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August  2019, 13(4): 755-786. doi: 10.3934/ipi.2019035

Total generalized variation regularization in data assimilation for Burgers' equation

Juan Carlos De los Reyes 1, and  Estefanía Loayza-Romero 2,

1. 

Research Center of Mathematical Modelling (MODEMAT), Escuela Politécnica Nacional, Quito, Ecuador
2. 

Faculty of Mathematics, Professorship Numerical Mathematics (Partial Differential Equations), Chemnitz University of Technology, Chemnitz, Germany

This paper was developed within the Master Program in Mathematical Optimization at Escuela Politécnica Nacional de Ecuador

Received  May 2018 Revised  November 2018 Published  May 2019

We propose a second-order total generalized variation (TGV) regularization for the reconstruction of the initial condition in variational data assimilation problems. After showing the equivalence between TGV regularization and a Bayesian MAP estimator, we focus on the detailed study of the inviscid Burgers' data assimilation problem. Due to the difficult structure of the governing hyperbolic conservation law, we consider a discretize–then–optimize approach and rigorously derive a first-order optimality condition for the problem. For the numerical solution, we propose a globalized reduced Newton-type method together with a polynomial line-search strategy, and prove convergence of the algorithm to stationary points. The paper finishes with some numerical experiments where, among others, the performance of TGV–regularization compared to TV–regularization is tested.

Keywords: Data Assimilation, inviscid Burgers' equation, second-order methods, Primal-dual methods.
Mathematics Subject Classification: Primary: 76B75, 49M15; Secondary: 49K20.
Citation: Juan Carlos De los Reyes, Estefanía Loayza-Romero. Total generalized variation regularization in data assimilation for Burgers' equation. Inverse Problems & Imaging, 2019, 13 (4) : 755-786. doi: 10.3934/ipi.2019035
References:
[1]

A. Apte, D. Auroux and M. Ramaswamy, Variational data assimilation for discrete Burgers equation, Proc. Electron. J. Differential Equations, 19 (2010), 15–30. Avaliable form http://ejde.math.txstate.edu Google Scholar

[2]

K. Bredies, Y. Dong and M. Hintermüller, Spatially dependent regularization parameter selection in total generalized variation models for image restoration, Int. J. Comput. Math., 90 (2013), 109-123.  doi: 10.1080/00207160.2012.700400.  Google Scholar

[3]

K. Bredies, K. Kunisch and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2010), 492-526.  doi: 10.1137/090769521.  Google Scholar

[4]

K. Bredies and T. Valkonen, Inverse problems with second-order total generalized variation constraints, Proc. SampTA, 201 (2011). Avaliable from: https://imsc.uni-graz.at/bredies/papers/SampTA2011.pdf Google Scholar

[5]

C. J. Budd, M. A. Freitag and N. K. Nichols, Regularization techniques for ill-posed inverse problems in data assimilation, Comput. & Fluids, 46 (2011), 168-173.  doi: 10.1016/j.compfluid.2010.10.002.  Google Scholar

[6]

T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.  doi: 10.1137/S1064827596299767.  Google Scholar

[7]

P. G. Ciarlet, Linear and nonlinear functional analysis with applications, SIAM, 2013.  Google Scholar

[8]

J. C. De los Reyes, Numerical PDE-constrained optimization, Springer, 2015. doi: 10.1007/978-3-319-13395-9.  Google Scholar

[9]

J. C. De los Reyes, C. B. Schönlieb and T. Valkonen, Bilevel parameter learning for higher-order total variation regularisation models, J. Math. Imaging Vision, 57 (2017), 1-25.  doi: 10.1007/s10851-016-0662-8.  Google Scholar

[10]

J. E. Dennis Jr. and R. B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, SIAM, 1996. doi: 10.1137/1.9781611971200.  Google Scholar

[11]

M. A. Freitag, N. K. Nichols and C. J. Budd, Resolution of sharp fronts in the presence of model error in variational data assimilation, Q. J. Royal Meteorol. Soc., 139 (2010), 742-757.  doi: 10.1002/qj.2002.  Google Scholar

[12]

M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration, SIAM J. Sci. Comput., 28 (2006), 1-23.  doi: 10.1137/040613263.  Google Scholar

[13]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE constraints, Springer Science & Business Media, 2008.  Google Scholar

[14]

