Abstract
This paper presents work on solving elliptic BVPs problems based on quasi-random walks, by using a subset of uniformly distributed sequences—completely uniformly distributed (c.u.d.) sequences. This approach is novel for solving elliptic boundary value problems. The enhanced uniformity of c.u.d. sequences leads to faster convergence. We demonstrate that c.u.d. sequences can be a viable alternative to pseudorandom numbers when solving elliptic boundary value problems. Analysis of a simple problem in this paper showed that c.u.d. sequences achieve better numerical results than pseudorandom numbers, but also have the potential to converge faster and so reduce the computational burden.
Supported by the Ministry of Education and Science of Bulgaria under Grant No. I1405/04 and by the EC FP6 under Grant No: INCO-CT-2005-016639 of the project BIS-21++.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Caflisch, R.E.: Monte Carlo and quasi-Monte Carlo methods. Acta Numerica 7, 1–49 (1998)
Chaudhary, S.: Acceleration of Monte Carlo Methods using Low Discrepancy Sequences. Dissertation, University of California, Los Angeles (2004)
Dimov, I.T., Gurov, T.V.: Estimates of the computational complexity of iterative Monte Carlo algorithm based on Green’s function approach. Mathematics and Computers in Simulation 47, 183–199 (1998)
Ermakov, S., Nekrutkin, V., Sipin, V.: Random Processes for solving classical equations of the mathematical physics. Nauka, Moscow (1984)
Entacher, K., Hellekalek, P., L’Ecuyer, P.: Quasi-Monte Carlo Node Sets from Linear Congruential Generators. In: Niederreiter, H., Spanier, J. (eds.) Monte Carlo and Quasi-Monte Carlo methods 1998, pp. 188–198. Springer, Berlin (2000)
Hofmann, N., Mathe, P.: On Quasi-Monte Carlo Simulation of stochastic Differential equations. Mathematics of Computation 66, 573–589 (1997)
Karaivanova, A., Mascagni, M., Simonov, N.A.: Solving BVPs using quasirandom walks on the boundary. In: Lirkov, I., et al. (eds.) LSSC 2003. LNCS, vol. 2907, pp. 162–169. Springer, Heidelberg (2004)
Knuth, D.E.: The Art of Computer Programming, vol. 2: Seminumerical Algorithms. Addison-Wesley, Reading (1997)
Knuth, D.E.: Construction of a random sequence. BIT 4, 250–264 (1965)
Korobov, N.M.: Bounds of trigonometric sums involving completely uniformly distributed functions. Soviet. Math. Dokl. 1, 923–926 (1964)
Korobov, N.M.: On some topics of uniform distribution. Izvestia Akademii Nauk SSSR, Seria Matematicheskaja 14, 215–238 (1950)
Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. John Wiley and Sons, New York (1974)
Lecot, C., Tuffin, B.: Quasi-Monte Carlo Methods for Estimating Transient Measures of Discrete Time Markov Chains. In: Fifth International Conference on Monte Carlo and Quasi- Monte Carlo Methods in Scientific Computing, pp. 329–344. Springer, Heidelberg (2002)
Levin, M.B.: Discrepancy Estimates of Completely Uniformly Distributed and Pseudorandom Number Sequences. International Mathematics Research Notices 22, 1231–1251 (1999)
Loh, W.L.: On the asymptotic distribution of scrambled net quadrature. Annals of Statistics 31, 1282–1324 (2003)
Mikhailov, G.A.: New Monte Carlo Methods with Estimating Derivatives, Utrecht, The Netherlands (1995)
Mascagni, M., Karaivanova, A., Hwang, C.: Monte Carlo Methods for elliptic boundary Value Problems. In: Proceedings of MCQMC 2002: Monte Carlo and Quasi-Monte Carlo Methods 2002, pp. 345–355. Springer, Heidelberg (2004)
Morokoff, W.: Generating Quasi-Random Paths for Stochastic Processes. SIAM Rev. 40(4), 765–788 (1998)
Miranda, C.: Equasioni alle dirivate parziali di tipo ellittico. Springer, Berlin (1955)
Morokoff, W., Caflisch, R.E.: A quasi-Monte Carlo approach to particle simulation of the heat equation. SIAM J. Numer. Anal. 30, 1558–1573 (1993)
Niederreiter, H.: Random Number Generations and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)
Owen, A.B.: Scrambled Net Variance for Integrals of Smooth Functions. Annals of Statistics 25(4), 1541–1562 (1997)
Owen, A.B., Tribble, S.: A quasi-Monte Carlo Metroplis Algorithm. Proceedings of the National Academy of Sciences of the United States of America 102, 8844–8849 (2005)
Sabelfeld, K.: Monte Carlo Methods in Boundary Value Problems. Springer, Heidelberg (1991)
Shparlinskii, I.E.: On a Completely Uniform Distribution. USSR Computational Mathematics and Mathematical Physics 19, 249–253 (1979)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Karaivanova, A., Chi, H., Gurov, T. (2007). Quasi-random Walks on Balls Using C.U.D. Sequences. In: Boyanov, T., Dimova, S., Georgiev, K., Nikolov, G. (eds) Numerical Methods and Applications. NMA 2006. Lecture Notes in Computer Science, vol 4310. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70942-8_19
Download citation
DOI: https://doi.org/10.1007/978-3-540-70942-8_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70940-4
Online ISBN: 978-3-540-70942-8
eBook Packages: Computer ScienceComputer Science (R0)