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International Journal for Uncertainty Quantification
VARIABLE-SEPARATION BASED ITERATIVE ENSEMBLE SMOOTHER FOR BAYESIAN INVERSE PROBLEMS IN ANOMALOUS DIFFUSION REACTION MODELS
College of Mathematics and Econometrics, Hunan University 1, Changsha 410082, China
School of Mathematical Sciences, Tongji University, Shanghai 200092, China
College of Mathematics and Econometrics, Hunan University 1, Changsha 410082, China
The iterative ensemble smoother (IES) has been widely used to estimate parameters and states of dynamic models
where the data are collected at all observation steps simultaneously. A large number of IES ensemble samples may
be required in the estimation. This implies that we need to repeatedly compute the forward model corresponding to
the ensemble samples. This leads to slow efficiency for large-scale and strongly nonlinear models. To accelerate the
posterior inference in the estimation, a low rank approximation using a variable-separation (VS) method is presented to reduce the cost of computing the forward model. It will be efficient to construct a surrogate model based on the low rank approximation, which gives a separated representation of the solution for the stochastic partial differential equations (SPDEs). The separated representation is the product of deterministic basis functions and stochastic basis functions. For the anomalous diffusion reaction equations, the solution of the next moment depends on all of the previous moments, and this causes expensive computation for the Bayesian inverse problem. The presented VS can avoid this process through a few deterministic basis functions. The surrogate model can work well as the iteration moves on because the stochastic basis becomes more accurate when the uncertainty of random parameters decreases. To enhance the applicability in Bayesian inverse problems, we apply the VS-based IES method to complex structure patterns, which can be parameterized by discrete cosine transform (DCT). The post-processing technique based on a regularization method is employed after the iterations to improve the connectivity of the main features. In the paper, we focus on the time fractional diffusion reaction models in porous media and investigate their Bayesian inverse problems using the VS-based IES. A few numerical examples are presented to show the performance of the proposed IES method by taking account of structure inversion in permeability fields, parameters in permeability and reaction fields, and source functions.
Oldham, K. and Spanier, J., The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 111, Amsterdam: Elsevier, 1974.
Wyss, W., The Fractional Diffusion Equation, J. Math. Phys., 27(11):2782-2785, 1986.
Miller, K.S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: Wiley, 1993.
Li, C., Zhao, Z., and Chen, Y., Numerical Approximation of Nonlinear Fractional Differential Equations with Subdiffusion and Superdiffusion, Comput. Math. Appl., 62(3):855-875, 2011.
Miller, L. and Yamamoto, M., Coefficient Inverse Problem for a Fractional Diffusion Equation, Inverse Probl., 29(7):075013, 2013.
Hansen, P.C., Analysis of Discrete Ill-Posed Problems by Means of the l-Curve, SIAMRev., 34(4):561-580, 1992.
Haber, E., Ascher, U.M., and Oldenburg, D., On Optimization Techniques for Solving Nonlinear Inverse Problems, Inverse Probl, 16(5):1263, 2000.
Jiang, L. and Ou, N., Bayesian Inference Using Intermediate Distribution based on Coarse Multiscale Model for Time Fractional Diffusion Equations, Multiscale Model. Simul., 16(1):327-355, 2018.
Ou, N., Jiang, L., and Lin, G., A New Bi-Fidelity Model Reduction Method for Bayesian Inverse Problems, Int. J. Numer. Methods Eng., submitted, 2018.
Evensen, G., Sequential Data Assimilation with a Nonlinear Quasi-Geostrophic Model Using Monte Carlo Methods to Forecast Error Statistics, J. Geophys. Res.: Oceans, 99(C5):10143-10162, 1994.
Chang, H., Liao, Q., and Zhang, D., Surrogate Model based Iterative Ensemble Smoother for Subsurface Flow Data Assimilation, Adv. Water Res, 100:96-108, 2017.
