Recent Trends on Nonlinear Filtering for Inverse Problems

Michael Herty; Elisa Iacomini; Giuseppe Visconti

doi:10.2478/caim-2022-0002

Abstract

Among the class of nonlinear particle filtering methods, the Ensemble Kalman Filter (EnKF) has gained recent attention for its use in solving inverse problems. We review the original method and discuss recent developments in particular in view of the limit for infinitely particles and extensions towards stability analysis and multi–objective optimization. We illustrate the performance of the method by using test inverse problems from the literature.

References

1. M. Dashti and A. M. Stuart, The Bayesian Approach to Inverse Problems, pp. 311–424. Springer International Publishing, 2016.10.1007/978-3-319-12385-1_7
Search in Google Scholar Back to article
2. J. O. Berger, Statistical Decision Theory and Bayesian Analysis. Springer, 2nd ed., 1985.10.1007/978-1-4757-4286-2
Search in Google Scholar Back to article
3. M. Burger and F. Lucka, Maximum a posteriori estimates in linear inverse problems with log-concave priors are proper Bayes estimators, Inverse Problems, vol. 30, p. 114004, 2014.10.1088/0266-5611/30/11/114004
Search in Google Scholar Back to article
4. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, vol. 375. Springer Science and Business Media, 1996.10.1007/978-94-009-1740-8
Search in Google Scholar Back to article
5. J. Carrillo, F. Hoffmann, A. Stuart, and U. Vaes, Consensus-based sampling, Studies in Applied Mathematics, vol. 148, no. 3, pp. 1069–1140, 2022.10.1111/sapm.12470
Search in Google Scholar Back to article
6. M. Iglesias, K. Law, and A. M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Probl., vol. 29, no. 4, p. 045001, 2013.10.1088/0266-5611/29/4/045001
Search in Google Scholar Back to article
7. N. K. Chada, C. Schillings, and S. Weissmann, On the incorporation of box-constraints for ensemble Kalman inversion, Foundations of Data Science, vol. 1, no. 2639-8001_2019_4_433, p. 433, 2019.10.3934/fods.2019018
Search in Google Scholar Back to article
8. M. Herty and G. Visconti, Continuous limits for constrained ensemble Kalman filter, Inverse Probl., 2020.10.1088/1361-6420/ab8bc5
Search in Google Scholar Back to article
9. D. J. Albers, P.-A. Blancquart, M. E. Levine, E. E. Seylabi, and A. M. Stuart, Ensemble Kalman methods with constraints, Inverse Probl., vol. 35, no. 9, p. 095007, 2019.10.1088/1361-6420/ab1c09
Search in Google Scholar Back to article
10. K. Bergemann and S. Reich, An ensemble Kalman-Bucy filter for continuous data assimilation, Meteorologische Zeitschrift, vol. 21, no. 3, pp. 213–219, 2012.10.1127/0941-2948/2012/0307
Search in Google Scholar Back to article
11. Y. Chen and D. S. Oliver, Parameterization techniques to improve mass conservation and data assimilation for ensemble Kalman filter, 2010.10.2118/133560-MS
Search in Google Scholar Back to article
12. A. A. Emerick and A. C. Reynolds, Ensemble smoother with multiple data assimilation, Computers and Geosciences, vol. 55, pp. 3–15, 2013.10.1016/j.cageo.2012.03.011
Search in Google Scholar Back to article
13. G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res, vol. 99, pp. 10143–10162, 1994.10.1029/94JC00572
Search in Google Scholar Back to article
14. G. Evensen and P. J. Van Leeuwen, Assimilation of geosat altimeter data for the agulhas current using the ensemble Kalman filter with a quasi-geostrophic model, Monthly Weather, vol. 128, pp. 85–96, 1996.10.1175/1520-0493(1996)124<0085:AOGADF>2.0.CO;2
Search in Google Scholar Back to article
15. S. I. Aanonsen, G. Naevdal, D. S. Oliver, A. C. Reynolds, and B. Valles, The ensemble Kalman filter in reservoir engineering–a review, SPE J., vol. 14, no. 3, pp. 393–412, 2009.10.2118/117274-PA
Search in Google Scholar Back to article
