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The Multiscale Maximum Change Algorithm for Subsurface Characterization

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Computational Science and Its Applications – ICCSA 2023 Workshops (ICCSA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14105))

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Abstract

The characterization of subsurface formations is a formidable task due to the high dimension of the stochastic space involved in the solution of inverse problems. To make the task computationally manageable, one needs to apply a dimensional reduction technique. In this paper we are considering the Karhunen Loève expansion (KLE) as the aforementioned technique. Considering the subsurface properties of interest, such as permeability and porosity, it may be suitable to localize the sampling method so that it can better accommodate the large variation in rock properties. In a Bayesian framework we investigate the solution of an inverse problem involving an elliptic partial differential equation for porous media flows. We propose a new multiscale sampling algorithm in which the prior distribution is expressed in terms of local KL expansions in non-overlapping subdomains of the domain of the inverse problem. We solve the inverse problem using multiple Markov Chain Monte Carlo (MCMC) simulations performed on a multi-GPU cluster. The simulation results indicate that the proposed algorithm significantly improves the convergence of a preconditioned MCMC method.

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References

  1. Akbarabadi, M., Borges, M., Jan, A., Pereira, F., Piri, M.: A Bayesian framework for the validation of models for subsurface flows: synthetic experiments. Comput. Geosci. 19(6), 1231–1250 (2015). https://doi.org/10.1007/s10596-015-9538-z

    Article  MathSciNet  MATH  Google Scholar 

  2. Al-Mamun, A., Barber, J., Ginting, V., Pereira, F., Rahunanthan, A.: Contaminant transport forecasting in the subsurface using a Bayesian framework. Appl. Math. Comput. 387, 124980 (2020)

    MathSciNet  MATH  Google Scholar 

  3. Ali, A., Al-Mamun, A., Pereira, F., Rahunanthan, A.: Multiscale sampling for the inverse modeling of partial differential equations (2023). https://doi.org/10.48550/ARXIV.2302.11149

  4. Ali, A.A.H.: Multiscale Sampling for Subsurface Characterization. The University of Texas at Dallas (2021)

    Google Scholar 

  5. Brooks, S., Gelman, A.: General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat. 7, 434–455 (1998)

    MathSciNet  Google Scholar 

  6. Christen, J.A., Fox, C.: Markov chain Monte Carlo using an approximation. J. Comput. Graph. Stat. 14(4), 795–810 (2005)

    Article  MathSciNet  Google Scholar 

  7. Contreras, A.A., Mycek, P., Maítre, O.P., Rizzi, F., Debusschere, B., Knio, O.M.: Parallel domain decomposition strategies for stochastic elliptic equations. part a: local Karhunen-Loève representations. SIAM J. Sci. Statist. Comput. 40(4), C520–C546 (2018)

    Google Scholar 

  8. Durlofsky, L.: Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res. 27(5), 699–708 (1991)

    Article  MathSciNet  Google Scholar 

  9. Efendiev, Y., Hou, T., Luo, W.: Preconditioning Markov chain Monte Carlo simulations using coarse-scale models. SIAM J. Sci. Comput. 28, 776–803 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Francisco, A., Ginting, V., Pereira, F., Rigelo, J.: Design and implementation of a multiscale mixed method based on a nonoverlapping domain decomposition procedure. Math. Comput. Simul. 99, 125–138 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ginting, V., Pereira, F., Rahunanthan, A.: Multiple Markov chains Monte Carlo approach for flow forecasting in porous media. Procedia Comput. Sci. 9, 707–716 (2012)

    Article  Google Scholar 

  12. Ginting, V., Pereira, F., Rahunanthan, A.: A multi-stage Bayesian prediction framework for subsurface flows. Int. J. Uncertain. Quantif. 3(6), 499–522 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ginting, V., Pereira, F., Rahunanthan, A.: A prefetching technique for prediction of porous media flows. Comput. Geosci. 18(5), 661–675 (2014). https://doi.org/10.1007/s10596-014-9413-3

    Article  MathSciNet  MATH  Google Scholar 

  14. Ginting, V., Pereira, F., Rahunanthan, A.: Rapid quantification of uncertainty in permeability and porosity of oil reservoirs for enabling predictive simulation. Math. Comput. Simul. 99, 139–152 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ginting, V., Pereira, F., Rahunanthan, A.: Multi-physics Markov chain Monte Carlo methods for subsurface flows. Math. Comput. Simul. 118, 224–238 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guiraldello, R.T., Ausas, R.F., Sousa, F.S., Pereira, F., Buscaglia, G.C.: The multiscale Robin coupled method for flows in porous media. J. Comput. Phys. 355, 1–21 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Guiraldello, R.T., Ausas, R.F., Sousa, F.S., Pereira, F., Buscaglia, G.C.: Velocity postprocessing schemes for multiscale mixed methods applied to contaminant transport in subsurface flows. Comput. Geosci. 24(3), 1141–1161 (2020). https://doi.org/10.1007/s10596-019-09930-8

    Article  MathSciNet  MATH  Google Scholar 

  18. Jaramillo, A., et al.: Towards HPC simulations of billion-cell reservoirs by multiscale mixed methods. arXiv preprint arXiv:2103.08513 (2021)

