Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T15:55:01.906Z Has data issue: false hasContentIssue false
  • English
  • Français

A Numerical Comparison Between Quasi-Monte Carlo and Sparse Grid Stochastic Collocation Methods

Published online by Cambridge University Press:  20 August 2015

Juarez dos Santos Azevedo  and
Saulo Pomponet Oliveira
Juarez dos Santos Azevedo*
Affiliation:
CETEC-UFRB, Centro, 44380-000, Cruz das Almas-BA, Brazil
Saulo Pomponet Oliveira*
Affiliation:
DMAT-UFPR, Centro Politécnico, 81531-980, Curitiba-PR, Brazil
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

Abstract

Quasi-Monte Carlo methods and stochastic collocation methods based on sparse grids have become popular with solving stochastic partial differential equations. These methods use deterministic points for multi-dimensional integration or interpolation without suffering from the curse of dimensionality. It is not evident which method is best, specially on random models of physical phenomena. We numerically study the error of quasi-Monte Carlo and sparse grid methods in the context of ground-water flow in heterogeneous media. In particular, we consider the dependence of the variance error on the stochastic dimension and the number of samples/collocation points for steady flow problems in which the hydraulic conductivity is a lognormal process. The suitability of each technique is identified in terms of computational cost and error tolerance.

Keywords

60H1565C0565C2065C20Karhunen-Loève expansionMonte Carloquasi-Monte Carlosparse grid
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 4 , October 2012 , pp. 1051 - 1069
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Andrieu, C., Freitas, N., and Doucet, A.. An Introduction to MCMC for Machine Learning. Machine Learning, 50(1-2):5–43, 2003.CrossRefGoogle Scholar
[2]Babuška, I., Nobile, F., and Tempone, R.. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal., 45(3):1005–1034,2007.CrossRefGoogle Scholar
[3]Babuška, I., Tempone, R., and Zouraris, R. E.. Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal., 42(2):800–825, 2004.CrossRefGoogle Scholar
[4]Barthelmann, V., Novak, E., and Ritter, K.. High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math., 12(4):273–288, 2000.Google Scholar
[5]Borges, M. R., Murad, M. A., Pereira, F., and Furtado, F.. A new multiscale scheme for computing statistical moments in single phase flow in heterogeneous porous media. Adv. Water Resour., 32(3):361–382, 2009.Google Scholar
[6]Bungartz, H.‐J. and Dirnstorfer, S.. Multivariate quadrature on adaptive sparse grids. Computing, 71(1):89–114, 2003.Google Scholar
[7]Correa, M. R. and Loula, A. F. D.. Unconditionally stable mixed finite element methods for Darcy flow. Comput. Methods Appl. Mech. Engrg., 197(17-18):1525–1540, 2008.Google Scholar
[8]Drmota, M. and Tichy, R. F.. Sequences, discrepancies and applications. Lecture Notes in Mathematics. Springer, Berlin, 1997.Google Scholar
[9]Galvis, J. and Sarkis, M.. Approximating infinity-dimensional stochastic Darcy’s equations without uniform ellipticity. SIAM J. Numer. Anal., 47(5):3624–3651, 2009.CrossRefGoogle Scholar
[10]Ganapathysubramanian, B. and Zabaras, N.. Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys., 225(1):652x2013;685, 2007.CrossRefGoogle Scholar
[11]Gerstner, T. and Griebel, M.. Dimension-adaptive tensor-product quadrature. Computing, 71(1):65–87, 2003.Google Scholar
[12]Gerstner, T., Griebel, M., and Holtz, M.. Efficient deterministic numerical simulation of stochastic asset-liability management models in life insurance. Insurance: Mathematics and Economics, 44(3):434–446, 2009.Google Scholar
