# American Institute of Mathematical Sciences

December  2019, 1(4): 433-456. doi: 10.3934/fods.2019018

## On the incorporation of box-constraints for ensemble Kalman inversion

 1 Department of Statistics and Applied Probability, National University of Singapore, 119077, Singapore 2 Institute of Mathematics, University of Mannheim, 68131 Mannheim, Germany

* Corresponding author: Neil K. Chada

Published  December 2019

Fund Project: The first author is supported by a Singapore Ministry of Education Academic Research Funds Tier 2 grant [MOE2016-T2-2-135]. The third author is grateful to the DFG RTG1953 "Statistical Modeling of Complex Systems and Processes" for funding of this research.

The Bayesian approach to inverse problems is widely used in practice to infer unknown parameters from noisy observations. In this framework, the ensemble Kalman inversion has been successfully applied for the quantification of uncertainties in various areas of applications. In recent years, a complete analysis of the method has been developed for linear inverse problems adopting an optimization viewpoint. However, many applications require the incorporation of additional constraints on the parameters, e.g. arising due to physical constraints. We propose a new variant of the ensemble Kalman inversion to include box constraints on the unknown parameters motivated by the theory of projected preconditioned gradient flows. Based on the continuous time limit of the constrained ensemble Kalman inversion, we discuss a complete convergence analysis for linear forward problems. We adopt techniques from filtering, such as variance inflation, which are crucial in order to improve the performance and establish a correct descent. These benefits are highlighted through a number of numerical examples on various inverse problems based on partial differential equations.

