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October  2019, 12(5): 1109-1130. doi: 10.3934/krm.2019042

Kinetic methods for inverse problems

RWTH Aachen University, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52062 Aachen, Germany

* Corresponding author: Giuseppe Visconti

Received  November 2018 Revised  February 2019 Published  July 2019

The Ensemble Kalman Filter method can be used as an iterative numerical scheme for parameter identification ornonlinear filtering problems. We study the limit of infinitely large ensemble size and derive the corresponding mean-field limit of the ensemble method. The solution of the inverse problem is provided by the expected value of the distribution of the ensembles and the kinetic equation allows, in simple cases, to analyze stability of these solutions. Further, we present a slight but stable modification of the method which leads to a Fokker-Planck-type kinetic equation. The kinetic methods proposed here are able to solve the problem with a reduced computational complexity in the limit of a large ensemble size. We illustrate the properties and the ability of the kinetic model to provide solution to inverse problems by using examples from the literature.

Citation: Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
References:
[1]

S. I. AanonsenG. NævdalD. S. OliverA. C. Reynolds and B. Vallès, The Ensemble Kalman Filter in Reservoir Engineering–a Review, SPE Journal, 14 (2013), 393-412. doi: 10.2118/117274-PA. Google Scholar

[2]

G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., 11 (2013), 1-29. doi: 10.1137/120868748. Google Scholar

[3]

A. ApteM. HairerA. M. Stuart and J. Voss, Sampling the posterior: An approach to non-Gaussian data assimilation, Phys. D, 230 (2007), 50-64. doi: 10.1016/j.physd.2006.06.009. Google Scholar

[4]

J. O. Berger, Statistical Decision Theory and Bayesian Analysis, 2nd edition, Springer, 1985. doi: 10.1007/978-1-4757-4286-2. Google Scholar

[5]

D. Bianchi, A. Buccini, M. Donatelli and S. Serra-Capizzano, Iterated fractional Tikhonov regularization, Inverse Problems, 31 (2015), 055005, 34pp. doi: 10.1088/0266-5611/31/5/055005. Google Scholar

[6]

D. BloemkerC. Schillings and P. Wacker, A strongly convergent numerical scheme from ensemble kalman inversion, SIAM J. Numer. Anal., 56 (2018), 2537-2562. doi: 10.1137/17M1132367. Google Scholar

[7]

D. Bloemker, C. Schillings, P. Wacker and S. Weissman, Well Posedness and Convergence Analysis of the Ensemble Kalman Inversion, 2018, Preprint. arXiv: 1810.08463.Google Scholar

[8]

H. Bobovsky and H. Neunzert, On a simulation scheme for the boltzmann equation, Math. Methods Appl. Sci., 8 (1986), 223-233. doi: 10.1002/mma.1670080114. Google Scholar

[9]

M. Burger and F. Lucka, Maximum a posteriori estimates in linear inverse problems with log-concave priors are proper Bayes estimators, Inverse Problems, 30 (2014), 114004, 21pp. doi: 10.1088/0266-5611/30/11/114004. Google Scholar

[10]

R. E. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numerica, 7 (1998), 1-49. doi: 10.1017/S0962492900002804. Google Scholar

[11]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Chapter Particle, kinetic, and hydrodynamic models of swarming, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, 297–336, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, 2010. doi: 10.1007/978-0-8176-4946-3_12. Google Scholar

[12]

J. A. CarrilloL. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys., 25 (2019), 508-531. Google Scholar

[13]

N. K. Chada and X. T. Stuart A. M. Tong, Tikhonov regularization within ensemble Kalman inversion, 2019, arXiv.org/abs/1901.10382.Google Scholar

[14]

E. Cristiani, B. Piccoli and A. Tosin, MS & A: Modeling, Simulation and Applications, vol. 12, chapter Multiscale Modeling of Pedestrian Dynamics, Springer International Publishing, 2014. doi: 10.1007/978-3-319-06620-2. Google Scholar

