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p-kernel Stein variational gradient descent for data assimilation and history matching. (English) Zbl 1465.86017

Summary: A Bayesian method of inference known as “Stein variational gradient descent” was recently implemented for data assimilation problems, under the heading of “mapping particle filter”. In this manuscript, the algorithm is applied to another type of geoscientific inversion problems, namely history matching of petroleum reservoirs. In order to combat the curse of dimensionality, the commonly used Gaussian kernel, which defines the solution space, is replaced by a p-kernel. In addition, the ensemble gradient approximation used in the mapping particle filter is rectified, and the data assimilation experiments are re-run with more relevant settings and comparisons. Our experimental results in data assimilation are rather disappointing. However, the results from the subsurface inverse problem show more promise, especially as regards the use of p-kernels.

MSC:

86A32 Geostatistics
86A22 Inverse problems in geophysics

Software:

AdaGrad; FADBAD++
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[1] Anterion F, Eymard R, Karcher B (1989) Use of parameter gradients for reservoir history matching. In: SPE symposium on reservoir simulation. Society of Petroleum Engineers
[2] Bendtsen C, Stauning O (1996) Fadbad, a flexible c++ package for automatic differentiation. Technical report, Technical Report IMM-REP-1996-17, Department of Mathematical Modelling
[3] Bocquet, M.; Sakov, P., An iterative ensemble Kalman smoother, Q J R Meteorol Soc, 140, 682, 1521-1535 (2014)
[4] Carrassi, A.; Bocquet, M.; Bertino, L.; Evensen, G., Data assimilation in the geosciences: an overview of methods, issues, and perspectives, Wiley Interdiscip Rev Clim Change, 9, 5, e535 (2018)
[5] Chavent, G.; Dupuy, M.; Lemmonier, P., History matching by use of optimal theory, Soc Petrol Eng J, 15, 1, 74-86 (1975)
[6] Chen P, Wu K, Chen J, O’Leary-Roseberry T, Ghattas O (2019) Projected stein variational newton: a fast and scalable Bayesian inference method in high dimensions. arXiv:1901.08659
[7] Chen, Y.; Oliver, DS, Ensemble randomized maximum likelihood method as an iterative ensemble smoother, Math Geosci, 44, 1, 1-26 (2012)
[8] Chen, Y.; Oliver, DS, Levenberg-Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification, Comput Geosci, 17, 689-703 (2013) · Zbl 1382.65031
[9] Chwialkowski K, Strathmann H, Gretton A (2016) A kernel test of goodness of fit. In: Proceedings of the 33rd international conference on machine learning. JMRL
[10] de Moraes, RJ; Rodrigues, JR; Hajibeygi, H.; Jansen, JD, Computing derivative information of sequentially coupled subsurface models, Comput Geosci, 22, 6, 1527-1541 (2018) · Zbl 1404.86012
[11] Detommaso G, Cui T, Spantini A, Marzouk Y, Scheichl R (2018) A stein variational Newton method. In: 32nd Conference on neural information processing systems, Montreal, Canada. NeurIPS 2018
[12] Duchi, J.; Hazan, E.; Singer, Y., Adaptive subgradient methods for online learning and stochastic optimization, J Mach Learn Res, 12, 2121-2159 (2011) · Zbl 1280.68164
[13] El Moshely, TA; Marzouk, YM, Bayesian inference with optimal maps, J Comput Phys, 231, 23, 7815-7850 (2012) · Zbl 1318.62087
[14] Emerick, A.; Reynolds, A., Ensemble smoother with multiple data assimilation, Comput Geosci, 55, 3-15 (2013)
[15] Evensen, G., Sampling strategies and square root analysis schemes for the EnKF, Ocean Dyn, 54, 6, 539-560 (2004)
[16] Feng Y, Wang D, Liu Q (2017) Learning to draw samples with amortized stein variational gradient descent. arXiv:1707.06626v2
[17] Francois D, Wertz V, Verleysen M (2005) About locality of kernels in high-dimensional spaces. In: International symposium on applied stochastic models and data analysis. ASDMA
[18] Gao G, Jiang H, Van Hagen P, Vink JC, Wells T (2017) A gauss-newton trust region solver for large scale history matching problems. In: SPE reservoir simulation conference, 20-22 Feb, Montgomery, Texas. SPE-182602
[19] Han J, Liu Q (2018) Stein variational gradient descent without gradient. arXiv:1806.02775v1
[20] Hunt, BR; Kalnay, E.; Kostelich, EJ; Ott, E.; Patil, DJ; Sauer, T.; Szunyogh, I.; Yorke, JA; Zimin, AV, Four-dimensional ensemble Kalman filtering, Tellus A, 56, 4, 273-277 (2004)
[21] Jansen, JD, SimSim: a simple reservoir simulator (2011), Delft: Departement of Geotechnology, TU, Delft
[22] Jazwinski, AH, Stochastic processes and filtering theory (1970), London: Academic Press, London · Zbl 0203.50101
[23] Kitanidis, PK, Quasi-linear geostatistical theory for inversing, Water Resour Res, 31, 10, 2411-2419 (1995)
