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Cubature formulas for symmetric measures in higher dimensions with few points
Author(s):
Aicke
Hinrichs;
Erich
Novak.
Journal:
Math. Comp.
76
(2007),
1357-1372.
MSC (2000):
Primary 65D32
Posted:
February 16, 2007
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Abstract:
We study cubature formulas for  -dimensional integrals with an arbitrary symmetric weight function of product form. We present a construction that yields a high polynomial exactness: for fixed degree  or  and large dimension  the number of knots is only slightly larger than the lower bound of Möller and much smaller compared to the known constructions. We also show, for any odd degree  , that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.
References:
- [Berens, Schmid and Xu(1995)]
- Berens, H., Schmid, H. J., and Xu, Y. (1995): Multivariate Gaussian cubature formulae. Arch. Math. 64, 26-32 MR 1305657 (95i:41054)
- [Bungartz and Griebel(2004)]
- Bungartz, H.-J., Griebel, M. (2004): Sparse grids. Acta Numerica 13, 147-269. MR 2249147
- [Capstick and Keister(1996)]
- Capstick, S., Keister, B. D. (1996): Multidimensional quadrature algorithms at higher degree and/or dimension. J. of Computational Physics 123, 267-273 MR 1372373
- [Cools(1997)]
- Cools, R. (1997):Constructing cubature formulas: the science behind the art. Acta Numerica 6, 1-54 MR 1489255 (99f:65038)
- [Cools(2003)]
- Cools (2003): An encyclopedia of cubature formulas. J. Complexity 19, 445-453 MR 1984127 (2004e:41031)
- [Cools and Haegemans(1994)]
- Cools, R. and Haegemans, A. (1994): An imbedded family of cubature formulae for
 -dimensional product regions. J. Comput. Appl. Math. 51, 251-262 MR 1290241 (95e:65020)
- [Genz(1986)]
- Genz, A. C. (1986): Fully symmetric interpolatory rules for multiple integrals. SIAM J. Numer. Anal. 23, 1273-1283 MR 865956 (88b:65032)
- [Genz and Keister(1996)]
- Genz, A. C., Keister, B. D. (1996): Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. J. Comput. Appl. Math. 71, 299-309 MR 1399898 (97b:65028)
- [Gerstner and Griebel(1998)]
- Gerstner, T., Griebel, M. (1998): Numerical integration using sparse grids. Numer. Algorithms 18, 209-232 MR 1669959 (99m:65042)
- [Kuperberg(2004)]
- Kuperberg, G. (2004): Numerical cubature using error-correcting codes. Preprint, arXiv:math.NA/0402047 MR 2231848
- [Lu and Darmofal(2004)]
- Lu, J., Darmofal, D. L. (2004): Higher-dimensional integration with Gaussian weight for applications in probabilistic design. SIAM J. Sci. Comput. 26, 613-624 MR 2116364 (2006a:65032)
- [Lyness(1965a)]
- Lyness, J. N. (1965a): Symmetric integration rules for hypercubes I-III. Math. Comp. 19, 260-276, 394-407, 625-637 MR 0201067 (34:952); MR 0201068 (34:953); MR 201069 (34:954)
- [Lyness(1965b)]
- Lyness J. N. (1965b): Limits on the number of function evaluations required by certain high-dimensional integration rules of hypercubic symmetry. Math. Comp. 19, 638-643 MR 0199961 (33:8101)
- [McNamee and Stenger(1967)]
- McNamee, J., Stenger, F. (1967): Construction of fully symmetric numerical integration formulas. Numer. Math. 10, 327-344 MR 0219241 (36:2324)
- [Möller(1979)]
- Möller, H. M. (1979): Lower bounds for the number of nodes in cubature formulae. In: Hämmerlin, G., ed., Numerische Integration, ISNM 45, pp. 221-230. Birkhäuser, Basel MR 561295 (81j:65053)
- [Mysovkikh(1968)]
- Mysovkikh, I. P. (1968): On the construction of cubature formulas with the smallest number of nodes. Soviet Math. Dokl. 9, 277-280. [Russian original: Dokl. Akad. Nauk SSSR 178, 1252-1254.] MR 0224284 (36:7328)
- [Mysovskikh(1981)]
- Mysovskikh, I. P. (1981): Interpolatory Cubature Formulas. Nauka, Moscow. [In Russian.] MR 656522 (83i:65025)
- [Novak and Ritter(1996)]
- Novak, E., Ritter, K. (1996): High dimensional integration of smooth functions over cubes. Numer. Math. 75, 79-97 MR 1417864 (97k:65057)
- [Novak and Ritter(1999)]
- Novak, E., Ritter, K. (1999): Simple cubature formulas with high polynomial exactness. Constr. Approx. 15, 499-522 MR 1702807 (2000k:65050)
- [Novak, Ritter, Schmitt and Steinbauer(1999)]
- Novak, E., Ritter, K., Schmitt, R., Steinbauer A. (1999): On a recent interpolatory method for high dimensional integration. J. Comput. Appl. Math. 112, 215-228 MR 1728461
- [Petras(2003)]
- Petras, K. (2003): Smolyak cubature of given polynomial degree with few nodes for increasing dimension. Numer. Math. 93, 729-753 MR 1961886 (2004c:65017)
- [Schmid(1983)]
- Schmid, H. J. (1983): Interpolatorische Kubaturformeln. Dissertationes Mathematicae, CCXX MR 735919 (85i:65034)
- [Smolyak(1963)]
- Smolyak, S. A. (1963): Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4, 240-243
- [Stroud(1971)]
- Stroud, A. H. (1971): Approximate calculation of multiple integrals. Prentice-Hall, Englewood Cliffs, NJ MR 0327006 (48:5348)
- [Victoir(2004)]
- Victoir, N. (2004): Asymmetric cubature formulae with few points in high dimension for symmetric measures. SIAM J. Numer. Anal. 42, 209-227 MR 2051063 (2005g:65044)
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Additional Information:
Aicke
Hinrichs
Affiliation:
Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany
Email:
hinrichs@math.uni-jena.de
Erich
Novak
Affiliation:
Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany
Email:
novak@math.uni-jena.de
DOI:
10.1090/S0025-5718-07-01974-6
PII:
S 0025-5718(07)01974-6
Keywords:
Cubature formulas,
M\"oller bound,
Smolyak method,
polynomial exactness
Received by editor(s):
August 25, 2005
Received by editor(s) in revised form:
June 16, 2006
Posted:
February 16, 2007
Additional Notes:
Research of the first author was supported by the DFG Emmy-Noether grant Hi 584/2-4.
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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