Abstract
In this paper, we establish a deterministic error bound for estimating a functional of the solution of the integral transport equation via random walks that improves an earlier result of Chelson generalizing the Koksma-Hlawka inequality for finite dimensional quadrature. We solve such problems by simulation, using sequences that combine pseudorandom and quasirandom elements in the construction of the random walks in order to take advantage of the superior uniformity properties of quasirandom numbers and the statistical (independence) properties of pseudorandom numbers. We discuss implementation issues that arise when these hybrid sequences are used in practice. The quasi-Monte Carlo techniques described in this paper have the potential to improve upon the convergence rates of both (conventional) Monte Carlo and quasi-Monte Carlo simulations in many problems. Recent model problem computations confirm these improved convergence properties.
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
H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, CBMS-SIAM, 1992.
I.H. Sloan and S. Joe, Lattice Methods for Multiple Integration, Clarendon Press, Oxford, 1994.
N.M. Korobov, Number-Theoretic Methods in Approximate Analysis, Fizmatig, Moscow, 1963. (in Russian).
L.K. Hua and Y. Wang, Applications of Number Theory to Numerical Analysis, Springer, Berlin, 1981.
A. Keller, “A Quasi-Monte Carlo Algorithm for the Global Illumination Problem in the Radiosity Setting”, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P. Shiue, eds.), vol. 106, Springer, 1995.
P. Chelson,“Quasi-Random Techniques for Monte Carlo Methods,” Ph.D. dissertation, The Claremont Graduate School, 1976.
J. Spanier, “Quasi-Monte Carlo Methods for Particle Transport Problems”, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P. Shiue, eds.), vol. 106, Springer, 1995.
J. Spanier and E.M. Gelbard, Monte Carlo Principles and Neutron Transport Problems, Addison-Wesley Pub. Co., 1969.
L. Li, “Quasi-Monte Carlo Methods for Transport Equations”, Ph.D. dissertation, The Claremont Graduate School, 1995.
E. Hlawka and R. Mück, “Uber eine Transformation von Gleichverteilen Folgen II, Computing, (1972).
J. Spanier, “An Analytic Approach to Variance Reduction,” SIAM J. Appl. Math., 18, (1970).
E. Maize, “Contributions to the Theory of Error Reduction in Quasi-Monte Carlo Methods”, Ph.D. dissertation, The Claremont Graduate School, 1981.
J. Spanier and E.H. Maize, “Quasi-Random Methods for Estimating Integrals Using Relatively Small Samples”, Siam Rev., 36 18–44, (1994).
D.E. Knuth, The Art of Computer Programming, Vol.,’L: Seminumerical Algorithms, 2nd ed., Addison-Wesley, 1981.
G.A. Mikhailov, Optimization of Weighted Monte Carlo Methods, Springer-Verlag, 1992.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this paper
Cite this paper
Spanier, J., Li, L. (1998). Quasi-Monte Carlo Methods for Integral Equations. In: Niederreiter, H., Hellekalek, P., Larcher, G., Zinterhof, P. (eds) Monte Carlo and Quasi-Monte Carlo Methods 1996. Lecture Notes in Statistics, vol 127. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1690-2_28
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1690-2_28
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98335-6
Online ISBN: 978-1-4612-1690-2
eBook Packages: Springer Book Archive