16. T. Janjić, D. McLaughlin, S. E. Cohn, and M. Verlaan, Conservation of mass and preservation of positivity with ensemble-type Kalman filter algorithms, Monthly Weather Review, vol. 142, no. 2, pp. 755–773, 2014.10.1175/MWR-D-13-00056.1
Search in Google Scholar Back to article
17. M. Schwenzer, G. Visconti, M. Ay, T. Bergs, M. Herty, and D. Abel, Identifying trending coefficients with an ensemble Kalman filter, IFAC-PapersOnLine, vol. 53, no. 2, pp. 2292–2298, 2020.10.1016/j.ifacol.2020.12.1490
Search in Google Scholar Back to article
18. B. O. S. Teixeira, L. A. B. Târres, L. A. Aguirre, and D. S. Bernstein, On unscented Kalman filtering with state interval constraints, J. Process Contr., vol. 20, no. 1, pp. 45–57, 2010.10.1016/j.jprocont.2009.10.007
Search in Google Scholar Back to article
19. J. Keller, H.-J. Franssen, and W. Nowak, Investigating the pilot point ensemble kalman filter for geostatistical inversion and data assimilation, Adv. Water Resour., vol. 155, 2021.10.1016/j.advwatres.2021.104010
Search in Google Scholar Back to article
20. J. B. Muir and V. C. Tsai, Geometric and level set tomography using ensemble Kalman inversion, Geophysical Journal International, vol. 220, no. 2, pp. 967–980, 2019.10.1093/gji/ggz472
Search in Google Scholar Back to article
21. C.-H. M. Tso, M. Iglesias, P. Wilkinson, O. Kuras, J. Chambers, and A. Binley, Efficient multiscale imaging of subsurface resistivity with uncertainty quantification using ensemble Kalman inversion, Geophysical Journal International, vol. 225, no. 2, pp. 887–905, 2021.10.1093/gji/ggab013
Search in Google Scholar Back to article
22. Z. Li, An iterative ensemble kalman method for an inverse scattering problem in acoustics, Modern Physics Letters B, vol. 34, no. 28, p. 2050312, 2020.10.1142/S0217984920503121
Search in Google Scholar Back to article
23. E. Haber, F. Lucka, and L. Ruthotto, Never look back - A modified EnKF method and its application to the training of neural networks without back propagation. Preprint arXiv:1805.08034, 2018.
Search in Google Scholar Back to article
24. N. B. Kovachki and A. M. Stuart, Ensemble Kalman inversion: a derivative-free technique for machine learning tasks, Inverse Probl., vol. 35, no. 9, p. 095005, 2019.10.1088/1361-6420/ab1c3a
Search in Google Scholar Back to article
25. A. Yegenoglu, S. Diaz, K. Krajsek, and M. Herty, Ensemble Kalman filter optimizing deep neural networks, in Conference on Machine Learning, Optimization and Data Science, vol. 12514, 2020.
Search in Google Scholar Back to article
26. O. G. Ernst, B. Sprungk, and H.-J. Starkloff, Analysis of the ensemble and polynomial chaos Kalman filters in Bayesian inverse problems, SIAM/ASA J. Uncertain. Quantif., vol. 3, no. 1, pp. 823–851, 2015.10.1137/140981319
Search in Google Scholar Back to article
27. A. Garbuno-Inigo, F. Hoffmann, W. Li, and A. M. Stuart, Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler, SIAM J. Appl. Dyn. Syst., vol. 19, no. 1, pp. 412–441, 2020.10.1137/19M1251655
Search in Google Scholar Back to article
28. A. Apte, M. Hairer, A. M. Stuart, and J. Voss, Sampling the posterior: An approach to non-Gaussian data assimilation, Phys. D, vol. 230, pp. 50–64, 2007.10.1016/j.physd.2006.06.009
Search in Google Scholar Back to article
29. F. Le Gland, V. Monbet, and V.-D. Tran, Large sample asymptotics for the ensemble Kalman filter, Research Report RR-7014, INRIA, 2009.
Search in Google Scholar Back to article
30. D. Bloemker, C. Schillings, and P. Wacker, A strongly convergent numerical scheme from ensemble Kalman inversion, SIAM J. Numer. Anal., vol. 56, no. 4, pp. 2537–2562, 2018.10.1137/17M1132367
Search in Google Scholar Back to article
31. D. Bloemker, C. Schillings, P. Wacker, and S. Weissman, Well Posedness and Convergence Analysis of the Ensemble Kalman Inversion, Inverse Probl., vol. 35, no. 8, 2019.10.1088/1361-6420/ab149c
Search in Google Scholar Back to article
