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Parallel Multilevel Monte Carlo Algorithms for Elliptic PDEs with Random Coefficients

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Large-Scale Scientific Computing (LSSC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11958))

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Abstract

In this work, we developed and investigated Monte Carlo algorithms for elliptic PDEs with random coefficients. We considered groundwater flow as a model problem, where a permeability field represents random coefficients. The computational complexity is the main challenge in uncertainty quantification methods. The computation contains generating of a random coefficient and solving of partial differential equations. The permeability field was generated using the circulant embedding method. Multilevel Monte Carlo (MLMC) simulation can be based on different approximations of partial differential equations. We developed three MLMC algorithms based on finite volume, finite volume with renormalization and renormalization approximation. We compared numerical simulations and parallel performance of MLMC algorithms for 2D and 3D problems.

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References

  1. Barth, A., Schwab, C., Zollinger, N.: Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Numer. Math. 119(1), 123–161 (2011)

    Article  MathSciNet  Google Scholar 

  2. Brandt, A., Galun, M., Ron, D.: Optimal multigrid algorithms for calculating thermodynamic limits. J. Stat. Phys. 74(1–2), 313–348 (1994)

    Article  Google Scholar 

  3. Charrier, J., Scheichl, R., Teckentrup, A.L.: Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. SIAM J. Numer. Anal. 51(1), 322–352 (2013)

    Article  MathSciNet  Google Scholar 

  4. Cliffe, K.A., Giles, M.B., Scheichl, R., Teckentrup, A.L.: Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci. 14(1), 3 (2011)

    Article  MathSciNet  Google Scholar 

  5. Cliffe, K., Graham, I.G., Scheichl, R., Stals, L.: Parallel computation of flow in heterogeneous media modelled by mixed finite elements. J. Comput. Phys. 164(2), 258–282 (2000)

    Article  MathSciNet  Google Scholar 

  6. De Marsily, G., Delay, F., Gonçalvès, J., Renard, P., Teles, V., Violette, S.: Dealing with spatial heterogeneity. Hydrol. J. 13(1), 161–183 (2005)

    Google Scholar 

  7. Delhomme, J.: Spatial variability and uncertainty in groundwater flow parameters: a geostatistical approach. Water Resour. Res. 15(2), 269–280 (1979)

    Article  Google Scholar 

  8. Dietrich, C.R., Newsam, G.N.: Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput. 18(4), 1088–1107 (1997)

    Article  MathSciNet  Google Scholar 

  9. Dimov, I., Georgieva, R., Todorov, V.: Balancing of systematic and stochastic errors in Monte Carlo algorithms for integral equations. In: Dimov, I., Fidanova, S., Lirkov, I. (eds.) NMA 2014. LNCS, vol. 8962, pp. 44–51. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-15585-2_5

    Chapter  MATH  Google Scholar 

  10. Drzisga, D., Gmeiner, B., Rüde, U., Scheichl, R., Wohlmuth, B.: Scheduling massively parallel multigrid for multilevel Monte Carlo methods. SIAM J. Sci. Comput. 39(5), S873–S897 (2017)

    Article  MathSciNet  Google Scholar 

  11. Frigo, M., Johnson, S.G.: The design and implementation of FFTW3. Proc. IEEE 93(2), 216–231 (2005)

    Article  Google Scholar 

  12. Ghanem, R.G., Spanos, P.D.: Stochastic finite element method: response statistics. In: Ghanem, R.G., Spanos, P.D. (eds.) Stochastic Finite Elements: A Spectral Approach, pp. 101–119. Springer, New York (1991). https://doi.org/10.1007/978-1-4612-3094-6_4

    Chapter  MATH  Google Scholar 

  13. Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)

    Article  MathSciNet  Google Scholar 

  14. Giles, M.: Improved multilevel Monte Carlo convergence using the milstein scheme. In: Keller, A., Heinrich, S., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods, pp. 343–358. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-74496-2_20

