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Journal articles on the topic 'Cubature formulae'

Author: Grafiati

Published: 4 June 2021

Last updated: 28 January 2023

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1

Cools, Ronald. "Constructing cubature formulae: the science behind the art." Acta Numerica 6 (January 1997): 1–54. http://dx.doi.org/10.1017/s0962492900002701.

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In this paper we present a general, theoretical foundation for the construction of cubature formulae to approximate multivariate integrals. The focus is on cubature formulae that are exact for certain vector spaces of polynomials. Our main quality criteria are the algebraic and trigonometric degrees. The constructions using ideal theory and invariant theory are outlined. The known lower bounds for the number of points are surveyed and characterizations of minimal cubature formulae are given. We include references to all known minimal cubature formulae. Finally, some methods to construct cubature formulae illustrate the previously introduced concepts and theorems.

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2

Gushev, Vesselin, and Geno Nikolov. "Modified product cubature formulae." Journal of Computational and Applied Mathematics 224, no. 2 (February 2009): 465–75. http://dx.doi.org/10.1016/j.cam.2008.05.031.

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3

Berens, H., H. J. Schmid, and Y. Xu. "Multivariate Gaussian cubature formulae." Archiv der Mathematik 64, no. 1 (January 1995): 26–32. http://dx.doi.org/10.1007/bf01193547.

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4

Teichmann, Josef. "Calculating the Greeks by cubature formulae." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2066 (December 14, 2005): 647–70. http://dx.doi.org/10.1098/rspa.2005.1583.

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We provide cubature formulae for the calculation of derivatives of expected values in the spirit of Terry Lyons and Nicolas Victoir. In financial mathematics derivatives of option prices with respect to initial values, so called Greeks, are of particular importance as hedging parameters. The proof of existence of cubature formulae for Greeks is based on universal formulae, which lead to the calculation of Greeks in an asymptotic sense—even without Hörmander's condition. Cubature formulae then allow to calculate these quantities very quickly. Simple examples are added to the theoretical exposition.

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5

Schmid, H. J., and Yuan Xu. "On bivariate Gaussian cubature formulae." Proceedings of the American Mathematical Society 122, no. 3 (March 1, 1994): 833. http://dx.doi.org/10.1090/s0002-9939-1994-1209428-0.

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6

Gismalla, D. A. "Schmid's approach on cubature formulae." International Journal of Computer Mathematics 32, no. 1-2 (January 1990): 75–83. http://dx.doi.org/10.1080/00207169008803816.

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7

Dubinin, V. V. "Cubature formulae for Besov classes." Izvestiya: Mathematics 61, no. 2 (April 30, 1997): 259–83. http://dx.doi.org/10.1070/im1997v061n02abeh000114.

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8

Cools, R., I. P. Mysovskikh, and H. J. Schmid. "Cubature formulae and orthogonal polynomials." Journal of Computational and Applied Mathematics 127, no. 1-2 (January 2001): 121–52. http://dx.doi.org/10.1016/s0377-0427(00)00495-7.

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9

Wang, Xiaoqun. "On generalized invariant cubature formulae." Journal of Computational and Applied Mathematics 130, no. 1-2 (May 2001): 271–81. http://dx.doi.org/10.1016/s0377-0427(99)00377-5.

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10

Degani, Ilan, Jeremy Schiff, and David J. Tannor. "Commuting extensions and cubature formulae." Numerische Mathematik 101, no. 3 (July 18, 2005): 479–500. http://dx.doi.org/10.1007/s00211-005-0628-z.

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11

Xu, Yuan. "Cubature Formulae and Polynomial Ideals." Advances in Applied Mathematics 23, no. 3 (October 1999): 211–33. http://dx.doi.org/10.1006/aama.1999.0652.

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12

Jandrlic, Davorka, Miodrag Spalevic, and Jelena Tomanovic. "Error estimates for certain cubature formulae." Filomat 32, no. 20 (2018): 6893–902. http://dx.doi.org/10.2298/fil1820893j.

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We estimate the errors of selected cubature formulae constructed by the product of Gauss quadrature rules. The cases of multiple and (hyper-)surface integrals over n-dimensional cube, simplex, sphere and ball are considered. The error estimates are obtained as the absolute value of the difference between cubature formula constructed by the product of Gauss quadrature rules and cubature formula constructed by the product of corresponding Gauss-Kronrod or corresponding generalized averaged Gaussian quadrature rules. Generalized averaged Gaussian quadrature rule ?2l+1 is (2l + 1)-point quadrature formula. It has 2l + 1 nodes and the nodes of the corresponding Gauss rule Gl with l nodes form a subset, similar to the situation for the (2l + 1)-point Gauss-Kronrod rule H2l+1 associated with Gl. The advantages of bG2l+1 are that it exists also when H2l+1 does not, and that the numerical construction of ?2l+1, based on recently proposed effective numerical procedure, is simpler than the construction of H2l+1.

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13

Cools, Ronald, and Ian H. Sloan. "Minimal cubature formulae of trigonometric degree." Mathematics of Computation 65, no. 216 (October 1, 1996): 1583–601. http://dx.doi.org/10.1090/s0025-5718-96-00767-3.

