A robust and accurate quasi-Monte Carlo algorithm for estimating eigenvalue of homogeneous integral equations. (English) Zbl 1306.65293
Summary: We present an efficient numerical algorithm for computing the eigenvalue of the linear homogeneous integral equations. The proposed algorithm is based on antithetic Monte Carlo algorithm and a low-discrepancy sequence, namely, Faure sequence. To reduce the computational time we reduce the variance by using the antithetic variance reduction procedure. Numerical results show that our scheme is robust and accurate.
MSC:
| 65R20 | Numerical methods for integral equations |
| 62C05 | General considerations in statistical decision theory |
| 45A05 | Linear integral equations |
| 45C05 | Eigenvalue problems for integral equations |
| 11K38 | Irregularities of distribution, discrepancy |
References:
| [1] | I. T. Dimov and T. V. Gurov, “Monte Carlo algorithm for solving integral equations with polynomial nonlinearity: parallel implementation,” Pliska Studia Mathematica Bulgarica, vol. 13, pp. 117-132, 2000. · Zbl 0965.60024 |
| [2] | B. Fathi and E. Radmoghaddam, “Optimal Faure sequence via mix Faure with the best scrambling schemes,” Computer Science and Its Applications, vol. 2, no. 1, pp. 2166-2924, 2012. |
| [3] | B. Vandewoestyne, H. Chi, and R. Cools, “Computational investigations of scrambled faure sequences,” Mathematics and Computers in Simulation, vol. 81, no. 3, pp. 522-535, 2010. · Zbl 1207.65010 · doi:10.1016/j.matcom.2009.09.007 |
| [4] | B. Fathi and F. Mehrdoust, “Quasi Monte Carlo algorithm for computing smallest and largest generalized eigenvalues,” ANZIAM Journal E, vol. 52, pp. 41-58, 2011. · Zbl 1333.65037 |
| [5] | F. Mehrdoust and B. Fathi, “A reliable stochastic algorithm for estimating eigenvalue of homogeneous integral equations,” Journal of Advanced Research in Applied Mathematics, vol. 5, no. 2, p. 13, 2013. · Zbl 1306.65293 |
| [6] | H. Faure, “Discrepancy of sequences associated with a number system (in dimension s),” Acta Arithmetica, vol. 41, no. 4, pp. 337-351, 1982 (French). · Zbl 0442.10035 |
| [7] | H. Niederreiter, Random Number Generations and Quasi-Monte Carlo Methods, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1992. · Zbl 0761.65002 |
| [8] | J. Spanier and E. H. Maize, “Quasi-random methods for estimating integrals using relatively small samples,” SIAM Review, vol. 36, no. 1, pp. 18-44, 1994. · Zbl 0824.65009 · doi:10.1137/1036002 |
| [9] | F. J. Hickernell, “Mean square discrepancy of randomized nets,” ACM Transactions on Modeling and Computer Simulation, vol. 6, pp. 274-296, 1996. · Zbl 0887.65030 · doi:10.1145/240896.240909 |
| [10] | I. M. Sobol, “Uniformly distributed sequences with an additional uniform property,” USSR Computational Mathematics and Mathematical Physics, vol. 16, no. 5, pp. 236-242, 1976. · Zbl 0391.10033 |
| [11] | S. Tezuka, “Polynomial arithmetic analogue of halton sequences,” ACM Transactions on Modeling and Computer Simulation, vol. 3, no. 2, pp. 99-107, 1993. · Zbl 0846.11045 · doi:10.1145/169702.169694 |
| [12] | M. S. Joshi, The Concepts and Practice of Mathematical Finance, Cambridge University Press, 2nd edition, 2003. · Zbl 1052.91001 |
| [13] | H. Chi, Scrambled Quasi-Random Sequences and their Application, The Florida state university college of arts and science, 2004. |
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.
