Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Stochastic methods for the prediction of complex multiscale phenomena

Authors: James Glimm and David Sharp
Journal: Quart. Appl. Math. 56 (1998), 741-765
MSC: Primary 76F25; Secondary 62A01, 76F65, 76M25, 76S05
DOI: https://doi.org/10.1090/qam/1668736
MathSciNet review: MR1668736
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to develop a general framework for the prediction of complex multiscale phenomena and to illustrate this framework through comparison to two examples of current interest to the authors. Prediction involves a two-step process of inverse prediction to describe the system, given observations of its behavior, and forward prediction, to specify system behavior, given its description.

References [Enhancements On Off] (What's this?)

  • [1] K. Abbaspour, R. Schulin, M. Th. van Genuchten, and E. Schlappi, Accounting for uncertainty in geostatistical parameters within a stochastic simulation, Technical report, Swiss Federal Institute of Technology, Department of Soil Protection, 1997
  • [2] K. Abbaspour, R. Schulin, M. Th. van Genuchten, and E. Schlappi, Procedures for uncertainty analysis applied to a landfill leacheate plume, Technical report, Swiss Federal Institute of Technology, Department of Soil Protection, 1997
  • [3] U. Alon, J. Hecht, D. Ofer, and D. Shvarts, Power laws and similarity of Rayleigh-Taylor and Richtmyer-Meshkov mixing fronts at all density ratios, Phys. Rev. Lett. 74, 534-538 (1995)
  • [4] U. Alon and D. Shvarts, Two-phase flow model for Rayleigh-Taylor and Richtmyer-Meshkov mixing, Proceedings of the Fifth International Workshop on Compressible Turbulent Mixing (Stony Brook, NY, 1995) World Sci. Publ., River Edge, NJ, 1996, pp. 14–22. MR 1661878
  • [5] L. An, J. Glimm, D. H. Sharp, and Q. Zhang, Scale up of flow in porous media, in A. Burgeat, C. Carasso, S. Luckhaus, and A. Mikelic, editors, Mathematical Modeling of Flow Through Porous Media, World Scientific, 1995, pp. 26-44
  • [6] R. C. Bissel, Y. Sharma, and J. E. Killough, History matching using the method of gradients: Two case studies, SPE 28590, 1994. 69th Annual Technical Conference and Exhibition, New Orleans, 25-28 September, 1994
  • [7] Y. Chen, J. Glimm, D. H. Sharp, and Q. Zhang, A two-phase flow model of the Rayleigh-Taylor mixing zone, Phys. Fluids 8(3), 816-825 (1996)
  • [8] I-Liang Chern and Tai-Ping Liu, Convergence to diffusion waves of solutions for viscous conservation laws, Comm. Math. Phys. 110 (1987), no. 3, 503–517. MR 891950
  • [9] G. Christakos, Random Field Models in Earth Sciences, Academic Press, New York, 1992
  • [10] M. Christie, Private communication, 1997
  • [11] M. A. Christie and P. J. Clifford, A fast procedure for upscaling in compositional simulation, SPE 37986, 1997
  • [12] S. Cohn, An introduction to estimation theory, J. Meteorological Soc. Japan 77, 257-288 (1997)
  • [13] A. Datta-Gupta, L. W. Lake, and G. A. Pope, Characterizing heterogeneous permeable media with spatial statistics and tracer data using sequential simulated annealing, Mathematical Geology 27, 763-787 (1995)
  • [14] C. V. Deusch and A. G. Journel, Geostatistical Software Library and User's Guide, Oxford University Press, Oxford, 1992
  • [15] L. J. Durlofsky, R. A. Behrens, R. C. Jones, and A. Bernath, Scale up of heterogeneous three dimensional reservoir descriptions, SPE Proceedings (SPE 30709), October 1995. Presented at SPE Annual Technical Conference and Exhibition, Dallas, USA, 22-25 October 1995
  • [16] J. R. Eggleston, S. A. Rojstaczer, and J. J. Pierce, Identification of hydraulic conductivity structure in sand and gravel aquifers: Cape cod data set, Water Res. Research 32, 1209-1222 (1996)