  19. Laloy, E., Rogiers, B., Vrugt, J., Mallants, D., Jacques, D.: Efficient posterior exploration of a high-dimensional groundwater model from two-stage Markov chain Monte Marlo simulation and polynomial chaos expansion. Water Resour. 49(5), 2664–2682 (2013)

    Article  Google Scholar 

  20. Liao, Q., Willcox, K.: A domain decomposition approach for uncertainty analysis. SIAM J. Sci. Comput. 37(1), A103–A133 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Loève, M.: Probability Theory. Springer, Berlin (1997)

    MATH  Google Scholar 

  22. Cowles, M.K., Carlin, B.P.: Markov chain Monte Carlo convergence diagnostics: a comparative review. J. Am. Stat. Assoc. 91, 883–904 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mamun, A., Pereira, F., Rahunanthan, A.: Convergence analysis of MCMC methods for subsurface flow problems. In: Gervasi, O., et al. (eds.) ICCSA 2018. LNCS, vol. 10961, pp. 305–317. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-95165-2_22

    Chapter  Google Scholar 

  24. Mengersen, K.L., Robert, C.P., Guihenneuc-Jouyaux, C.: MCMC convergence diagnostics: a review. In: Bernardo, M., Berger, J.O., Dawid, A.P., Smtith, A.F.M. (eds.) Bayesian Statistics, vol. 6, pp. 415–440. Oxford University Press (1999)

    Google Scholar 

  25. Metropolis, N., Ulam, S.: The Monte Carlo method. J. Am. Stat. Assoc. 44(247), 335–341 (1949)

    Article  MATH  Google Scholar 

  26. Pereira, F., Rahunanthan, A.: A semi-discrete central scheme for the approximation of two-phase flows in three space dimensions. Math. Comput. Simul. 81(10), 2296–2306 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Polson, N.G.: Convergence of Markov chain Monte Carlo algorithms. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics, vol. 5, pp. 297–322 (1996)

    Google Scholar 

  28. Quarteroni, A.M., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  29. Rosenthal, J.S.: Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Am. Stat. Assoc. 90(430), 558–566 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Roy, V.: Convergence diagnostics for Markov chain Monte Carlo. Annu. Rev. Stat. Appl. 7, 387–412 (2019)

    Article  MathSciNet  Google Scholar 

  31. Brooks, S.P., Roberts, G.O.: Convergence assessment techniques for Markov chain Monte Carlo. Stat. Comput. 8, 319–335 (1998)

    Article  Google Scholar 

  32. Smith, B.J.: boa: an R package for MCMC output convergence assessment and posterior inference. J. Stat. Softw. 21, 1–37 (2007)

    Article  Google Scholar 

  33. Stuart, G.K., Minkoff, S.E., Pereira, F.: A two-stage Markov chain Monte Carlo method for seismic inversion and uncertainty quantification. Geophysics 84(6), R1003–R1020 (2019)

    Article  Google Scholar 

  34. Tong, X.T., Morzfeld, M., Marzouk, Y.M.: MALA-within-Gibbs samplers for high-dimensional distributions with sparse conditional structure. SIAM J. Sci. Comput. 42(3), A1765–A1788 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

A. Rahunanthan research was supported by NIFA/USDA through Central State University’s Evans-Allen Research Program.

All the numerical simulations presented in this paper were performed on the GPU Computing cluster housed in the Department of Mathematics and Computer Science at Central State University.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX, 75080, USA

    Abdullah Al Mamun, Alsadig Ali & Felipe Pereira

  2. Institute of Natural Sciences, United International University, Dhaka, Bangladesh

    Abdullah Al-Mamun

  3. Department of Mathematics and Computer Science, Central State University, Wilberforce, OH, 45384, USA

    Arunasalam Rahunanthan

Authors
  1. Abdullah Al Mamun
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  2. Alsadig Ali
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  3. Abdullah Al-Mamun
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  4. Felipe Pereira
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  5. Arunasalam Rahunanthan
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Corresponding author

Correspondence to Arunasalam Rahunanthan .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Italy

    Beniamino Murgante

  3. University of Minho, Braga, Portugal

    Ana Maria A. C. Rocha

  4. University of Cagliari, Cagliari, Italy

    Chiara Garau

  5. University of Basilicata, Potenza, Italy

    Francesco Scorza

  6. University of Massachusetts Medical School, Worcester, MA, USA

    Yeliz Karaca

  7. Polytechnic University of Bari, Bari, Italy

    Carmelo M. Torre

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Mamun, A.A., Ali, A., Al-Mamun, A., Pereira, F., Rahunanthan, A. (2023). The Multiscale Maximum Change Algorithm for Subsurface Characterization. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2023 Workshops. ICCSA 2023. Lecture Notes in Computer Science, vol 14105. Springer, Cham. https://doi.org/10.1007/978-3-031-37108-0_8

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  • DOI: https://doi.org/10.1007/978-3-031-37108-0_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-37107-3

  • Online ISBN: 978-3-031-37108-0

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