[15]

M. Dashti and A. M. Stuart, The bayesian approach to inverse problems, Handbook of Uncertainty Quantification, Vol. 1, 2, 3,311–428, Springer, Cham, 2017. Google Scholar

[16]

P. Del MoralA. Kurtzmann and J. Tugaut, On the stability and the uniform propagation of chaos of a class of Extended Ensemble Kalman-Bucy filters, SIAM J. Control Optim., 55 (2017), 119-155. doi: 10.1137/16M1087497. Google Scholar

[17]

P. Del Moral and J. Tugaut, On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters, Ann. Appl. Probab., 28 (2018), 790-850. doi: 10.1214/17-AAP1317. Google Scholar

[18]

L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transport Theor. Stat., 21 (1992), 259-276. doi: 10.1080/00411459208203923. Google Scholar

[19]

R. J. DiPerna and P. L. Lions, On the Fokker-Planck-Boltzmann equation, Commun. Math. Phys., 120 (1988), 1-23. doi: 10.1007/BF01223204. Google Scholar

[20]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, vol. 375, Springer Science and Business Media, 1996. Google Scholar

[21]

O. G. ErnstB. Sprungk and H.-J. Starkloff, Analysis of the ensemble and polynomial chaos kalman filters in bayesian inverse problems, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 823-851. doi: 10.1137/140981319. Google Scholar

[22]

G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics, J. Geophys. Res, 99 (1994), 10143-10162. doi: 10.1029/94JC00572. Google Scholar

[23]

G. Evensen, Data Assimilation: The Ensemble Kalman Filter, Springer Verlag, 2009. doi: 10.1007/978-3-642-03711-5. Google Scholar

[24]

M. FornasierJ. Haskovec and J. Vybíral, Particle systems and kinetic equations modeling interacting agents in high dimension, Multiscale Model. Simul., 9 (2011), 1727-1764. doi: 10.1137/110830617. Google Scholar

[25]

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, vol. 105, Pitman Advanced Publishing Program, 1984. Google Scholar

[26]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415. Google Scholar

[27]

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, 1998. doi: 10.1137/1.9780898719697. Google Scholar

[28]

M. Herty and C. Ringhofer, Averaged kinetic models for flows on unstructured networks, Kinet. Relat. Models, 4 (2011), 1081-1096. doi: 10.3934/krm.2011.4.1081. Google Scholar

[29]

M. Iglesias, Iterative regularization for ensemble data assimilation in reservoir models, Computational Geosciences, 19 (2015), 177-212. doi: 10.1007/s10596-014-9456-5. Google Scholar

[30]

M. Iglesias, K. Law and A. M. Stuart, Analysis of the Ensamble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 045001, 20pp. doi: 10.1088/0266-5611/29/4/045001. Google Scholar

[31]

M. IglesiasK. Law and A. M. Stuart, Evaluation of Gaussian approximations for data assimilation in reservoir models, Comput. Geosci., 17 (2013), 851-885. doi: 10.1007/s10596-013-9359-x. Google Scholar

[32]

R. E. Kalman, A new approach to linear filtering and prediction problems, J. Basic Eng.-T. ASME, 82 (1960), 35-45. doi: 10.1115/1.3662552. Google Scholar

[33]

E. Klann and R. Ramlau, Regularization by fractional filter methods and data smoothing, Inverse Problems, 24 (2008), 0125018, 26pp. doi: 10.1088/0266-5611/24/2/025018. Google Scholar

[34]

E. Kwiatkowski and J. Mandel, Convergence of the square root ensemble Kalman filter in the large ensemble limit, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 1-17. doi: 10.1137/140965363. Google Scholar

[35]

T. Lange and W. Stannat, On the continuous time limit of the ensemble Kalman filter, 2019, arXiv.org/abs/1901.05204.Google Scholar

[36]

K. J. H. Law and A. M. Stuart, Evaluating data assimilation algorithms, Mon. Weather Rev., 140 (2012), 3757-3782. doi: 10.1175/MWR-D-11-00257.1. Google Scholar

