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Constrained probabilistic collocation method for uncertainty quantification of geophysical models

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Abstract

The traditional probabilistic collocation method (PCM) uses either polynomial chaos expansion (PCE) or Lagrange polynomials to represent the model output response. Since the PCM relies on the regularity of the response, it may generate nonphysical realizations or inaccurate estimations of the statistical properties under strongly nonlinear/unsmooth conditions. In this study, we develop a new constrained PCM (CPCM) to quantify the uncertainty of geophysical models accurately and efficiently, where the PCE coefficients are solved via inequality constrained optimization considering the physical constraints of model response, different from that in the traditional PCM where the PCE coefficients are solved using spectral projection or least-square regression. Through solute transport and multiphase flow tests in porous media, we show that the CPCM achieves higher accuracy for statistical moments as well as probability density functions, and produces more reasonable realizations than does the PCM, while the computational effort is greatly reduced compared to the Monte Carlo approach.

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References

  1. Babuska, I., Tempone, R., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic differential equations. SIAM J. Numer. Anal. 42, 800–825 (2004)

    Article  Google Scholar 

  2. Ballio, F., Guadagnini, A.: Convergence assessment of numerical Monte Carlo simulations in groundwater hydrology. Water Resour. Res. 40, W04603 (2004)

    Google Scholar 

  3. Bear, J.: Dynamics of fluids in porous media. Dover, New York (1972)

    Google Scholar 

  4. Bellin, A., Rubin, Y., Rinaldo, A.: Eulerian–Lagrangian approach for modeling of flow and transport in heterogeneous geological formations. Water Resour. Res. 30(11), 2913–2924 (1994)

    Article  Google Scholar 

  5. Berveiller, M., Sudret, B., Lemaire, M.: Stochastic finite elements: a non intrusive approach by regression. Eur. J. Comput. Mech. 15, 81–92 (2006)

    Google Scholar 

  6. Chang, H., Zhang, D.: A comparative study of stochastic collocation methods for flow in spatially correlated random fields. Commun. Comput. Phys. 6, 509–535 (2009)

    Google Scholar 

  7. Coats, K.H., Smith, B.D.: Dead-end pore volume and dispersion in porous media. SPE J 4(1), 73–84 (1964)

    Article  Google Scholar 

  8. Dagan, G.: Stochastic modeling of groundwater flow by unconditional and conditional probabilities: 2. The solute transport. Water Resour. Res. 18(4), 835–848 (1982)

    Article  Google Scholar 

  9. Genz, A., Keister, B.D.: Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. J. Comput. Appl. Math. 71, 299–309 (1996)

    Article  Google Scholar 

  10. Ghanem, R.: Scales of fluctuation and the propagation of uncertainty in random porous media. Water Resour. Res. 34(9), 2123–36 (1998)

    Article  Google Scholar 

  11. Ghanem, R., Spanos, S.: Stochastic finite element: A spectral approach. Springer, New York (1991)

    Book  Google Scholar 

  12. Hosder, S., Walters, R.W., Balch, M.: Efficient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables. In: Proceeding of the 48th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference (2007)

  13. Isukapalli, S.S., Roy, A., Georgopoulos, P.G.: Stochastic response surface methods (SRSMs) for uncertainty propagation: application to environmental and biological systems. Risk Anal. 18(3), 351–363 (1998)

    Article  Google Scholar 

  14. Janjić, T., McLaughlin, D., Cohn, S.E., Verlaan, M.: Conservation of mass and preservation of positivity with ensemble-type Kalman filter algorithms. Monthly Weather Rev. 142(2), 755–773 (2014)

    Article  Google Scholar 

  15. Le Maitre, O., Reagan, M., Najm, H., Ghanem, R., Knio, O.: A stochastic projection method for fluid flow: II. Random process. J. Comput. Phys. 181, 9–44 (2002)

    Article  Google Scholar 

  16. Le Maitre, O., Ghanem, R., Knio, O., Najm, H.: Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197, 28–57 (2004)

    Article  Google Scholar 

  17. Le Maitre, O., Knio, O.: Spectral methods for uncertainty quantification: with applications to computational fluid dynamics. Springer, New York (2010)