32. N. K. Chada, A. M. Stuart, and X. T. Tong, Tikhonov regularization within ensemble Kalman inversion, SIAM J. Numer. Anal., vol. 58, no. 2, pp. 1263–1294, 2020.10.1137/19M1242331
Search in Google Scholar Back to article
33. C. Schillings and A. M. Stuart, Analysis of the Ensamble Kalman Filter for Inverse Problems, SIAM J. Numer. Anal., vol. 55, no. 3, pp. 1264–1290, 2017.10.1137/16M105959X
Search in Google Scholar Back to article
34. C. Schillings and A. M. Stuart, Convergence analysis of ensemble Kalman inversion: the linear, noisy case, Appl. Anal., vol. 97, no. 1, pp. 107–123, 2018.10.1080/00036811.2017.1386784
Search in Google Scholar Back to article
35. J. A. Carrillo and U. Vaes, Wasserstein stability estimates for covariance-preconditioned Fokker-Planck equations, Nonlinearity, vol. 34, no. 4, p. 2275, 2021.10.1088/1361-6544/abbe62
Search in Google Scholar Back to article
36. Z. Ding and Q. Li, Ensemble Kalman Inversion: mean-field limit and convergence analysis, Stat. Comput., vol. 31, p. 9, 2021.10.1007/s11222-020-09976-0
Search in Google Scholar Back to article
37. M. Herty and G. Visconti, Kinetic methods for inverse problems, Kinet. Relat. Models, vol. 12, no. 5, pp. 1109–1130, 2019.10.3934/krm.2019042
Search in Google Scholar Back to article
38. N. K. Chada, Limit analysis of hierarchical ensemble Kalman inversion, J. Inverse Ill-Posed Probl., 2020. In press.
Search in Google Scholar Back to article
39. Z. Ding, Q. Li, and J. Lu, Ensemble Kalman inversion for nonlinear problems: Weights, consistency, and variance bounds, Found. Data Sci., vol. 3, no. 3, pp. 371–411, 2021.10.3934/fods.2020018
Search in Google Scholar Back to article
40. A. Armbruster, M. Herty, and G. Visconti, A stabilization of a continuous limit of the ensemble Kalman inversion, SIAM J. Numer. Anal., 2022. Accepted. Preprint arXiv:2006.15390.10.1137/21M1414000
Search in Google Scholar Back to article
41. J. A. Carrillo, M. Fornasier, G. Toscani, and F. Vecil, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, ch. Particle, kinetic, and hydrodynamic models of swarming, pp. 297–336. Modeling and Simulation in Science, Engineering and Technology, Birkh¨auser Boston, 2010.10.1007/978-0-8176-4946-3_12
Search in Google Scholar Back to article
42. F. Golse, On the dynamics of large particle systems in the mean field limit, in Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity, pp. 1–144, Springer, 2016.10.1007/978-3-319-26883-5_1
Search in Google Scholar Back to article
43. P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinetic & Related Models, vol. 7, no. 4, pp. 661–711, 2014.10.3934/krm.2014.7.661
Search in Google Scholar Back to article
44. L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic equations and Monte Carlo methods. Oxford University Press, 2013.
Search in Google Scholar Back to article
45. G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., vol. 11, no. 1, pp. 1–29, 2013.10.1137/120868748
Search in Google Scholar Back to article
46. M. Ehrgott, Multicriteria optimization, vol. 491. Springer Science & Business Media, 2005.
Search in Google Scholar Back to article
47. K. Miettinen, Nonlinear multiobjective optimization, vol. 12. Springer Science & Business Media, 2012.
Search in Google Scholar Back to article
48. P. M. Pardalos, A.Žilinskas, J.Žilinskas, et al., Non-convex multi-objective optimization. Springer, 2017.10.1007/978-3-319-61007-8
Search in Google Scholar Back to article
49. M. Herty and E. Iacomini, Filtering methods for coupled inverse problems. Preprint. arXiv:2203.09841, 2022.
Search in Google Scholar Back to article
50. K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE transactions on evolutionary computation, vol. 6, no. 2, pp. 182–197, 2002.10.1109/4235.996017
Search in Google Scholar Back to article

Recent Trends on Nonlinear Filtering for Inverse Problems

Abstract

Paradigm

My account