[37]

K. J. H. Law, H. Tembine and R. Tempone, Deterministic mean-field ensemble kalman filtering, SIAM J. Sci. Comput., 38 (2016), 1251–1279. doi: 10.1137/140984415. Google Scholar

[38]

F. Le Gland, V. Monbet and V.-D. Tran, Large Sample Asymptotics for the Ensemble Kalman Filter, Research Report RR-7014, INRIA, 2009.Google Scholar

[39]

M. Lemou, Multipole expansions for the Fokker-Planck equation, Numer. Math., 78 (1998), 597-618. doi: 10.1007/s002110050327. Google Scholar

[40]

A. J. Majda and X. T. Tong, Performance of Ensemble Kalman filters in large dimensions, Commun. Pur. Appl. Math., 71 (2018), 892-937. doi: 10.1002/cpa.21722. Google Scholar

[41]

C. Mouhot and L. Pareschi, Fast algorithms for computing the Boltzmann collision operator, Math. Comput., 75 (2006), 1833-1852. doi: 10.1090/S0025-5718-06-01874-6. Google Scholar

[42] D. S. OliverA. C. Reynolds and N. Liu, Inverse Theory for Petroleum Reservoir Characterization and History Matching, Cambridge University Press, 2008. doi: 10.1017/CBO9780511535642. Google Scholar
[43]

L. Pareschi and G. Russo, An introduction to Monte Carlo methods for the Boltzmann equation, in CEMRACS 1999 (Orsay), ESAIM Proc., Soc. Math. Appl. Indust., Paris, 10 (1999), 35–76. doi: 10.1051/proc:2001004. Google Scholar

[44] L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic equations and Monte Carlo methods, Oxford University Press, 2013. Google Scholar
[45]

L. PareschiG. Toscani and C. Villani, Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit, Numer. Math., 93 (2003), 527-548. doi: 10.1007/s002110100384. Google Scholar

[46]

C. Schillings and A. M. Stuart, Analysis of the ensamble kalman filter for inverse problems, SIAM J. Numer. Anal., 55 (2017), 1264-1290. doi: 10.1137/16M105959X. Google Scholar

[47]

C. Schillings and A. M. Stuart, Convergence analysis of ensemble Kalman inversion: The linear, noisy case, Applicable Analysis, 97 (2018), 107-123. doi: 10.1080/00036811.2017.1386784. Google Scholar

[48]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559. doi: 10.1017/S0962492910000061. Google Scholar

[49]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1. Google Scholar

[50]

T. Trimborn, L. Pareschi and M. Frank, Portfolio optimization and model predictive control: A kinetic approach, 2018, Preprint. arXiv: 1711.03291.Google Scholar

[51]

C. Villani, Conservative forms of Boltzmann's collision operator: Landau revisited, ESAIM Math. Model. Numer. Anal., 33 (1999), 209-227. doi: 10.1051/m2an:1999112. Google Scholar

show all references

References:
[1]

S. I. AanonsenG. NævdalD. S. OliverA. C. Reynolds and B. Vallès, The Ensemble Kalman Filter in Reservoir Engineering–a Review, SPE Journal, 14 (2013), 393-412. doi: 10.2118/117274-PA. Google Scholar

[2]

G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., 11 (2013), 1-29. doi: 10.1137/120868748. Google Scholar

[3]

A. ApteM. HairerA. M. Stuart and J. Voss, Sampling the posterior: An approach to non-Gaussian data assimilation, Phys. D, 230 (2007), 50-64. doi: 10.1016/j.physd.2006.06.009. Google Scholar

[4]

J. O. Berger, Statistical Decision Theory and Bayesian Analysis, 2nd edition, Springer, 1985. doi: 10.1007/978-1-4757-4286-2. Google Scholar

[5]