  • [17] J. T. Fredrich and W. B. Lindquist, Statistical characterization of the three-dimensional microgeometry of porous media and correlation with macroscopic transport properties, Internat. J. Rock Mech. and Min. Sci. 34, 3-4 (1997)
  • [18] Nathan Freed, Dror Ofer, Dov Shvarts, and Steven A. Orszag, Two-phase flow analysis of self-similar turbulent mixing by Rayleigh-Taylor instability, Phys. Fluids A 3 (1991), no. 5, 912–918. MR 1205479, https://doi.org/10.1063/1.857967
  • [19] F. Furtado, J. Glimm, W. B. Lindquist, and F. Pereira, Multi length scale calculations of mixing length growth in tracer floods, in F. Kovarik, editor, Proceedings of the Emerging Technologies Conference, Institute for Improved Oil Recovery, Univ. Houston, Houston, TX, 1990, pp. 251-259
  • [20] Frederico Furtado and Felipe Pereira, Scaling analysis for two-phase immiscible flow in heterogeneous porous media, Comput. Appl. Math. 17 (1998), no. 3, 237–263. MR 1687873
  • [21] Alan E. Gelfand and Adrian F. M. Smith, Sampling-based approaches to calculating marginal densities, J. Amer. Statist. Assoc. 85 (1990), no. 410, 398–409. MR 1141740
  • [22] S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence 6, 721-741 (1984)
  • [23] W. R. Gilks, S. Richardson, and D. J. Spiegelhalter (eds.), Markov chain Monte Carlo in practice, Interdisciplinary Statistics, Chapman & Hall, London, 1996. MR 1397966
  • [24] J. Glimm, J. Grove, X.-L. Li, and D. C. Tan, Robust computational algorithms for dynamic interface tracking in three dimensions, Technical Report, in preparation, 1998
  • [25] James Glimm, John W. Grove, Xiao Lin Li, Keh-Ming Shyue, Yanni Zeng, and Qiang Zhang, Three-dimensional front tracking, SIAM J. Sci. Comput. 19 (1998), no. 3, 703–727. MR 1616658, https://doi.org/10.1137/S1064827595293600
  • [26] J. Glimm, H. Kim, D. Sharp, and T. Wallstrom, A stochastic analysis of the scale up problem for flow in porous media, Comput. Appl. Math. 17 (1998), no. 1, 67–79. MR 1687828
  • [27] James Glimm, Brent Lindquist, Felipe Pereira, and Ron Peierls, The multi-fractal hypothesis and anomalous diffusion, Mat. Apl. Comput. 11 (1992), no. 2, 189–207 (English, with English and Portuguese summaries). MR 1235739
  • [28] J. Glimm, W. B. Lindquist, F. Pereira, and Q. Zhang, A theory of macrodispersion for the scale up problem, Transport in Porous Media 13, 97-122 (1993)
  • [29] James Glimm, David Saltz, and David H. Sharp, Renormalization group solution of two-phase flow equations for Rayleigh-Taylor mixing, Phys. Lett. A 222 (1996), no. 3, 171–176. MR 1418599, https://doi.org/10.1016/0375-9601(96)00648-2
  • [30] Gui-Qiang G. Chen, Xiaolin Li, and David H. Sharp, Preface [Special issue dedicated to Professor James Glimm on the occasion of his seventy-fifth birthday], Acta Math. Sci. Ser. B Engl. Ed. 30 (2010), no. 2, i–ii. MR 2656545
  • [31] J. Glimm, D. Saltz, and D. H. Sharp, The statistical evolution of chaotic fluid mixing, Technical Report SUNYSB-AMS-97-05, SUNY at Stony Brook, 1997. Submitted to Phys. Rev. Lett.
  • [32] J. Glimm, D. Saltz, and D. H. Sharp, Two-phase modelling of a fluid mixing layer, J. Fluid Mech. 378 (1999), 119–143. MR 1671752, https://doi.org/10.1017/S0022112098003127
  • [33] J. Glimm, D. Saltz, and D. H. Sharp, Two-pressure two-phase flow, Advances in nonlinear partial differential equations and related areas (Beijing, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 124–148. MR 1690826
  • [34] J. Glimm and D. H. Sharp, Chaotic mixing as a renormalization-group fixed point, Phys. Rev. Lett. 64 (1990), no. 18, 2137–2139. MR 1049577, https://doi.org/10.1103/PhysRevLett.64.2137