[13]Gerstner, T. and Griebel, M.. Numerical integration using sparse grids. Numer. Algorithms, 18:209–232, 1988.Google Scholar
[14]Graham, I. G., Kuo, F. Y., Nuyens, D., Scheichl, R., and Sloan, I. H.. Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications. J. Comput. Phys., 230(10):3668– 3694, 2010.Google Scholar
[15]Heiss, F. and Winschel, V.. Likelihood approximation by numerical integration on sparse grids. Journal of Econometrics, 144(1):62–80, 2008.Google Scholar
[16]Holden, H., Øksendal, B., Ubøe, J., and Zhang, T.. Stochastic Partial Differential Equations: A Modeling White Noise Functional Approach. Birkhäuser, 1996.Google Scholar
[17] L Kuipers and Niederreiter, H.. Uniform distribution of sequences. Dover, New York, 1995.Google Scholar
[18]Ma, X. and Zabaras, N.. An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations. J. Comput. Phys., 229(10):3884–3915, 2010.Google Scholar
[19]Morokoff, W. J. and Caflisch, R. E.. Quasi-random sequences and their discrepancies. SIAM J. Sci. Comput., 15(6):1251–1279, 1994.Google Scholar
[20]Morokoff, W. J. and Caflisch, R. E.. Quasi-Monte Carlo integration. J. Comput. Phys., 122(2):218–230, 1995.CrossRefGoogle Scholar
[21]Niederreiter, H.. Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Amer. Math. Soc., 84(6):957–1041, 1978.Google Scholar
[22]Niederreiter, H.. Random number generation and quasi-Monte Carlo methods. SIAM, Philadelphia, 1992.Google Scholar
[23]Nobile, F., Tempone, R., and Webster, C. G.. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal., 46(5):2309– 2345, 2008.Google Scholar
[24]Novak, E. and Ritter, K.. The curse of dimension and a universal method for numerical integration. In Nürnberger, G., W. Schmidt, J., and Walz, G., editors, Multivariate Approximation and Splines, pages 177–188. Birkhäuser, 1997.Google Scholar
[25]Ravalec, M., Noetinger, B., and Hu, L.. The FFT Moving Average (FFT-MA) Generator: An Efficient Numerical Method for Generating and Conditioning Gaussian Simulations. Mathematical Geology, 32(6):701–723, 2000.Google Scholar
[26]Schürer, R.. A comparison between (quasi-)Monte Carlo and cubature rule based methods for solving high-dimensional integration problems. Math. Comput. Simul., 62(3-6):509–517, 2003.Google Scholar
[27]Shewchuk, J. R.. Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Trian-gulator. In Lin, M. C. and Manocha, D., editors, Applied Computational Geometry: Towards Geometric Engineering, volume 1148 of Lecture Notes in Computer Science, pages 203–222. Springer-Verlag, 1996.Google Scholar
[28]Tsuji, P., Xiu, D., and Ying, L.. Fast method for high-frequency acoustic scattering from random scatterers. Int. J. Uncertain Quantification, 1(2):99–117, 2011.Google Scholar
[29]Tuffin, B.. Randomization of Quasi-Monte Carlo Methods for Error Estimation: Survey and Normal Approximation. Monte Carlo Methods and Applications, 10(3-4):617–628, 2004.Google Scholar
[30]Xiu, D.. Fast Numerical Methods for Stochastic Computations: A Review. Commun. Comput. Phys, 5(2-4):242–272, 2009.Google Scholar
[31]Xiu, D. and Hesthaven, J.. Higher-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput., 27(3):1118–1139,2005.Google Scholar
[32]Zhang, D. and Lu, Z.. An efficient, high-order pertubation approach for flow in random porous media via Karhunen-Loeve and polynomial expansions. J. Comput. Phys., 194(2):773–794, 2004.CrossRefGoogle Scholar
[33]Zhang, D., Shi, L., Chang, H., and Yang, J.. A comparative study of numerical approaches to risk assessment of contaminant transport. Stoch. Environ. Res. Risk Assess., 24(7):971–984, 2010.Google Scholar