Citation: Neil K. Chada, Claudia Schillings, Simon Weissmann. On the incorporation of box-constraints for ensemble Kalman inversion. Foundations of Data Science, 2019, 1 (4) : 433-456. doi: 10.3934/fods.2019018
##### References:
 [1] E. E. S. D. Albers, P.-A. Blacquart and A. M. Stuart, Ensemble Kalman Methods with Constraints, , In preparation, 2019. [2] N. Amor, G. Rasool and N. C. Bouaynaya, Constrained State Estimation - A Review, arXiv preprint, arXiv: 1807.03463. [3] J. L. Anderson, An adaptive covariance inflation error correction algorithm for ensemble filters, Tellus A, 59 (2007), 210-224. [4] ————, Spatially and temporally varying adaptive covariance inflation for ensemble filters, Tellus A, 61 (2009), 72–83. [5] K. Bergemann and S. Reich, A localization technique for ensemble Kalman filters, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 701-707. [6] ————, A mollified ensemble Kalman filter, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 1636–1643. [7] D. Bertsekas, Projected newton methods for optimization problems with simple constraints, SIAM Journal on Control and Optimization, 20 (1982), 221-246.  doi: 10.1137/0320018. [8] D. P. Bertsekas, Nonlinear Programming, Athena Scientific, Belmont, MA, 1999. [9] A. N. Bishop and P. Del Moral, On the stability of matrix-valued Riccati diffusions, Electron. J. Probab., 24 (2019), Paper No. 84, 40 pp, arXiv: 1808.00235. [10] D. Blömker, C. Schillings and P. Wacker, A strongly convergent numerical scheme from ensemble kalman inversion, SIAM Journal on Numerical Analysis, 56 (2018), 2537-2562.  doi: 10.1137/17M1132367. [11] D. Blömker, C. Schillings, P. Wacker and S. Weissmann, Well posedness and convergence analysis of the ensemble kalman inversion, Inverse Problems, 35 (2019), 085007, 32pp. doi: 10.1088/1361-6420/ab149c. [12] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, New York, NY, USA, 2004.  doi: 10.1017/CBO9780511804441. [13] N. K. Chada, M. A. Iglesias, L. Roininen and A. M. Stuart, Parameterizations for ensemble Kalman inversion, Inverse Problems, 34 (2018), 055009, 31pp. doi: 10.1088/1361-6420/aab6d9. [14] X. Chen, J. D. Lee, X. T. Tong and Y. Zhang, Statistical Inference for Model Parameters in Stochastic Gradient Descent, arXiv e-prints, 2016, arXiv: 1610.08637. [15] H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. [16] G. Evensen, The ensemble Kalman filter: Theoretical formulation and practical implementation, Ocean Dynamics, 53 (2003), 343-367. [17] G. Evensen, Data Assimilation: The Ensemble Kalman Filter, Earth and Environmental Science, Springer Berlin Heidelberg, 2009. doi: 10.1007/978-3-642-03711-5. [18] B. K. P. Hungerlander and F. Rendl, Regularization of inverse problems via box constrained minimization, arXiv preprint, arXiv: 1807.11316, (2018). [19] M. A. Iglesias, A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems, Inverse Problems, 32 (2016), 025002, 45pp. doi: 10.1088/0266-5611/32/2/025002. [20] M. A. Iglesias, K. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 045001, 20pp. doi: 10.1088/0266-5611/29/4/045001. [21] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160. Springer-Verlag, New York, 2005. [22] R. Kandepu, L. Imsland and B. A. Foss, Constrained State Estimation Using the Unscented Kalman Filter, 16th Mediterranean Conference on Control and Automation, 2008. [23] D. Kelly, K. Law and A. M. Stuart, Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time, Nonlinearity, 27 (2014), 2579-2604.  doi: 10.1088/0951-7715/27/10/2579. [24] D. Kelly, A. J. Majda and X. T. Tong, Concrete Ensemble Kalman Filters with Rigorous Catastrophic Filter Divergence, Proceedings of the National Academy of Sciences, 2015. [25] E. Kwiatkowski and J. Mandel, Convergence of the square root ensemble Kalman filter in the large ensemble limit, SIAM/ASA Journal on Uncertainty Quantification, 3 (2015), 1-17.  doi: 10.1137/140965363. [26] K. Law, A. M. Stuart and K. Zygalakis, Data Assimilation: A Mathematical Introduction, Texts in Applied Mathematics, 62. Springer, Cham, 2015. doi: 10.1007/978-3-319-20325-6. [27] F. Le Gland, V. Monbet and V.-D. Tran, Large sample asymptotics for the ensemble Kalman filter, The Oxford Handbook of Nonlinear Filtering, 598–631, Oxford Univ. Press, Oxford, 2011. [28] G. Li and A. C. Reynolds, Iterative Ensemble Kalman Filters for Data Assimilation, Society of Petroleum Engineers, 2009. [29] D. M. Livings, S. L. Dance and N. K. Nichols, Unbiased ensemble square root filters, Physica D: Nonlinear Phenomena, 237 (2008), 1021-1028.  doi: 10.1016/j.physd.2008.01.005. [30] S. Reich, Data assimilation, Acta Numer., 28 (2019), 635-711.  doi: 10.1017/S0962492919000011. [31] S. Reich and C. J. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.  doi: 10.1017/CBO9781107706804. [32] C. Schillings and A. M. Stuart, Analysis of the ensemble Kalman filter for inverse problems, SIAM Journal on Numerical Analysis, 55 (2017), 1264-1290.  doi: 10.1137/16M105959X. [33] M. Schmidt, D. Kim and S. Sra, Projected Newton-type Methods in Machine Learning, 2013. [34] V. Shikhman and O. Stein, Constrained optimization: Projected gradient flows, Journal of Optimization Theory and Applications, 140 (2009), 117-130.  doi: 10.1007/s10957-008-9445-8. [35] D. Simon, Kalman filtering with state constraints: A survey of linear and nonlinear algorithms, IET Control Theory Appl., 4 (2010), 1303-1318.  doi: 10.1049/iet-cta.2009.0032. [36] A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.  doi: 10.1017/S0962492910000061. [37] M. K. Tippett, J. L. Anderson, C. H. Bishop, T. M. Hamill and J. S. Whitaker, Ensemble square root filters, Monthly Weather Review, 131 (2003), 1485-1490. [38] D. Wang, Y. Chen and X. Cai, State and Parameter Estimation of Hydrologic Models Using the Constrained Ensemble Kalman Filter, Monthly Resources Research, 2009.