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14

Guessab, A. "Numerical cubature formulae with preassigned knots." Numerische Mathematik 52, no. 4 (July 1988): 467–78. http://dx.doi.org/10.1007/bf01462240.

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15

Nozaki, Hiroshi, and Masanori Sawa. "Note on Cubature Formulae and Designs Obtained from Group Orbits." Canadian Journal of Mathematics 64, no. 6 (December 1, 2012): 1359–77. http://dx.doi.org/10.4153/cjm-2011-069-5.

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Abstract In 1960, Sobolev proved that for a finite reflection group G, a G-invariant cubature formula is of degree t if and only if it is exact for all G-invariant polynomials of degree at most t . In this paper, we make some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and, moreover, gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007), which classifies tight Euclidean designs invariant under the Weyl group of type B, to other finite reflection groups.

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16

Bojanov, Borislav D., and Dimitar K. Dimitrov. "Gaussian extended cubature formulae for polyharmonic functions." Mathematics of Computation 70, no. 234 (February 23, 2000): 671–84. http://dx.doi.org/10.1090/s0025-5718-00-01206-0.

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17

MAEZTU, J. I. "On Symmetric Cubature Formulae for Planar Regions." IMA Journal of Numerical Analysis 9, no. 2 (1989): 167–83. http://dx.doi.org/10.1093/imanum/9.2.167.

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18

Luo, Zhongxuan, Zhaoliang Meng, and Fengshan Liu. "Construction of cubature formulae with preassigned nodes§." Applicable Analysis 87, no. 2 (February 2008): 233–45. http://dx.doi.org/10.1080/00036810701272254.

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19

Heo, Sangwoo, and Yuan Xu. "Constructing cubature formulae for spheres and balls." Journal of Computational and Applied Mathematics 112, no. 1-2 (November 1999): 95–119. http://dx.doi.org/10.1016/s0377-0427(99)00216-2.

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20

Petrova, Guergana. "Cubature formulae for spheres, simplices and balls." Journal of Computational and Applied Mathematics 162, no. 2 (January 2004): 483–96. http://dx.doi.org/10.1016/j.cam.2003.08.036.

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21

Ben Salem, Néjib, and Kamel Touahri. "Cubature Formulae Associated with the Dunkl Laplacian." Results in Mathematics 58, no. 1-2 (April 14, 2010): 119–44. http://dx.doi.org/10.1007/s00025-010-0035-3.

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22

Dryanov, Dimiter, and Petar Petrov. "On Trigonometric Blending Interpolation and Cubature Formulae." Results in Mathematics 62, no. 3-4 (August 18, 2012): 249–64. http://dx.doi.org/10.1007/s00025-012-0280-8.

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23

Benouahmane, Brahim, Cuyt Annie, and Irem Yaman. "Near-minimal cubature formulae on the disk." IMA Journal of Numerical Analysis 39, no. 1 (December 8, 2017): 297–314. http://dx.doi.org/10.1093/imanum/drx069.

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24

Verlinden, P., and A. Haegemans. "The construction of cubature formulae by continuation." Computing 45, no. 2 (June 1990): 145–55. http://dx.doi.org/10.1007/bf02247880.

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25

Bailey, S. B., and And Cohen. "The Construction Of Some Optimal 2D Cubature Formulae." International Journal of Computer Mathematics 79, no. 11 (January 2002): 1233–42. http://dx.doi.org/10.1080/00207160213941.

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26

Heo, Sangwoo, and Yuan Xu. "Constructing fully symmetric cubature formulae for the sphere." Mathematics of Computation 70, no. 233 (March 3, 2000): 269–80. http://dx.doi.org/10.1090/s0025-5718-00-01198-4.

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27

Bojanov, Borislav, and Guergana Petrova. "On minimal cubature formulae for product weight functions." Journal of Computational and Applied Mathematics 85, no. 1 (November 1997): 113–21. http://dx.doi.org/10.1016/s0377-0427(97)00133-7.

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28

De Marchi, Stefano, Marco Vianello, and Yuan Xu. "New cubature formulae and hyperinterpolation in three variables." BIT Numerical Mathematics 49, no. 1 (January 30, 2009): 55–73. http://dx.doi.org/10.1007/s10543-009-0210-7.

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29

Guessab, Allal, and Gerhard Schmeisser. "Negative Definite Cubature Formulae, Extremality and Delaunay Triangulation." Constructive Approximation 31, no. 1 (April 3, 2009): 95–113. http://dx.doi.org/10.1007/s00365-009-9049-z.

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30

Beckers, Marc, and Ann Haegemans. "The construction of three-dimensional invariant cubature formulae." Journal of Computational and Applied Mathematics 35, no. 1-3 (June 1991): 109–18. http://dx.doi.org/10.1016/0377-0427(91)90200-4.

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31

Pecher, Radek. "Efficient cubature formulae for MLPG and related methods." International Journal for Numerical Methods in Engineering 65, no. 4 (2005): 566–93. http://dx.doi.org/10.1002/nme.1458.