[24] Liu Q (2017) Stein variational gradient descent as gradient flow. In: 31st Conference on neural information processing systems. NIPS
[25] Liu Q, Wang D (2016) Stein variational gradient descent: a general purpose Bayesian inference algorithm. In: Lee D, Sugiyama M, Luxburg U, Guyon I, Garnett R (eds) Advances in neural information processing systems, vol 29. Curran Associates, Inc. https://proceedings.neurips.cc/paper/2016/file/b3ba8f1bee1238a2f37603d90b58898d-Paper.pdf
[26] Liu Q, Lee JD, Jordan M (2016) A kernelized stein discrepancy for goodness of fit tests. In: Proceedings of the 33rd international conference on machine learning. JMRL
[27] Lorenz, EN, Deterministic nonperiodic flow, J Atmos Sci, 20, 2, 130-141 (1963) · Zbl 1417.37129
[28] Lorenz EN (1996) Predictability: a problem partly solved. In: Proceedings ECMWF seminar on predictability, vol 1, pp 1-18, Reading, UK
[29] Marzouk, Y.; Moselhy, T.; Parno, M.; Spantini, A., Sampling via measure transport: an introduction, 785-825 (2017), Cham: Springer International Publishing, Cham
[30] Minh, HQ, Some properties of gaussian reproducing kernel hilbert spaces and their implications for function approximation and learning theory, Constr Approx, 32, 2, 307-338 (2010) · Zbl 1204.68157
[31] Oliver DS, He N, Reynolds AC (1996) Conditioning permeability fields to pressure data. In: Conference proceedings, ECMOR V - 5th European conference on the mathematics of oil recovery, Sep 1996, cp-101-00023. European Association of Geoscientists & Engineers (EAGE). doi:10.3997/2214-4609.201406884
[32] Oliver DS, Reynolds AC, Liu N (2008) Inverse theory for petroleum reservoir characterization and history matching. Cambridge University Press, Cambridge
[33] Ott, E.; Hunt, BR; Szunyogh, I.; Zimin, AV; Kostelich, EJ; Corazza, M.; Kalnay, E.; Patil, DJ; Yorke, JA, A local ensemble Kalman filter for atmospheric data assimilation, Tellus A, 56, 5, 415-428 (2004)
[34] Pulido, M.; van Leeuwen, PJ, Sequential Monte Carlo with kernel embedded mappings: the mapping particle filter, J Comput Phys, 396, 400-415 (2019) · Zbl 1452.65009
[35] Pulido, M.; van Leeuwen, PJ; Posselt, DJ; Rodrigues, JMF; Cardoso, PJS; Monteiro, J.; Lam, R.; Krzhizhanovskaya, VV; Lees, MH; Dongarra, JJ; Sloot, PM, Kernel embedded nonlinear observational mappings in the variational mapping particle filter, Computational science—ICCS, 141-155 (2019), Cham: Springer International Publishing, Cham
[36] Reich S (2013) A guided sequential Monte Carlo method for theassimilationof data into stochastic dynamical systems. In: Johann A, Kruse HP, Rupp F, Schmitz S (eds) Recent trends in dynamical systems. Springer proceedings in mathematics & statistics, vol 35. Springer, Basel · Zbl 1314.37059
[37] Rodrigues, JRP, Calculating derivatives for automatic history matching, Comput Geosci, 10, 1, 119-136 (2006) · Zbl 1096.65054
[38] Sakov, P.; Oke, PR, Implications of the form of the ensemble transformation in the ensemble square root filters, Mon Weather Rev, 136, 3, 1042-1053 (2008)
[39] Sakov, P.; Oliver, DS; Bertino, L., An iterative EnKF for strongly nonlinear systems, Mon Weather Rev, 140, 6, 1988-2004 (2012)
[40] Silverman, BW, Density estimation for statistics and data analysis (1986), Boca Raton: Chapman and Hall, Boca Raton
[41] Skauvold, J.; Eidsvik, J.; van Leeuwen, PJ; Amezcua, J., A revised implicit equal-weights particle filter, Q J R Meteorol Soc (2019)
[42] Stein C (1972) A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proceedings of the sixth Berkeley symposium on mathematical statistics and probability, vol 2: probability theory. The Regents of the University of California · Zbl 0278.60026
[43] Stordal, A., Iterative Bayesian inversion with gaussian mixtures: finite sample implementation and large sample asymptotics, Comput Geosci, 19, 1, 1-15 (2015) · Zbl 1327.62348
[44] Stordal, A.; Elsheikh, A., Iterative ensemble smoothers in the annealed importance sampling framework, Adv Water Resour, 86, 231-239 (2015)
[45] Stordal, AS; Karlsen, HA, Large sample properties of the adaptive gaussian mixture filter, Mon Weather Rev, 145, 7, 2533-2553 (2017)
[46] Vlasov AA (1961) Many-particle theory and its application to plasma. Gordon & Breach Science Publishers, Inc
[47] Wikle, CK; Berliner, LM, A Bayesian tutorial for data assimilation, Physica D, 230, 1-2, 1-16 (2007) · Zbl 1113.62032
[48] Zhang J, Zhang R, Chen C (2018) Stochastic particle-optimization sampling and the non-asymptotic convergence theory. arXiv:1809.01293v2
[49] Zhou, D., Derivative reproducing properties for kernel methods in learning theory, Journal od computational and Applied Mathematics, 220, 456-463 (2008) · Zbl 1152.68049
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