Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-08T14:28:27.154Z Has data issue: false hasContentIssue false
  • English
  • Français

A multilevel approach towards unbiased sampling of random elliptic partial differential equations

Part of: Equilibrium statistical mechanics Probabilistic methods, simulation and stochastic differential equations Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  29 November 2018

Xiaoou Li ,
Jingchen Liu  and
Shun Xu
Xiaoou Li*
Affiliation:
University of Minnesota
Jingchen Liu*
Affiliation:
Columbia University
Shun Xu*
Affiliation:
Columbia University
*
* Postal address: School of Statistics, University of Minnesota, 224 Church Street SE, Minneapolis, MN 55455, USA. Email address: [email protected]
** Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, 10027, USA.
** Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, 10027, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Partial differential equations are powerful tools for used to characterizing various physical systems. In practice, measurement errors are often present and probability models are employed to account for such uncertainties. In this paper we present a Monte Carlo scheme that yields unbiased estimators for expectations of random elliptic partial differential equations. This algorithm combines a multilevel Monte Carlo method (Giles (2008)) and a randomization scheme proposed by Rhee and Glynn (2012), (2013). Furthermore, to obtain an estimator with both finite variance and finite expected computational cost, we employ higher-order approximations.

Keywords

Unbiased samplingpartial differential equations with random coefficientsMonte Carlo method

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations 82B80: Numerical methods (Monte Carlo, series resummation, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1007 - 1031
Copyright
Copyright © Applied Probability Trust 2018 

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]Charrier, J. (2012).Strong and weak error estimates for elliptic partial differential equations with random coefficients.SIAM J. Numer. Anal. 50,216–246.Google Scholar
[2]Charrier, J.,Scheichl, R. and Teckentrup, A. L. (2013).Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods.SIAM J. Numer. Anal. 51,322–352.Google Scholar
[3]Ciarlet, P. (1991).Basic error estimates for elliptic problems. In Handbook of Numerical Analysis, Vol. 2.North-Holland,Amsterdam, pp. 17–351.Google Scholar
[4]Cliffe, K.,Giles, M. B.,Scheichl, R. and Teckentrup, A. L. (2011).Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients.Comput. Visualization Sci. 14,3–15.Google Scholar
[5]Delhomme, J. P. (1979).Spatial variability and uncertainty in groundwater flow parameters: a geostatistical approach.Water Resources Res. 15,269–280.Google Scholar
[6]De Marsily, G. et al. (2005).Dealing with spatial heterogeneity.Hydrogeol. J. 13,161–183.Google Scholar
[7]Evans, L. C. (1998).Partial Differential Equations.American Mathematical Society,Providence, RI.Google Scholar
[8]Giles, M. B. (2008).Multilevel Monte Carlo path simulation.Operat. Res. 56,607–617.Google Scholar
[9]Graham, I. G. et al. (2011).Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications.J. Comput. Phys. 230,3668–3694.Google Scholar
[10]Knabner, P. and Angermann, L. (2003).Numerical Methods for Elliptic and Parabolic Partial Differential Equations.Springer,New York.Google Scholar
[11]Ostoja-Starzewski, M. (2008).Microstructural Randomness and Scaling in Mechanics of Materials.Chapman and Hall/CRC Press.Google Scholar
[12]Rhee, C.-H. and Glynn, P. W. (2012).A new approach to unbiased estimation for SDE's. In Proc. 2012 Winter Simul. Conf.,IEEE, 7pp.Google Scholar
[13]Rhee, C.-H. and Glynn, P. W. (2013).Unbiased estimation with square root convergence for SDE models.Operat. Res. 63,1026–1043.Google Scholar
[14]Sobczyk, K. and Kirkner, D. J. (2001).Stochastic Modeling of Microstructures.Birkhäuser.Google Scholar
[15]Teckentrup, A. L.,Scheichl, R.,Giles, M. B. and Ullmann, E. (2013).Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients.Numer. Math. 125,569–600.Google Scholar