    Book  Google Scholar 

  18. Li, H., Zhang, D.: Probabilistic collocation method for flow in porous media: comparisons with other stochastic methods. Water Resour. Res. 43, W09409 (2007)

    Google Scholar 

  19. Li, H., Zhang, D.: Efficient and accurate quantification of uncertainty for multiphase flow with probabilistic collocation method. SPE J. 14(4), 665–679 (2009). SPE–114802–PA

    Article  Google Scholar 

  20. Li, W., Lu, Z., Zhang, D.: Stochastic analysis of unsaturated flow with probabilistic collocation method. Water Resour. Res. 45, W08425 (2009)

    Google Scholar 

  21. Liao, Q., Zhang, D.: Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location. Water Resour. Res. 49(12), 7911–7928 (2013)

    Article  Google Scholar 

  22. Liao, Q., Zhang, D.: Probabilistic collocation method for strongly nonlinear problems: 2. Transform by displacement Water Resour. Res. 50(11), 8736–8759 (2014)

  23. Lin, G., Tartakovsky, A.M.: An efficient, high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media. Adv. Water Resour. 32(5), 712–722 (2009)

    Article  Google Scholar 

  24. Mathelin, L., Hussaini, M.: A stochastic collocation algorithm for uncertainty analysis. NASA Tech Rep NASA/CR–2003–212153 (2003)

  25. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)

    Book  Google Scholar 

  26. Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse theory for petroleum reservoir characterization and history matching. Cambridge, New York (2008)

    Book  Google Scholar 

  27. Phale, H.A., Oliver, D.S.: Data assimilation using the constrained ensemble Kalman filter. SPE J. 16(2), 331–342 (2011)

    Article  Google Scholar 

  28. Shi, L., Yang, J., Zhang, D., Li, H.: Probabilistic collocation method for unconfined flow in heterogeneous media. J. Hydrol 365, 4–10 (2009)

    Article  Google Scholar 

  29. Shi, L., Zhang, D., Lin, L., Yang, J.: A multiscale probabilistic collocation method for subsurface flow in heterogeneous media. Water Resour. Res. 46, W11562 (2010)

    Google Scholar 

  30. Snyman, J.A.: Practical mathematical optimization. Springer, New York (2005)

    Google Scholar 

  31. Tatang, M.A., Pan, W., Prinn, R.G., McRae, G.J.: An efficient method for parametric uncertainty analysis of numerical geophysical models. J. Geophys. Res. 102(D18), 21925–21931 (1997)

    Article  Google Scholar 

  32. Wan, X., Karniadakis, G.E.: An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys 209, 617–642 (2005)

    Article  Google Scholar 

  33. Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)

    Article  Google Scholar 

  34. Xiu, D.: Numerical methods for stochastic computations: a spectral method approach. Princeton University, New Jersey (2010)

    Google Scholar 

  35. Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)

    Article  Google Scholar 

  36. Xiu, D., Karniadakis, G.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput 24(2), 619–644 (2002)

    Article  Google Scholar 

  37. Zeng, L., Chang, H., Zhang, D.: A probabilistic collocation-based Kalman filter for history matching. SPE J. 16(2), 294–306 (2011)

    Article  Google Scholar 

  38. Zhang, D.: Stochastic methods for flow in porous media: coping with uncertainties. Academic, San Diego (2002)

    Google Scholar 

  39. Zhang, D., Shi, L., Chang, H., Yang, J.: A comparative study of numerical approaches to risk assessment. Stoch. Environ. Res. Risk Assess 24, 971–984 (2010)

    Article  Google Scholar 

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Authors and Affiliations

  1. Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, CA, 90089, USA

    Qinzhuo Liao

  2. ERE & SKLTCS, College of Engineering, Peking University, Beijing, 100871, China

    Dongxiao Zhang

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  1. Qinzhuo Liao
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  2. Dongxiao Zhang
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Correspondence to Dongxiao Zhang.

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Liao, Q., Zhang, D. Constrained probabilistic collocation method for uncertainty quantification of geophysical models. Comput Geosci 19, 311–326 (2015). https://doi.org/10.1007/s10596-015-9471-1

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  • DOI: https://doi.org/10.1007/s10596-015-9471-1

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