
-
Previous Article
Partitioned integrators for thermodynamic parameterization of neural networks
- FoDS Home
- This Issue
-
Next Article
Quantum topological data analysis with continuous variables
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 |
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.
References:
[1] |
E. E. S. D. Albers, P.-A. Blacquart and A. M. Stuart, Ensemble Kalman Methods with Constraints, , In preparation, 2019. Google Scholar |
[2] |
N. Amor, G. Rasool and N. C. Bouaynaya, Constrained State Estimation - A Review, arXiv preprint, arXiv: 1807.03463. Google Scholar |
[3] |
J. L. Anderson, An adaptive covariance inflation error correction algorithm for ensemble filters, Tellus A, 59 (2007), 210-224. Google Scholar |
[4] |
————, Spatially and temporally varying adaptive covariance inflation for ensemble filters, Tellus A, 61 (2009), 72–83. Google Scholar |
[5] |
K. Bergemann and S. Reich, A localization technique for ensemble Kalman filters, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 701-707. Google Scholar |
[6] |
————, A mollified ensemble Kalman filter, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 1636–1643. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
show all references
References:
[1] |
E. E. S. D. Albers, P.-A. Blacquart and A. M. Stuart, Ensemble Kalman Methods with Constraints, , In preparation, 2019. Google Scholar |
[2] |
N. Amor, G. Rasool and N. C. Bouaynaya, Constrained State Estimation - A Review, arXiv preprint, arXiv: 1807.03463. Google Scholar |
[3] |
J. L. Anderson, An adaptive covariance inflation error correction algorithm for ensemble filters, Tellus A, 59 (2007), 210-224. Google Scholar |
[4] |
————, Spatially and temporally varying adaptive covariance inflation for ensemble filters, Tellus A, 61 (2009), 72–83. Google Scholar |
[5] |
K. Bergemann and S. Reich, A localization technique for ensemble Kalman filters, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 701-707. Google Scholar |
[6] |
————, A mollified ensemble Kalman filter, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 1636–1643. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |










[1] |
Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020 doi: 10.3934/fods.2020018 |
[2] |
Zi Xu, Siwen Wang, Jinjin Huang. An efficient low complexity algorithm for box-constrained weighted maximin dispersion problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 971-979. doi: 10.3934/jimo.2020007 |
[3] |
Håkon Hoel, Gaukhar Shaimerdenova, Raúl Tempone. Multilevel Ensemble Kalman Filtering based on a sample average of independent EnKF estimators. Foundations of Data Science, 2020, 2 (4) : 351-390. doi: 10.3934/fods.2020017 |
[4] |
George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003 |
[5] |
Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 |
[6] |
C. J. Price. A modified Nelder-Mead barrier method for constrained optimization. Numerical Algebra, Control & Optimization, 2020 doi: 10.3934/naco.2020058 |
[7] |
Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021006 |
[8] |
Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 987-999. doi: 10.3934/dcdss.2020381 |
[9] |
Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 |
[10] |
Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020446 |
[11] |
Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072 |
[12] |
Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074 |
[13] |
Xinlin Cao, Huaian Diao, Jinhong Li. Some recent progress on inverse scattering problems within general polyhedral geometry. Electronic Research Archive, 2021, 29 (1) : 1753-1782. doi: 10.3934/era.2020090 |
[14] |
Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117 |
[15] |
Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105 |
[16] |
Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 |
[17] |
Vadim Azhmyakov, Juan P. Fernández-Gutiérrez, Erik I. Verriest, Stefan W. Pickl. A separation based optimization approach to Dynamic Maximal Covering Location Problems with switched structure. Journal of Industrial & Management Optimization, 2021, 17 (2) : 669-686. doi: 10.3934/jimo.2019128 |
[18] |
Qing-Hu Hou, Yarong Wei. Telescoping method, summation formulas, and inversion pairs. Electronic Research Archive, , () : -. doi: 10.3934/era.2021007 |
[19] |
Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020053 |
[20] |
Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 |
Impact Factor:
Tools
Article outline
Figures and Tables
[Back to Top]