D. Bianchi, A. Buccini, M. Donatelli and S. Serra-Capizzano, Iterated fractional Tikhonov regularization, Inverse Problems, 31 (2015), 055005, 34pp. doi: 10.1088/0266-5611/31/5/055005. Google Scholar

[6]

D. BloemkerC. Schillings and P. Wacker, A strongly convergent numerical scheme from ensemble kalman inversion, SIAM J. Numer. Anal., 56 (2018), 2537-2562. doi: 10.1137/17M1132367. Google Scholar

[7]

D. Bloemker, C. Schillings, P. Wacker and S. Weissman, Well Posedness and Convergence Analysis of the Ensemble Kalman Inversion, 2018, Preprint. arXiv: 1810.08463.Google Scholar

[8]

H. Bobovsky and H. Neunzert, On a simulation scheme for the boltzmann equation, Math. Methods Appl. Sci., 8 (1986), 223-233. doi: 10.1002/mma.1670080114. Google Scholar

[9]

M. Burger and F. Lucka, Maximum a posteriori estimates in linear inverse problems with log-concave priors are proper Bayes estimators, Inverse Problems, 30 (2014), 114004, 21pp. doi: 10.1088/0266-5611/30/11/114004. Google Scholar

[10]

R. E. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numerica, 7 (1998), 1-49. doi: 10.1017/S0962492900002804. Google Scholar

[11]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Chapter Particle, kinetic, and hydrodynamic models of swarming, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, 297–336, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, 2010. doi: 10.1007/978-0-8176-4946-3_12. Google Scholar

[12]

J. A. CarrilloL. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys., 25 (2019), 508-531. Google Scholar

[13]

N. K. Chada and X. T. Stuart A. M. Tong, Tikhonov regularization within ensemble Kalman inversion, 2019, arXiv.org/abs/1901.10382.Google Scholar

[14]

E. Cristiani, B. Piccoli and A. Tosin, MS & A: Modeling, Simulation and Applications, vol. 12, chapter Multiscale Modeling of Pedestrian Dynamics, Springer International Publishing, 2014. doi: 10.1007/978-3-319-06620-2. Google Scholar

[15]

M. Dashti and A. M. Stuart, The bayesian approach to inverse problems, Handbook of Uncertainty Quantification, Vol. 1, 2, 3,311–428, Springer, Cham, 2017. Google Scholar

[16]

P. Del MoralA. Kurtzmann and J. Tugaut, On the stability and the uniform propagation of chaos of a class of Extended Ensemble Kalman-Bucy filters, SIAM J. Control Optim., 55 (2017), 119-155. doi: 10.1137/16M1087497. Google Scholar

[17]

P. Del Moral and J. Tugaut, On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters, Ann. Appl. Probab., 28 (2018), 790-850. doi: 10.1214/17-AAP1317. Google Scholar

[18]

L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transport Theor. Stat., 21 (1992), 259-276. doi: 10.1080/00411459208203923. Google Scholar

[19]

R. J. DiPerna and P. L. Lions, On the Fokker-Planck-Boltzmann equation, Commun. Math. Phys., 120 (1988), 1-23. doi: 10.1007/BF01223204. Google Scholar

[20]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, vol. 375, Springer Science and Business Media, 1996. Google Scholar

[21]

O. G. ErnstB. Sprungk and H.-J. Starkloff, Analysis of the ensemble and polynomial chaos kalman filters in bayesian inverse problems, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 823-851. doi: 10.1137/140981319. Google Scholar

[22]

G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics, J. Geophys. Res, 99 (1994), 10143-10162. doi: 10.1029/94JC00572. Google Scholar

[23]

G. Evensen, Data Assimilation: The Ensemble Kalman Filter, Springer Verlag, 2009. doi: 10.1007/978-3-642-03711-5. Google Scholar

[24]

M. FornasierJ. Haskovec and J. Vybíral, Particle systems and kinetic equations modeling interacting agents in high dimension, Multiscale Model. Simul., 9 (2011), 1727-1764. doi: 10.1137/110830617. Google Scholar

