A quasi-Monte Carlo method based on the Karhunen–Loeve expansion for groundwater flow simulation

Abstract

To study the effects of spatial variability on groundwater flow simulation, some stochastic methods are proposed in this study. The Karhunen–Loeve expansion is utilized to generate the random field of the parameter conductivity. The conventional and pseudo-random, Latin hypercube, and quasi-random sampling methods are employed to generate random vectors in the Karhunen–Loeve expansion. Various Monte Carlo methods based on different sampling methods are compared and analyzed in groundwater flow simulation. It is shown that the quasi-Monte Carlo method has better convergence and efficiency compared to other methods, especially in cases with large input variance and short correlation length. The Karhunen–Loeve expansion possesses the ability to generate a random field of general type. The type of stochastic input has a great impact on uncertainty in groundwater simulations.

Keywords

Monte Carlo simulation Karhunen–Loeve expansion golden rule low-discrepancy sequence Latin hypercube sampling groundwater flow

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References

Zhang

. Spatial variation theory and its application. Beijing: Science Press, 2005.

Freeze

. A Stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media. Water Resour Res 1975; 11: 725–741.

Yang

Zhang

. Stochastic analysis of saturated-unsaturated flow in heterogeneous media by combining Karhunen-Loeve expansion and perturbation method. Water Resour Res 2003; 294: 18–38.

Zhang

. Stochastic methods for flow in porous media. Pittsburgh, USA:Academic Press, 2002.

Zhang

. On importance sampling Monte Carlo approach to uncertainty analysis for flow and transport in porous media. Adv Water Resourc 2003; 26: 1177–1188.

Morokoff

Caffisch

. Quasi-random sequences and their discrepancies. SIAM Journal on Scientific Computing 1994; 15: 1251–1279.

Morokoff

Caffisch

. Quasi-Monte Carlo integration. Journal of Computational Physics 1995: 122: 218–230.

Mckay

Beckman

Conover

. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 1979; 21: 239–245.

Mello

. On rates of convergence for stochastic optimization problems under non-independent and identically distributed sampling. SIAM J Optimiz 2008; 19: 524–551.

10.

Ghanem

Spanos

. Stochastic finite elements: a spectral approach. New York: Springer, 1991.

11.

Agrawal

Abbadi

. Applying the golden rule of sampling for query estimation. ACM SIGMOD Record 2001; 30: 1–12.

12.

Garrido

. Estimation performance of low discrepancy sequences in stated preferences. In: 10th international conference on travel behaviour research, Lucerne, 10–15 August 2003.

13.

Zsolt

Andras

. Alternative sampling methods for estimating multivariate normal probabilities. J Econometr 2004; 120: 207–234.

14.

Loh

. On Latin hypercube sampling. Ann Stat 1996; 24: 2058–2080.

15.

Keramat

Kielbasa

. Latin hypercube sampling Monte Carlo estimation of average quality index for integrated circuits. Analog Integrat Circ Signal Proc 1997; 14: 131–142.

16.

Palmer

Tsui

. A minimum bias Latin hypercube design. IIE Trans 2001; 33: 793–808.

17.

Phoon

Quek

. Comparison between Karhunen-Loève expansion and translation-based simulation of non-Gaussian processes. Comput Struct 2007; 85: 264–276.