    Chapter  MATH  Google Scholar 

  15. Graham, I.G., Kuo, F.Y., Nuyens, D., Scheichl, R., Sloan, I.H.: Analysis of circulant embedding methods for sampling stationary random fields. SIAM J. Numer. Anal. 56(3), 1871–1895 (2018)

    Article  MathSciNet  Google Scholar 

  16. Graham, I.G., Kuo, F.Y., Nuyens, D., Scheichl, R., Sloan, I.H.: Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications. J. Comput. Phys. 230(10), 3668–3694 (2011)

    Article  MathSciNet  Google Scholar 

  17. Hoeksema, R.J., Kitanidis, P.K.: Analysis of the spatial structure of properties of selected aquifers. Water Resour. Res. 21(4), 563–572 (1985)

    Article  Google Scholar 

  18. Iliev, O., Mohring, J., Shegunov, N.: Renormalization based MLMC method for scalar elliptic SPDE. In: Lirkov, I., Margenov, S. (eds.) LSSC 2017. LNCS, vol. 10665, pp. 295–303. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73441-5_31

    Chapter  Google Scholar 

  19. Lunati, I., Bernard, D., Giudici, M., Parravicini, G., Ponzini, G.: A numerical comparison between two upscaling techniques: non-local inverse based scaling and simplified renormalization. Adv. Water Resour. 24(8), 913–929 (2001)

    Article  Google Scholar 

  20. Mohring, J., et al.: Uncertainty quantification for porous media flow using multilevel Monte Carlo. In: Lirkov, I., Margenov, S.D., Waśniewski, J. (eds.) LSSC 2015. LNCS, vol. 9374, pp. 145–152. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-26520-9_15

    Chapter  Google Scholar 

  21. Renard, P., De Marsily, G.: Calculating equivalent permeability: a review. Adv. Water Resour. 20(5–6), 253–278 (1997)

    Article  Google Scholar 

  22. Šukys, J., Mishra, S., Schwab, C.: Static load balancing for multi-level Monte Carlo finite volume solvers. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds.) PPAM 2011. LNCS, vol. 7203, pp. 245–254. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31464-3_25

    Chapter  Google Scholar 

  23. Teckentrup, A.L., Scheichl, R., Giles, M.B., Ullmann, E.: Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients. Numer. Math. 125(3), 569–600 (2013)

    Article  MathSciNet  Google Scholar 

  24. Wen, X.H., Gómez-Hernández, J.J.: Upscaling hydraulic conductivities in heterogeneous media: an overview. J. Hydrol. 183(1–2), ix–xxxii (1996)

    Google Scholar 

  25. Whitaker, S.: Flow in porous media I: a theoretical derivation of Darcy’s law. Transp. Porous Media 1(1), 3–25 (1986)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. North-Eastern Federal University, Yakutsk, Russia

    Petr Zakharov

  2. Fraunhofer ITWM, Kaiserslautern, Germany

    Oleg Iliev & Jan Mohring

  3. Sofia University, Sofia, Bulgaria

    Nikolay Shegunov

  4. Institute of Mathematics and Informatics, Bulgarian Academy of Science, Sofia, Bulgaria

    Oleg Iliev

Authors
  1. Petr Zakharov
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  2. Oleg Iliev
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  3. Jan Mohring
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  4. Nikolay Shegunov
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Corresponding author

Correspondence to Petr Zakharov .

Editor information

Editors and Affiliations

  1. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

  2. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Svetozar Margenov

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Zakharov, P., Iliev, O., Mohring, J., Shegunov, N. (2020). Parallel Multilevel Monte Carlo Algorithms for Elliptic PDEs with Random Coefficients. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2019. Lecture Notes in Computer Science(), vol 11958. Springer, Cham. https://doi.org/10.1007/978-3-030-41032-2_53

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  • DOI: https://doi.org/10.1007/978-3-030-41032-2_53

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-41031-5

  • Online ISBN: 978-3-030-41032-2

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