[25]

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, vol. 105, Pitman Advanced Publishing Program, 1984. Google Scholar

[26]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415. Google Scholar

[27]

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, 1998. doi: 10.1137/1.9780898719697. Google Scholar

[28]

M. Herty and C. Ringhofer, Averaged kinetic models for flows on unstructured networks, Kinet. Relat. Models, 4 (2011), 1081-1096. doi: 10.3934/krm.2011.4.1081. Google Scholar

[29]

M. Iglesias, Iterative regularization for ensemble data assimilation in reservoir models, Computational Geosciences, 19 (2015), 177-212. doi: 10.1007/s10596-014-9456-5. Google Scholar

[30]

M. Iglesias, K. Law and A. M. Stuart, Analysis of the Ensamble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 045001, 20pp. doi: 10.1088/0266-5611/29/4/045001. Google Scholar

[31]

M. IglesiasK. Law and A. M. Stuart, Evaluation of Gaussian approximations for data assimilation in reservoir models, Comput. Geosci., 17 (2013), 851-885. doi: 10.1007/s10596-013-9359-x. Google Scholar

[32]

R. E. Kalman, A new approach to linear filtering and prediction problems, J. Basic Eng.-T. ASME, 82 (1960), 35-45. doi: 10.1115/1.3662552. Google Scholar

[33]

E. Klann and R. Ramlau, Regularization by fractional filter methods and data smoothing, Inverse Problems, 24 (2008), 0125018, 26pp. doi: 10.1088/0266-5611/24/2/025018. Google Scholar

[34]

E. Kwiatkowski and J. Mandel, Convergence of the square root ensemble Kalman filter in the large ensemble limit, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 1-17. doi: 10.1137/140965363. Google Scholar

[35]

T. Lange and W. Stannat, On the continuous time limit of the ensemble Kalman filter, 2019, arXiv.org/abs/1901.05204.Google Scholar

[36]

K. J. H. Law and A. M. Stuart, Evaluating data assimilation algorithms, Mon. Weather Rev., 140 (2012), 3757-3782. doi: 10.1175/MWR-D-11-00257.1. Google Scholar

[37]

K. J. H. Law, H. Tembine and R. Tempone, Deterministic mean-field ensemble kalman filtering, SIAM J. Sci. Comput., 38 (2016), 1251–1279. doi: 10.1137/140984415. Google Scholar

[38]

F. Le Gland, V. Monbet and V.-D. Tran, Large Sample Asymptotics for the Ensemble Kalman Filter, Research Report RR-7014, INRIA, 2009.Google Scholar

[39]

M. Lemou, Multipole expansions for the Fokker-Planck equation, Numer. Math., 78 (1998), 597-618. doi: 10.1007/s002110050327. Google Scholar

[40]

A. J. Majda and X. T. Tong, Performance of Ensemble Kalman filters in large dimensions, Commun. Pur. Appl. Math., 71 (2018), 892-937. doi: 10.1002/cpa.21722. Google Scholar

[41]

C. Mouhot and L. Pareschi, Fast algorithms for computing the Boltzmann collision operator, Math. Comput., 75 (2006), 1833-1852. doi: 10.1090/S0025-5718-06-01874-6. Google Scholar

[42] D. S. OliverA. C. Reynolds and N. Liu, Inverse Theory for Petroleum Reservoir Characterization and History Matching, Cambridge University Press, 2008. doi: 10.1017/CBO9780511535642. Google Scholar
[43]

L. Pareschi and G. Russo, An introduction to Monte Carlo methods for the Boltzmann equation, in CEMRACS 1999 (Orsay), ESAIM Proc., Soc. Math. Appl. Indust., Paris, 10 (1999), 35–76. doi: 10.1051/proc:2001004. Google Scholar

[44] L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic equations and Monte Carlo methods, Oxford University Press, 2013. Google Scholar
[45]

L. PareschiG. Toscani and C. Villani, Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit, Numer. Math., 93 (2003), 527-548. doi: 10.1007/s002110100384. Google Scholar