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##### References:
 [1] E. E. S. D. Albers, P.-A. Blacquart and A. M. Stuart, Ensemble Kalman Methods with Constraints, , In preparation, 2019. [2] N. Amor, G. Rasool and N. C. Bouaynaya, Constrained State Estimation - A Review, arXiv preprint, arXiv: 1807.03463. [3] J. L. Anderson, An adaptive covariance inflation error correction algorithm for ensemble filters, Tellus A, 59 (2007), 210-224. [4] ————, Spatially and temporally varying adaptive covariance inflation for ensemble filters, Tellus A, 61 (2009), 72–83. [5] K. Bergemann and S. Reich, A localization technique for ensemble Kalman filters, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 701-707. [6] ————, A mollified ensemble Kalman filter, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 1636–1643. [7] D. Bertsekas, Projected newton methods for optimization problems with simple constraints, SIAM Journal on Control and Optimization, 20 (1982), 221-246.  doi: 10.1137/0320018. [8] D. P. Bertsekas, Nonlinear Programming, Athena Scientific, Belmont, MA, 1999. [9] A. N. Bishop and P. Del Moral, On the stability of matrix-valued Riccati diffusions, Electron. J. Probab., 24 (2019), Paper No. 84, 40 pp, arXiv: 1808.00235. [10] D. Blömker, C. Schillings and P. Wacker, A strongly convergent numerical scheme from ensemble kalman inversion, SIAM Journal on Numerical Analysis, 56 (2018), 2537-2562.  doi: 10.1137/17M1132367. [11] D. Blömker, C. Schillings, P. Wacker and S. Weissmann, Well posedness and convergence analysis of the ensemble kalman inversion, Inverse Problems, 35 (2019), 085007, 32pp. doi: 10.1088/1361-6420/ab149c. [12] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, New York, NY, USA, 2004.  doi: 10.1017/CBO9780511804441. [13] N. K. Chada, M. A. Iglesias, L. Roininen and A. M. Stuart, Parameterizations for ensemble Kalman inversion, Inverse Problems, 34 (2018), 055009, 31pp. doi: 10.1088/1361-6420/aab6d9. [14] X. Chen, J. D. Lee, X. T. Tong and Y. Zhang, Statistical Inference for Model Parameters in Stochastic Gradient Descent, arXiv e-prints, 2016, arXiv: 1610.08637. [15] H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. [16] G. Evensen, The ensemble Kalman filter: Theoretical formulation and practical implementation, Ocean Dynamics, 53 (2003), 343-367. [17] G. Evensen, Data Assimilation: The Ensemble Kalman Filter, Earth and Environmental Science, Springer Berlin Heidelberg, 2009. doi: 10.1007/978-3-642-03711-5. [18] B. K. P. Hungerlander and F. Rendl, Regularization of inverse problems via box constrained minimization, arXiv preprint, arXiv: 1807.11316, (2018). [19] M. A. Iglesias, A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems, Inverse Problems, 32 (2016), 025002, 45pp. doi: 10.1088/0266-5611/32/2/025002. [20] M. A. Iglesias, K. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 045001, 20pp. doi: 10.1088/0266-5611/29/4/045001. [21] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160. Springer-Verlag, New York, 2005. [22] R. Kandepu, L. Imsland and B. A. Foss, Constrained State Estimation Using the Unscented Kalman Filter, 16th Mediterranean Conference on Control and Automation, 2008. [23] D. Kelly, K. Law and A. M. Stuart, Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time, Nonlinearity, 27 (2014), 2579-2604.  doi: 10.1088/0951-7715/27/10/2579. [24] D. Kelly, A. J. Majda and X. T. Tong, Concrete Ensemble Kalman Filters with Rigorous Catastrophic Filter Divergence, Proceedings of the National Academy of Sciences, 2015. [25] E. Kwiatkowski and J. Mandel, Convergence of the square root ensemble Kalman filter in the large ensemble limit, SIAM/ASA Journal on Uncertainty Quantification, 3 (2015), 1-17.  