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An adaptive ANOVA-based PCKF for high-dimensional nonlinear inverse modeling. (English) Zbl 1349.65025

Summary: The probabilistic collocation-based Kalman filter (PCKF) is a recently developed approach for solving inverse problems. It resembles the ensemble Kalman filter (EnKF) in every aspect – except that it represents and propagates model uncertainty by polynomial chaos expansion (PCE) instead of an ensemble of model realizations. Previous studies have shown PCKF is a more efficient alternative to EnKF for many data assimilation problems. However, the accuracy and efficiency of PCKF depends on an appropriate truncation of the PCE series. Having more polynomial chaos basis functions in the expansion helps to capture uncertainty more accurately but increases computational cost. Selection of basis functions is particularly important for high-dimensional stochastic problems because the number of polynomial chaos basis functions required to represent model uncertainty grows dramatically as the number of input parameters (random dimensions) increases. In classic PCKF algorithms, the PCE basis functions are pre-set based on users’ experience. Also, for sequential data assimilation problems, the basis functions kept in PCE expression remain unchanged in different Kalman filter loops, which could limit the accuracy and computational efficiency of classic PCKF algorithms. To address this issue, we present a new algorithm that adaptively selects PCE basis functions for different problems and automatically adjusts the number of basis functions in different Kalman filter loops. The algorithm is based on adaptive functional ANOVA (analysis of variance) decomposition, which approximates a high-dimensional function with the summation of a set of low-dimensional functions. Thus, instead of expanding the original model into PCE, we implement the PCE expansion on these low-dimensional functions, which is much less costly. We also propose a new adaptive criterion for ANOVA that is more suited for solving inverse problems. The new algorithm was tested with different examples and demonstrated great effectiveness in comparison with non-adaptive PCKF and EnKF algorithms.

MSC:

65C20 Probabilistic models, generic numerical methods in probability and statistics
62M20 Inference from stochastic processes and prediction
62J10 Analysis of variance and covariance (ANOVA)

Software:

EnKF
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Full Text: DOI

References:

[1] Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E., Equation of state calculations by fast computing machines, J. Chem. Phys., 21, 1087 (1953) · Zbl 1431.65006
[2] Geman, S.; Geman, D., Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell., PAMI-6, 721-741 (1984) · Zbl 0573.62030
[3] Kalman, R. E., A new approach to linear filtering and prediction problems, J. Basic Eng., 82, 35-45 (1960)
[4] Evensen, G., Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res., Oceans, 99, 10143-10162 (1994)
[5] Evensen, G., Data Assimilation: The Ensemble Kalman Filter (2009), Springer · Zbl 1395.93534
[6] Evensen, G., The ensemble Kalman filter: theoretical formulation and practical implementation, Ocean Dyn., 53, 343-367 (2003)
[7] Wiener, N., The homogeneous chaos, Am. J. Math., 60, 897-936 (1938) · JFM 64.0887.02
[8] Ghanem, R. G.; Spanos, P. D., Stochastic Finite Elements: A Spectral Approach (2003), Courier Dover Publications
[9] Xiu, D.; Karniadakis, G. E., Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys., 187, 137-167 (2003) · Zbl 1047.76111
[10] 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, 351-363 (1998)
[11] 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, 21925-21926 (1997)
[12] Xiu, D.; Hesthaven, J. S., High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27, 1118-1139 (2005) · Zbl 1091.65006
[13] Eldred, M.; Burkardt, J., Comparison of non-intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification, (47th AIAA Aerosp. Sci. Meet. New Horizons Forum Aerosp. Expo. (2009))
[14] Webster, M. D.; Tatang, M. A.; McRae, G. J., Application of the probabilistic collocation method for an uncertainty analysis of a simple ocean model (1996), MIT Technical Report
[15] Hockenberry, J. R.; Lesieutre, B. C., Evaluation of uncertainty in dynamic simulations of power system models: The probabilistic collocation method, IEEE Trans. Power Syst., 19, 1483-1491 (2004)
[16] Li, H.; Zhang, D., Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods, Water Resour. Res., 43 (2007)
[17] Li, W.; Lu, Z.; Zhang, D., Stochastic analysis of unsaturated flow with probabilistic collocation method, Water Resour. Res., 45 (2009)
[18] Saad, G.; Ghanem, R., Characterization of reservoir simulation models using a polynomial chaos-based ensemble Kalman filter, Water Resour. Res., 45 (2009)
[19] Li, J.; Xiu, D., A generalized polynomial chaos based ensemble Kalman filter with high accuracy, J. Comput. Phys., 228, 5454-5469 (2009) · Zbl 1280.93084
[20] Zeng, L.; Zhang, D., A stochastic collocation based Kalman filter for data assimilation, Comput. Geosci., 14, 721-744 (2010) · Zbl 1381.86028
[21] Zeng, L.; Chang, H.; Zhang, D., A probabilistic collocation-based Kalman filter for history matching, SPE J., 16, 294-306 (2011)
[22] Li, W.; Oyerinde, A.; Stern, D.; Wu, X.; Zhang, D., Probabilistic collocation based Kalman filter for assisted history matching—a case study, (SPE Reserv. Simul. Symp. (2011))
[23] Rabitz, H.; Alis, Ö. F., General foundations of high-dimensional model representations, J. Math. Chem., 25, 197-233 (1999) · Zbl 0957.93004
[24] Li, G.; Wang, S.-W.; Rosenthal, C.; Rabitz, H., High dimensional model representations generated from low dimensional data samples. I. mp-Cut-HDMR, J. Math. Chem., 30, 1-30 (2001) · Zbl 1023.81521
[25] Foo, J.; Karniadakis, G. E., Multi-element probabilistic collocation method in high dimensions, J. Comput. Phys., 229, 1536-1557 (2010) · Zbl 1181.65014
[26] Ma, X.; Zabaras, N., An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations, J. Comput. Phys., 229, 3884-3915 (2010) · Zbl 1189.65019
[27] Yang, X.; Choi, M.; Lin, G.; Karniadakis, G. E., Adaptive ANOVA decomposition of stochastic incompressible and compressible flows, J. Comput. Phys., 231, 1587-1614 (2012) · Zbl 1408.76428
[28] Burgers, G.; Jan van Leeuwen, P.; Evensen, G., Analysis scheme in the ensemble Kalman filter, Mon. Weather Rev., 126, 1719-1724 (1998)
[29] Whitaker, J. S.; Hamill, T. M., Ensemble data assimilation without perturbed observations, Mon. Weather Rev., 130, 1913-1924 (2002)
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