×

Hybrid HDMR method with an optimized hybridity parameter in multivariate function representation. (English) Zbl 1310.41026

Summary: High dimensional model representation (HDMR) based methods are used to generate an approximation for a given multivariate function in terms of less variate functions. This paper focuses on hybrid HDMR which is composed of plain HDMR and logarithmic HDMR. The plain HDMR method works well for representing multivariate functions having additive nature. If the function under consideration has a multiplicative nature, then the logarithmic HDMR method produces better approximation. Hybrid HDMR method aims to successfully represent a multivariate function having neither purely additive nor purely multiplicative nature under a hybridity parameter. The performance of the hybrid HDMR method strongly depends on the value of this hybridity parameter because this parameter manages the contribution level of plain and logarithmic HDMR expansions. The main purpose of this work is to optimize the hybridity parameter to get the best approximations. Fluctuationlessness approximation theorem is used in this optimization process and in evaluating the multiple integrals appearing in HDMR based methods. A number of numerical implementations are given at the end of the paper to show the performance of our proposed method.

MSC:

41A63 Multidimensional problems

Software:

MuPAD
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Sobol I.M.: Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp. (MMCE) 1, 407–414 (1993) · Zbl 1039.65505
[2] Rabitz H., Alış Ö.: General foundations of high dimensional model representations. J. Math. Chem. 25, 197–233 (1999) · Zbl 0957.93004
[3] Alış Ö., Rabitz H.: Efficient implementation of high dimensional model representations. J. Math. Chem. 29, 127–142 (2001) · Zbl 1051.93502
[4] Li G., Rosenthal C., Rabitz H.: High dimensional model representations. J. Math. Chem. A 105, 7765–7777 (2001)
[5] Demiralp M.: High dimensional model representation and its application varieties. Math. Res. 9, 146–159 (2003) · Zbl 1237.93093
[6] Tunga M.A., Demiralp M.: A factorized high dimensional model representation on the partitioned random discrete data. Appl. Numer. Anal. Comput. Math. 1, 231–241 (2004) · Zbl 1064.65007
[7] Tunga M.A., Demiralp M.: A factorized high dimensional model representation on the nodes of a finite hyperprismatic regular grid. Appl. Math. Comput. 164, 865–883 (2005) · Zbl 1070.65009
[8] M. Demiralp, Logarithmic High Dimensional Model Representation, 6th WSEAS International Conference on Mathematics (MATH’06) (\.Istanbul, Turkey, May 27–29 2006), pp. 157–161
[9] Tunga B., Demiralp M.: Fluctuation free multivariate integration based logarithmic HDMR in multivariate function representation. J. Math. Chem. 49(4), 894–909 (2011) · Zbl 1301.41022
[10] Tunga B., Demiralp M.: Hybrid high dimensional model representation approximants and their utilization in applications. Math. Res. 9, 438–446 (2003) · Zbl 1237.93094
[11] Tunga M.A., Demiralp M.: Hybrid high dimensional model representation (HHDMR) on the partitioned data. J. Comput. Appl. Math. 185, 107–132 (2006) · Zbl 1077.65010
[12] Tunga B., Demiralp M.: Fluctuationlessness approximation based multivariate integration in hybrid high dimensional model representation. AIP Conf. Proc. 1048, 562–565 (2008)
[13] M. Demiralp, A New Fluctuation Expansion Based Method for the Univariate Numerical Integration Under Gaussian Weights, WSEAS-2005 Proceedings, WSEAS 8-th International Conference on Applied Mathematics, (Tenerife, Spain, 16–18 December 2005), pp. 68–73
[14] Demiralp M.: Convergence issues in the Gaussian weighted multidimensional fluctuation expansion for the univariate numerical integration. WSEAS Tracs. Math. 4, 486–492 (2005) · Zbl 1290.65019
[15] Demiralp M.: Fluctuationlessness theorem to approximate univariate functions Matrix Representations. WSEAS Trans. Math. 8(6), 258–267 (2009)
[16] Demiralp M.: Production for a multivariate function on an orthogonal hyperprismatic grid via fluctuation free matrix representation: completely filled grid case. IJEECE 1(1), 61–76 (2010)
[17] Chowdhury R., Adhikari S.: High dimensional model representation for stochastic finite element analysis. Appl. Math. Model. 34, 3917–3932 (2010) · Zbl 1201.65009
[18] Chowdhury R., Rao B.N., Prasad A.M.: High dimensional model representation for structural reliability analysis. Commun. Numer. Methods Eng. 25(4), 301–337 (2009) · Zbl 1169.74040
[19] Ziehn T., Tomlin A.S.: A global sensitivity study of sulfur chemistry in a premixed methane flame model using HDMR. Int. J. Chem. Kinet 40, 742–753 (2008)
[20] A. Gil, J. Segura, N.M. Temme, Gauss Quadrature, Numerical Methods for Special Functions (SIAM, Philadelphia, 2007)
[21] William H., Flannery B.P., Teukolsky S.A., Vetterling W.T.: Gaussian Quadratures and Orthogonal Polynomials Numerical Recipes in C, 2nd edn. Cambridge University Press, Cambridge (1988) · Zbl 0661.65001
[22] Tunga B., Demiralp M.: Constancy maximization based weight optimization in high dimensional model representation for multivariate functions. J. Math. Chem. 49(9), 1996–2012 (2011) · Zbl 1234.62105
[23] Oevel W., Postel F., Wehmeier S., Gerhard J.: The MuPAD Tutorial. Springer, Berlin (2000) · Zbl 1007.68211
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.