doi: 10.1137/140965363. [26] K. Law, A. M. Stuart and K. Zygalakis, Data Assimilation: A Mathematical Introduction, Texts in Applied Mathematics, 62. Springer, Cham, 2015. doi: 10.1007/978-3-319-20325-6. [27] F. Le Gland, V. Monbet and V.-D. Tran, Large sample asymptotics for the ensemble Kalman filter, The Oxford Handbook of Nonlinear Filtering, 598–631, Oxford Univ. Press, Oxford, 2011. [28] G. Li and A. C. Reynolds, Iterative Ensemble Kalman Filters for Data Assimilation, Society of Petroleum Engineers, 2009. [29] D. M. Livings, S. L. Dance and N. K. Nichols, Unbiased ensemble square root filters, Physica D: Nonlinear Phenomena, 237 (2008), 1021-1028.  doi: 10.1016/j.physd.2008.01.005. [30] S. Reich, Data assimilation, Acta Numer., 28 (2019), 635-711.  doi: 10.1017/S0962492919000011. [31] S. Reich and C. J. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015.  doi: 10.1017/CBO9781107706804. [32] C. Schillings and A. M. Stuart, Analysis of the ensemble Kalman filter for inverse problems, SIAM Journal on Numerical Analysis, 55 (2017), 1264-1290.  doi: 10.1137/16M105959X. [33] M. Schmidt, D. Kim and S. Sra, Projected Newton-type Methods in Machine Learning, 2013. [34] V. Shikhman and O. Stein, Constrained optimization: Projected gradient flows, Journal of Optimization Theory and Applications, 140 (2009), 117-130.  doi: 10.1007/s10957-008-9445-8. [35] D. Simon, Kalman filtering with state constraints: A survey of linear and nonlinear algorithms, IET Control Theory Appl., 4 (2010), 1303-1318.  doi: 10.1049/iet-cta.2009.0032. [36] A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.  doi: 10.1017/S0962492910000061. [37] M. K. Tippett, J. L. Anderson, C. H. Bishop, T. M. Hamill and J. S. Whitaker, Ensemble square root filters, Monthly Weather Review, 131 (2003), 1485-1490. [38] D. Wang, Y. Chen and X. Cai, State and Parameter Estimation of Hydrologic Models Using the Constrained Ensemble Kalman Filter, Monthly Resources Research, 2009.
Varying contour lines of the function $\Phi(x)$ defined in Example 2.2, with both the preconditioned descent direction in the unconstrained case and the projected preconditioned descent direction
Transformed EnKF estimation in comparison to the EnKF estimation and the projected EnKF estimation. $J = 5$ particles have been simulated
Ensemble spread in the transformed EnKF in comparison to the EnKF and the projected EnKF. $J = 5$ particles have been simulated
KKT-Residuals and difference of the misfit functional and the global minimum in the transformed EnKF in comparison to the projected EnKF. $J = 5$ particles have been simulated
Transformed EnKF estimation in comparison to the EnKF estimation and the projected EnKF estimation. $J = 5$ particles have been simulated
Difference of the misfit functional and the global minimum in the transformed EnKF in comparison to the EnKF and the projected EnKF. $J = 5$ particles have been simulated
Transformed EnKF parameter estimation in comparison to the EnKF estimation and the projected EnKF estimation. $J = 5$ particles have been simulated
Transformed EnKF observation estimation in comparison to the EnKF estimation and the projected EnKF estimation. $J = 5$ particles have been simulated
Ensemble spread in the transformed EnKF in comparison to the EnKF and the projected EnKF. $J = 5$ particles have been simulated
Difference of the misfit functional and the global minimum in the transformed EnKF in comparison to the projected EnKF. $J = 5$ particles have been simulated
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