  • [35] James Glimm and David H. Sharp, A random field model for anomalous diffusion in heterogeneous porous media, J. Statist. Phys. 62 (1991), no. 1-2, 415–424. MR 1105269, https://doi.org/10.1007/BF01020877
  • [36] James Glimm and David Sharp, Stochastic partial differential equations: selected applications in continuum physics, Stochastic partial differential equations: six perspectives, Math. Surveys Monogr., vol. 64, Amer. Math. Soc., Providence, RI, 1999, pp. 3–44. MR 1661762, https://doi.org/10.1090/surv/064/01
  • [37] J. Glimm and X.-L. Li, On the validation of the Sharp-Wheeler bubble merger model from experimental and computational data, Phys. Fluids 31, 2077-2085 (1988)
  • [38] J. Grove, R. Holmes, D. H. Sharp, Y. Yang, and Q. Zhang, Quantitative theory of Richtmyer-Meshkov instability, Phys. Rev. Lett 71(21), 3473-3476 (1993)
  • [39] W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika 87, 97-109 (1970)
  • [40] F. Hiller and G. Lieberman, Introduction to Operations Research, McGraw-Hill, New York, fifth edition, 1990
  • [41] John H. Holland, Adaptation in natural and artificial systems, University of Michigan Press, Ann Arbor, Mich., 1975. An introductory analysis with applications to biology, control, and artificial intelligence. MR 0441393
  • [42] Richard L. Holmes, John W. Grove, and David H. Sharp, Numerical investigation of Richtmyer-Meshkov instability using front tracking, J. Fluid Mech. 301 (1995), 51–64. MR 1361279, https://doi.org/10.1017/S002211209500379X
  • [43] A. G. Journel and Ch. J. Huijbregts, Mining Geostatistics, Academic Press, New York, 1978
  • [44] L. Lake and H. Carroll, editors, Reservoir Characterization, Academic Press, New York, 1986
  • [45] L. Lake, H. Carroll, and T. Wesson, editors, Reservoir Characterization II, Academic Press, New York, 1991
  • [46] J. Lam and J.-M. Delosme, An efficient simulated annealing schedule: Derivation, Report No. 8816, Yale Electrical Engineering Department, New Haven, CT, 1988
  • [47] J. Lam and J.-M. Delosme, An efficient simulated annealing schedule: Implementation and evaluation, Report No. 8817, Yale Electrical Engineering Department, New Haven, CT, 1988
  • [48] C. E. Leith, Nonlinear normal-mode initialization of numerical weather prediction models, Multiple time scales, Comput. Tech., vol. 3, Academic Press, Orlando, FL, 1985, pp. 59–71. MR 807603
  • [49] W. B. Lindquist, S.-M. Lee, D. A. Coker, K. W. Jones, and P. Spanne, Medial axis analysis of void structure in three-dimensional tomographic images of porous media, J. Geophys. Res. 101B, 8297-8310 (1996)
  • [50] Dimitri Mihalas and Barbara Weibel Mihalas, Foundations of radiation hydrodynamics, Oxford University Press, New York, 1984. MR 781346
  • [51] W. Mulder, S. Osher, and James A. Sethian, Computing interface motion in compressible gas dynamics, J. Comput. Phys. 100 (1992), no. 2, 209–228. MR 1167743, https://doi.org/10.1016/0021-9991(92)90229-R
  • [52] J. Tinsley Oden and Tarek I. Zohdi, Analysis and adaptive modeling of highly heterogeneous elastic structures, Comput. Methods Appl. Mech. Engrg. 148 (1997), no. 3-4, 367–391. MR 1465823, https://doi.org/10.1016/S0045-7825(97)00032-7
  • [53] D. S. Oliver, L. B. Cunha, and A. C. Reynolds, Markov chain Monte Carlo methods for conditioning a permeability field to pressure data, Math. Geo. 29, 61-91 (1997)
  • [54] Bradley J. Plohr and David H. Sharp, Instability of accelerated elastic metal plates, Z. Angew. Math. Phys. 49 (1998), no. 5, 786–804. MR 1652201, https://doi.org/10.1007/s000330050121