E. Kalnay, Atmospheric modeling, data assimilation, and predictability, Cambridge University Press, 2003. doi: 10.1017/CBO9780511802270.  Google Scholar

[15]

F. Knoll, K. Bredies, T. Pock and R. Stollberger, Second order total generalized variation (TGV) for MRI, Magn. Reson. Med., 65 (2011), 480-491.  doi: 10.1002/mrm.22595.  Google Scholar

[16]

J. Lee and P. K. Kitanidis, Bayesian inversion with total variation prior for discrete geologic structure identification, Water Resour. Res., 49 (2013), 7658-7669.  doi: 10.1002/2012WR013431.  Google Scholar

[17]

J. M. Lewis, S. Lakshmivarahan and S. K. Dhall, Dynamic data assimilation : a least squares approach, Cambridge University Press, 2006. doi: 10.1017/CBO9780511526480.  Google Scholar

[18]

J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, SIAM, 2012. doi: 10.1137/1.9781611972344.  Google Scholar

[19]

S. Pfaff and S. Ulbrich, Optimal boundary control of nonlinear hyperbolic conservation laws with switched boundary data, SIAM J. Control Optim., 53 (2015), 1250-1277.  doi: 10.1137/140995799.  Google Scholar

[20]

A. Quarteroni, R. Sacco and F. Saleri, Numerical mathematics, Springer Science & Business Media, 2010.  Google Scholar

[21]

C. R. Vogel, Computational Methods for Inverse Problems, SIAM, 2002. doi: 10.1137/1.9780898717570.  Google Scholar

[22]

Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861.  Google Scholar

show all references

References:
[1]

A. Apte, D. Auroux and M. Ramaswamy, Variational data assimilation for discrete Burgers equation, Proc. Electron. J. Differential Equations, 19 (2010), 15–30. Avaliable form http://ejde.math.txstate.edu Google Scholar

[2]

K. Bredies, Y. Dong and M. Hintermüller, Spatially dependent regularization parameter selection in total generalized variation models for image restoration, Int. J. Comput. Math., 90 (2013), 109-123.  doi: 10.1080/00207160.2012.700400.  Google Scholar

[3]

K. Bredies, K. Kunisch and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2010), 492-526.  doi: 10.1137/090769521.  Google Scholar

[4]

K. Bredies and T. Valkonen, Inverse problems with second-order total generalized variation constraints, Proc. SampTA, 201 (2011). Avaliable from: https://imsc.uni-graz.at/bredies/papers/SampTA2011.pdf Google Scholar

[5]

C. J. Budd, M. A. Freitag and N. K. Nichols, Regularization techniques for ill-posed inverse problems in data assimilation, Comput. & Fluids, 46 (2011), 168-173.  doi: 10.1016/j.compfluid.2010.10.002.  Google Scholar

[6]

T. F. Chan, G. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.  doi: 10.1137/S1064827596299767.  Google Scholar

[7]

P. G. Ciarlet, Linear and nonlinear functional analysis with applications, SIAM, 2013.  Google Scholar

[8]

J. C. De los Reyes, Numerical PDE-constrained optimization, Springer, 2015. doi: 10.1007/978-3-319-13395-9.  Google Scholar

[9]

J. C. De los Reyes, C. B. Schönlieb and T. Valkonen, Bilevel parameter learning for higher-order total variation regularisation models, J. Math. Imaging Vision, 57 (2017), 1-25.  doi: 10.1007/s10851-016-0662-8.  Google Scholar

[10]

J. E. Dennis Jr. and R. B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, SIAM, 1996. doi: 10.1137/1.9781611971200.  Google Scholar

[11]

M. A. Freitag, N. K. Nichols and C. J. Budd, Resolution of sharp fronts in the presence of model error in variational data assimilation, Q. J. Royal Meteorol. Soc., 139 (2010), 742-757.  doi: 10.1002/qj.2002.  Google Scholar

[12]

M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration, SIAM J. Sci. Comput., 28 (2006), 1-23.  doi: 10.1137/040613263.  Google Scholar

[13]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE constraints, Springer Science & Business Media, 2008.  Google Scholar

[14]

E. Kalnay, Atmospheric modeling, data assimilation, and predictability, Cambridge University Press, 2003. doi: 10.1017/CBO9780511802270.  Google Scholar

[15]