Iglesias, M.A., Law, K.J., and Stuart, A.M., Evaluation of Gaussian Approximations for Data Assimilation in Reservoir Models, Comput. Geosci., 17(5):851-885, 2013.
Ernst, O.G., Sprungk, B., and Starkloff, H.J., Bayesian Inverse Problems and Kalman Filters, in Extraction of Quantifiable Information from Complex Systems, pp. 133-159, Berlin: Springer, 2014.
Emerick, A.A. and Reynolds, A.C., Ensemble Smoother with Multiple Data Assimilation, Comput. Geosci., 55:3-15,2013.
Chen, Y. and Oliver, D.S., Ensemble Randomized Maximum Likelihood Method as an Iterative Ensemble Smoother, Math. Geosci, 44(1):1-26,2012.
Tarantola, A., Inverse Problem Theory and Methods for Model Parameter Estimation, vol. 89, Philadelphia: SIAM, 2005.
Chen, Y. and Oliver, D.S., Levenberg-Marquardt Forms of the Iterative Ensemble Smoother for Efficient History Matching and Uncertainty Quantification, Comput. Geosci., 17(4):689-703, 2013.
Oliver, D.S. and Chen, Y., Recent Progress on Reservoir History Matching: A Review, Comput. Geosci., 15(1):185-221,2011.
Li, J. and Xiu, D., A Generalized Polynomial Chaos based Ensemble Kalman Filter with High Accuracy, J. Comput. Phys, 228(15):5454-5469,2009.
Regis, R.G. and Shoemaker, C.A., A Stochastic Radial Basis Function Method for the Global Optimization of Expensive Functions, INFORMS J. Comput, 19(4):497-509,2007.
Li, Q. and Jiang, L., A Novel Variable-Separation Method based on Sparse Representation for Stochastic Partial Differential Equations, J Sci. Comput., 39(6):A2879-A2910, 2017.
Jafarpour, B. and McLaughlin, D.B., History Matching with an Ensemble Kalman Filter and Discrete Cosine Parameterization, Comput. Geosci., 12(2):227-244,2008.
Zhao, Y., Forouzanfar, F., and Reynolds, A.C., History Matching of Multi-Facies Channelized Reservoirs Using ES-MDA with Common Basis DCT, Computational Geosci., 21(5-6):1343-1364, 2017.
Vo, H.X. and Durlofsky, L.J., A New Differentiable Parameterization based on Principal Component Analysis for the Low-Dimensional Representation of Complex Geological Models, Math. Geosci., 46(7):775-813,2014.
Vo, H.X. and Durlofsky, L.J., Data Assimilation and Uncertainty Assessment for Complex Geological Models Using a New PCA-Based Parameterization, Comput. Geosci., 19(4):747-767, 2015.
Gu, Y. and Oliver, D.S., An Iterative Ensemble Kalman Filter for Multiphase Fluid Flow Data Assimilation, SPEJ., 12(4):438-446, 2007.
Li, G., Gu, W., and Jia, X., Numerical Inversions for Space-Dependent Diffusion Coefficient in the Time Fractional Diffusion Equation, J. Inverse Ill-PosedProbl., 20(3):339-366, 2012.
Ba, Y., Jiang, L., and Ou, N., A Two-Stage Ensemble Kalman Filter based on Multiscale Model Reduction for Inverse Problems in Time Fractional Diffusion-Wave Equations, J. Comput. Phys, 374:300-330, 2018.
Rozza, G., Huynh, D.B.P., and Patera, A.T., Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations, Arch. Comput. Methods Eng., 15(3):1,2007.
Jiang, L. and Li, Q., Reduced Multiscale Finite Element Basis Methods for Elliptic PDEs with Parameterized Inputs, J. Comput. Appl. Math., 301:101-120, 2016.
Yan, L. and Guo, L., Stochastic Collocation Algorithms Using F1-Minimization for Bayesian Solution of Inverse Problems, J. Sci. Comput., 37(3):A1410-A1435, 2015.
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