65021)
 [CoLy01]
 R. Cools and J. N. Lyness Three and fourdimensional Koptimal lattice rules of moderate trigonometric degree, Math. Comp. 70 (2001), no. 236, 15491567. MR 1836918 (2002b:41026)
 [CoSl96]
 R. Cools and I. H. Sloan, Minimal cubature formulae of trigonometric degree, Math. Comp. 65 (1996), no. 216, 15831600. MR 1361806 (97a:65025)
 [Lyn89]
 J. N. Lyness, An introduction to lattice rules and their generator matrices, IMA J. Numer. Anal. 9 (1989), 405419. MR 1011399 (91b:65029)
 [Lyn03]
 J. N. Lyness, Notes on lattice rules, Journal of Complexity 19 (2003), no. 3, 321331. MR 1984117 (2004d:65035)
 [LyCo00]
 J. N. Lyness and R. Cools, Notes on a search for optimal lattice rules, in Cubature Formulae and Their Applications (M. V. Noskov, ed.), 259273, Krasnoyarsk STU, 2000. Also available as Argonne National Laboratory Preprint ANL/MCSP8290600.
 [LySo91]
 J. N. Lyness and T. Sørevik, A search program for finding optimal integration lattices, Computing 47 (1991), 103120. MR 1139431 (92k:65037)
 [LySo92]
 J. N. Lyness and T. Sørevik, An algorithm for finding optimal integration lattices of composite order, BIT 32 (1992), no. 4, 665675. MR 1191020 (93i:65038)
 [LySo93]
 J. N. Lyness and T. Sørevik, Lattice rules by component scaling, Math. Comp. 61 (1993), no. 204, 799820. MR 1185247 (94a:65011)
 [LySo04]
 J. N. Lyness and T. Sørevik, Fourdimensional lattice rules generated by skewcirculant matrices Math. Comp. 73 (2004), no. 245, 279295. MR 2034122 (2004k:65041)
 [Min11]
 H. Minkowski, Gesammelte Abhandlungen, reprint (originally published in 2 volumes, Leipzig, 1911), Chelsea Publishing Company, 1967.
 [Mol79]
 H. M. Möller, Lower bounds for the number of nodes in cubature formulae, in Numerische Integration (Tagung, Math. Forschungsinst., Oberwolfach 1978) Internat. Ser. Numer. Math. 45, Birhäuser, Basel 1979, 221230. MR 0561295 (81j:65053)
 [Mys88]
 I. P. Mysovskikh, Cubature formulas that are exact for trigonometric polynomials, Metody Vychisl. 15 (1988), 719 (Russian).MR 0967440 (90a:65050)
 [Nos91]
 M. V. Noskov, Cubature formulas for functions that are periodic with respect to some of the variables, Zh. Vychisl. Mat. Mat. Fiz. 31 (1991), no. 9, 14141418 (Russian). MR 1145212 (92i:65052)
 [SoMy01]
 T. Sørevik and J. F. Myklebust, GRISK: An Internet based search for Koptimal lattice rules, in Proceedings of PARA2000, Lecture Notes in Computer Science 1947, 196205, Springer Verlag, 2001.
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Additional Information
J. N. Lyness
Affiliation:
Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 604394844 and School of Mathematics, The University of New South Wales, Sydney 2052, Australia
Email:
lyness@mcs.anl.gov
Tor Sørevik
Affiliation:
Department of Mathematics, University of Bergen, N5020 Bergen, Norway
Email:
tor.sorevik@mi.uib.no
DOI:
http://dx.doi.org/10.1090/S002557180601845X
PII:
S 00255718(06)01845X
Keywords:
Multidimensional cubature,
optimal lattice rules,
$K$optimal rules,
and optimal trigonometric rules.
Received by editor(s):
September 22, 2004
Received by editor(s) in revised form:
April 25, 2005
Published electronically:
March 13, 2006
Additional Notes:
The first author’s work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contract W31109Eng38
Article copyright:
© Copyright 2006
American Mathematical Society