F. Knoll, K. Bredies, T. Pock and R. Stollberger, Second order total generalized variation (TGV) for MRI, Magn. Reson. Med., 65 (2011), 480-491.  doi: 10.1002/mrm.22595.  Google Scholar

[16]

J. Lee and P. K. Kitanidis, Bayesian inversion with total variation prior for discrete geologic structure identification, Water Resour. Res., 49 (2013), 7658-7669.  doi: 10.1002/2012WR013431.  Google Scholar

[17]

J. M. Lewis, S. Lakshmivarahan and S. K. Dhall, Dynamic data assimilation : a least squares approach, Cambridge University Press, 2006. doi: 10.1017/CBO9780511526480.  Google Scholar

[18]

J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, SIAM, 2012. doi: 10.1137/1.9781611972344.  Google Scholar

[19]

S. Pfaff and S. Ulbrich, Optimal boundary control of nonlinear hyperbolic conservation laws with switched boundary data, SIAM J. Control Optim., 53 (2015), 1250-1277.  doi: 10.1137/140995799.  Google Scholar

[20]

A. Quarteroni, R. Sacco and F. Saleri, Numerical mathematics, Springer Science & Business Media, 2010.  Google Scholar

[21]

C. R. Vogel, Computational Methods for Inverse Problems, SIAM, 2002. doi: 10.1137/1.9780898717570.  Google Scholar

[22]

Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861.  Google Scholar

Figure 1.  Exact solution
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Figure 2.  SSIM index for the TV and TGV regularizations
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Figure 3.  Best solutions for TV (left) and TGV (right) regularizations
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Figure 4.  Observations vs Optimal state
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Figure 5.  Exact solutions and best solutions obtained for the experiment
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Figure 6.  Global convergence of the algorithm
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Figure 7.  Locally superlinear convergence of the algorithm
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Figure 8.  Exact solutions for the experiment
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Table 1.  Results of the experiments with different values of $ \mu $
Experiment $ \mu $ iter SSIM $ J_\gamma $ time (s)
3 0 – NaN NaN –
1e-6 13 0.9594 35.8330 7.2
1e-8 12 0.9594 35.8330 6.9
1e-10 12 0.9594 35.8330 6.9
1e-12 13 0.9594 35.8333 7.1
1e-14 13 0.9594 35.8329 7.1
4 0 – NaN NaN –
1e-6 14 0.9592 39.2436 7.6
1e-8 13 0.9592 39.2434 7.0
1e-10 13 0.9592 39.2434 7.0
1e-12 14 0.9591 39.2437 7.7
1e-14 16 0.9591 39.2439 9.1
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Experiment $ \mu $ iter SSIM $ J_\gamma $ time (s)
3 0 – NaN NaN –
1e-6 13 0.9594 35.8330 7.2
1e-8 12 0.9594 35.8330 6.9
1e-10 12 0.9594 35.8330 6.9
1e-12 13 0.9594 35.8333 7.1
1e-14 13 0.9594 35.8329 7.1
4 0 – NaN NaN –
1e-6 14 0.9592 39.2436 7.6
1e-8 13 0.9592 39.2434 7.0
1e-10 13 0.9592 39.2434 7.0
1e-12 14 0.9591 39.2437 7.7
1e-14 16 0.9591 39.2439 9.1
Table 2.  Summary of the experiment
Observations M N $ \sigma $
0.1 0.01 0.001
iter SSIM iter SSIM iter SSIM
4 75 40 20 0.9573 12 0.9895 12 0.9985
9 50 20 19 0.9576 12 0.9895 11 0.9985
30 25 10 20 0.9581 14 0.9896 11 0.9985
40 20 10 19 0.9584 15 0.9896 12 0.9985
100 15 5 15 0.9604 12 0.9896 11 0.9985
150 10 5 20 0.9617 13 0.9897 11 0.9985
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Observations M N $ \sigma $
0.1 0.01 0.001
iter SSIM iter SSIM iter SSIM
4 75 40 20 0.9573 12 0.9895 12 0.9985
9 50 20 19 0.9576 12 0.9895 11 0.9985
30 25 10 20 0.9581 14 0.9896 11 0.9985
40 20 10 19 0.9584 15 0.9896 12 0.9985
100 15 5 15 0.9604 12 0.9896 11 0.9985
150 10 5 20 0.9617 13 0.9897 11 0.9985
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