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32

Migliorati, Giovanni, and Fabio Nobile. "Stable high-order randomized cubature formulae in arbitrary dimension." Journal of Approximation Theory 275 (March 2022): 105706. http://dx.doi.org/10.1016/j.jat.2022.105706.

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33

Cools, Ronald, and Ann Haegemans. "On the construction of multi-dimensional embedded cubature formulae." Numerische Mathematik 55, no. 6 (November 1989): 735–45. http://dx.doi.org/10.1007/bf01389339.

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34

Meng, Zhaoliang, and Zhongxuan Luo. "Constructing cubature formulae of degree 5 with few points." Journal of Computational and Applied Mathematics 237, no. 1 (January 2013): 354–62. http://dx.doi.org/10.1016/j.cam.2012.06.004.

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35

Guessab, Allal, Otheman Nouisser, and Gerhard Schmeisser. "A Definiteness Theory for Cubature Formulae of Order Two." Constructive Approximation 24, no. 3 (March 15, 2006): 263–88. http://dx.doi.org/10.1007/s00365-005-0619-4.

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36

Xu, Yuan. "Constructing cubature formulae by the method of reproducing kernel." Numerische Mathematik 85, no. 1 (March 1, 2000): 155–73. http://dx.doi.org/10.1007/s002110050481.

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37

Chen, Chuanmiao, Michal Křížek, and Liping Liu. "Numerical Integration over Pyramids." Advances in Applied Mathematics and Mechanics 5, no. 03 (June 2013): 309–20. http://dx.doi.org/10.4208/aamm.12-m12110.

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AbstractPyramidal elements are often used to connect tetrahedral and hexahedral elements in the finite element method. In this paper we derive three new higher order numerical cubature formulae for pyramidal elements.

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38

Xu, Yuan. "Orthogonal Polynomials and Cubature Formulae on Spheres and on Balls." SIAM Journal on Mathematical Analysis 29, no. 3 (July 1998): 779–93. http://dx.doi.org/10.1137/s0036141096307357.

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39

Heo, Sangwoo, and Yuan Xu. "Invariant Cubature Formulae for Spheres and Balls by Combinatorial Methods." SIAM Journal on Numerical Analysis 38, no. 2 (January 2000): 626–38. http://dx.doi.org/10.1137/s003614299935543x.

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40

Xu, Yuan. "Orthogonal polynomials and cubature formulae on balls, simplices, and spheres." Journal of Computational and Applied Mathematics 127, no. 1-2 (January 2001): 349–68. http://dx.doi.org/10.1016/s0377-0427(00)00504-5.

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41

Ramazanov, M. D. "Asymptotically optimal unsaturated lattice cubature formulae with bounded boundary layer." Sbornik: Mathematics 204, no. 7 (July 31, 2013): 1003–27. http://dx.doi.org/10.1070/sm2013v204n07abeh004328.

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42

Cools, Ronald, and Ann Haegemans. "Optimal addition of knots to cubature formulae for planar regions." Numerische Mathematik 49, no. 2-3 (March 1986): 269–74. http://dx.doi.org/10.1007/bf01389629.

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43

Bernardo, Fernando P. "Performance of cubature formulae in probabilistic model analysis and optimization." Journal of Computational and Applied Mathematics 280 (May 2015): 110–24. http://dx.doi.org/10.1016/j.cam.2014.11.053.

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44

Stoyanova, Srebra B. "Invariant cubature formulae of degree 6 for the n-simplex." Journal of Computational and Applied Mathematics 193, no. 2 (September 2006): 446–59. http://dx.doi.org/10.1016/j.cam.2005.06.027.

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45

Xu, Yuan. "Orthogonal polynomials and cubature formulae on spheres and on simplices." Methods and Applications of Analysis 5, no. 2 (1998): 169–84. http://dx.doi.org/10.4310/maa.1998.v5.n2.a5.

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46

Bernardo, Fernando P. "Model analysis and optimization under uncertainty using thinned cubature formulae." Computers & Chemical Engineering 92 (September 2016): 133–42. http://dx.doi.org/10.1016/j.compchemeng.2016.05.006.

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47

Orive, Ramón, Juan C. Santos-León, and Miodrag M. Spalević. "Cubature formulae for the Gaussian weight. Some old and new rules." ETNA - Electronic Transactions on Numerical Analysis 53 (2020): 426–38. http://dx.doi.org/10.1553/etna_vol53s426.

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48

Xu, Yuan. "On Zeros of Multivariate Quasi-Orthogonal Polynomials and Gaussian Cubature Formulae." SIAM Journal on Mathematical Analysis 25, no. 3 (May 1994): 991–1001. http://dx.doi.org/10.1137/s0036141092237200.

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49

Fliege, J. "The distribution of points on the sphere and corresponding cubature formulae." IMA Journal of Numerical Analysis 19, no. 2 (April 1, 1999): 317–34. http://dx.doi.org/10.1093/imanum/19.2.317.

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50

Stoyanova, Srebra B. "Cubature formulae of the seventh degree of accuracy for the hypersphere." Journal of Computational and Applied Mathematics 84, no. 1 (October 1997): 15–21. http://dx.doi.org/10.1016/s0377-0427(97)00094-0.

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