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

Total generalized variation regularization in data assimilation for Burgers' equation

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.

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

[2]

K. BrediesY. 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.

[3]

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

[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

[5]

C. J. BuddM. 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.

[6]

T. F. ChanG. 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.

[7]

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

[8]

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

[9]

J. C. De los ReyesC. 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.

[10]

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

[11]

M. A. FreitagN. 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.

[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.

[13]

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

[14]

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

[15]

F. KnollK. BrediesT. Pock and R. Stollberger, Second order total generalized variation (TGV) for MRI, Magn. Reson. Med., 65 (2011), 480-491. doi: 10.1002/mrm.22595.

[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.

[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.

[18]

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

[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.

[20]

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

[21]

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

[22]

Z. WangA. C. BovikH. 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.

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

[2]

K. BrediesY. 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.

[3]

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

[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

[5]

C. J. BuddM. 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.

[6]

T. F. ChanG. 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.

[7]

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

[8]

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

[9]

J. C. De los ReyesC. 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.

[10]

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

[11]

M. A. FreitagN. 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.

[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.

[13]

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

[14]

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

[15]

F. KnollK. BrediesT. Pock and R. Stollberger, Second order total generalized variation (TGV) for MRI, Magn. Reson. Med., 65 (2011), 480-491. doi: 10.1002/mrm.22595.

[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.

[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.

[18]

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

[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.

[20]

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

[21]

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

[22]

Z. WangA. C. BovikH. 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.

Figure 1.  Exact solution
Figure 2.  SSIM index for the TV and TGV regularizations
Figure 3.  Best solutions for TV (left) and TGV (right) regularizations
Figure 4.  Observations vs Optimal state
Figure 5.  Exact solutions and best solutions obtained for the experiment
Figure 6.  Global convergence of the algorithm
Figure 7.  Locally superlinear convergence of the algorithm
Figure 8.  Exact solutions for the experiment
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
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
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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