[46]

C. Schillings and A. M. Stuart, Analysis of the ensamble kalman filter for inverse problems, SIAM J. Numer. Anal., 55 (2017), 1264-1290. doi: 10.1137/16M105959X. Google Scholar

[47]

C. Schillings and A. M. Stuart, Convergence analysis of ensemble Kalman inversion: The linear, noisy case, Applicable Analysis, 97 (2018), 107-123. doi: 10.1080/00036811.2017.1386784. Google Scholar

[48]

A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010), 451-559. doi: 10.1017/S0962492910000061. Google Scholar

[49]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1. Google Scholar

[50]

T. Trimborn, L. Pareschi and M. Frank, Portfolio optimization and model predictive control: A kinetic approach, 2018, Preprint. arXiv: 1711.03291.Google Scholar

[51]

C. Villani, Conservative forms of Boltzmann's collision operator: Landau revisited, ESAIM Math. Model. Numer. Anal., 33 (1999), 209-227. doi: 10.1051/m2an:1999112. Google Scholar

Figure 1.  Left: vector field of the ODE system (16) with $ (y, G) = (2, 1) $. Red lines are the nullclines. Right: trajectory behavior around the equilibrium $ (\frac{y}{G}, \frac{y^2}{G^2}) = (2, 4) $
Figure 2.  Left: vector field of the ODE system (24) with $ (y, G) = (2, 1) $. Red lines are the nullclines. Right: trajectory behavior around the equilibrium $ \tilde{F}_1^+ = (2, 8) $
Figure 3.  Elliptic problem - Test case 1 with $ \gamma = 0.01 $. Top row: plots of the residual $ r $, the projected residual $ R $ and the misfit $ \vartheta $ for $ M = 250,500, 1000 $. Bottom row: plots of the noisy data, the reconstruction of $ p(x) $ and the reconstruction of the control $ u(x) $ at final iteration for $ M = 250,500, 1000 $
Figure 4.  Elliptic problem - Test case 1 with $ \gamma = 0.1 $. Top row: plots of the residual $ r $, the projected residual $ R $ and the misfit $ \vartheta $ for $ M = 250,500, 1000 $. Bottom row: plots of the noisy data, the reconstruction of $ p(x) $ and the reconstruction of the control $ u(x) $ at final iteration for $ M = 250,500, 1000 $
Figure 5.  Elliptic problem - Test case 1. Left: spectrum of $ \boldsymbol{{ \mathcal{C}}}(t) G^T \boldsymbol{{\Gamma}} G $ for the initial data with $ \gamma = 0.01 $ and $ \gamma = 0.1 $. Right: adaptive $ \epsilon $ and spectral radius of $ \boldsymbol{{ \mathcal{C}}}(t) G^T \boldsymbol{{\Gamma}} G $ over iterations with $ \gamma = 0.01 $ and $ \gamma = 0.1 $
Figure 6.  Elliptic problem - Test case 2 with $ \gamma = 0.01 $. Top row: plots of the residual $ r $, the projected residual $ R $ and the misfit $ \vartheta $ for $ J = 25, 25\cdot2^9 $. Middle row: plots of the noisy data and of the reconstruction of $ p(x) $ at final iteration for $ J = 25, 25\cdot2^9 $. Bottom row: plots of the reconstruction of the control $ u(x) $ at final iteration for $ J = 25, 25\cdot2^9 $ and behavior of the relative error $ \frac{\|\overline{\mathbf{{u}}}-\mathbf{{u}}\|_2^2}{\|\overline{\mathbf{{u}}}\|_2^2} $
Figure 7.  Nonlinear problem. Top row: plots of the density estimation of the initial samples (left) and position of the samples at final iteration (right). Middle row: Marginals of $ u_1 $ (left) and $ u_2 $ (right) as relative frequency plot. Bottom row: residual errors $ r $ and $ R $ (left) and misfit error (right)
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