  • [55] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Optimization by simulated annealing, Technical report, 1982
  • [56] M. B. Schneider, G. Dimonte, and B. Remington, Large and small scale structure in Rayleigh-Taylor mixing, Phys. Rev. Lett., 1997, Submitted
  • [57] G. A. F. Seber, Multivariate observations, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1984. MR 746474
  • [58] M. K. Sen, A. Datta-Gupta, P. L. Stoffa, L. Lake, and G. A. Pope, Stochastic reservoir modeling using simulated annealing and genetic algorithms, SPE Formation and Evaluation 10, 49-56 (1995)
  • [59] D. M. Snider and M. J. Andrews, Rayleigh-Taylor and shear driven mixing with an unstable thermal stratification, Phys. Fluids 6(10), 3324-3334 (1994)
  • [60] P. Spanne, J. F. Thovert, C. J. Jacquin, W. B. Lindquist, K. W. Jones, and P. M. Adler, Synchrotron computed microtomography of porous media: Topology and transports, Phys. Rev. Lett. 73, 2001-2004 (1994)
  • [61] D. W. Vasco and A. Datta-Gupta, Integrating multiphase production history in stochastic reservoir characterization, SPE Formation and Evaluation 12, 149-156 (1997)
  • [62] T. Wallstrom, S. Hou, D. H. Sharp, M. A. Christie, and L. J. Durlofsky, Improved reservoir simulation using upscaled relative permeabilities and flexible grids (in preparation)
  • [63] Yumin Yang, Qiang Zhang, and David H. Sharp, Small amplitude theory of Richtmyer-Meshkov instability, Phys. Fluids 6 (1994), no. 5, 1856–1873. MR 1270862, https://doi.org/10.1063/1.868245
  • [64] David L. Youngs, Modelling turbulent mixing by Rayleigh-Taylor instability, Phys. D 37 (1989), no. 1-3, 270–287. Advances in fluid turbulence (Los Alamos, NM, 1988). MR 1024395, https://doi.org/10.1016/0167-2789(89)90135-8
  • [65] D. L. Youngs, Three-dimensional numerical simulation of turbulent mixing by Rayleigh-Taylor instability, Phys. Fluids A, 3, 1312-1319 (1991)
  • [66] Qiang Zhang, Validation of the chaotic mixing renormalization group fixed point, Phys. Lett. A 151 (1990), no. 1-2, 18–22. MR 1085169, https://doi.org/10.1016/0375-9601(90)90839-G
  • [67] Q. Zhang and M. J. Graham, A numerical and theoretical study of Richtmyer-Meshkov instability driven by cylindrical shocks, Report No. SUNYSB-AMS-96-15, State Univ. of New York at Stony Brook, 1996
  • [68] Q. Zhang and M. J. Graham, Scaling laws for unstable interfaces driven by strong shocks in cylindrical geometry, Physical Review Letters, 1997, Submitted
  • [69] Qiang Zhang and Sung-Ik Sohn, An analytical nonlinear theory of Richtmyer-Meshkov instability, Phys. Lett. A 212 (1996), no. 3, 149–155. MR 1378608, https://doi.org/10.1016/0375-9601(96)00021-7
  • [70] Q. Zhang and S.-I. Sohn, Padé approximation to an interfacial fluid mixing problem, Appl. Math. Lett. 10 (1997), no. 5, 121–127. MR 1471327, https://doi.org/10.1016/S0893-9659(97)00094-3
  • [71] Qiang Zhang and Sung-Ik Sohn, Nonlinear theory of unstable fluid mixing driven by shock wave, Phys. Fluids 9 (1997), no. 4, 1106–1124. MR 1437567, https://doi.org/10.1063/1.869202
  • [72] Tarek I. Zohdi, J. Tinsley Oden, and Gregory J. Rodin, Hierarchical modeling of heterogeneous bodies, Comput. Methods Appl. Mech. Engrg. 138 (1996), no. 1-4, 273–298. MR 1422334, https://doi.org/10.1016/S0045-7825(96)01106-1

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 76F25, 62A01, 76F65, 76M25, 76S05

Retrieve articles in all journals with MSC: 76F25, 62A01, 76F65, 76M25, 76S05

Additional Information

DOI: https://doi.org/10.1090/qam/1668736
Article copyright: © Copyright 1998 American Mathematical Society

Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website