American Institute of Mathematical Sciences

Advanced
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.eduGoogle 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.pdfGoogle 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.eduGoogle 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.pdfGoogle 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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  SSIM index for the TV and TGV regularizations
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Best solutions for TV (left) and TGV (right) regularizations
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Observations vs Optimal state
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Exact solutions and best solutions obtained for the experiment
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Global convergence of the algorithm
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Locally superlinear convergence of the algorithm
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Exact solutions for the experiment
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
Table Options

Download as excel

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
[1]

Guoqiang Wang, Zhongchen Wu, Zhongtuan Zheng, Xinzhong Cai. Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 101-113. doi: 10.3934/naco.2015.5.101
[2]

Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 817-840. doi: 10.3934/dcdsb.2005.5.817
[3]

Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial & Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723
[4]

Gianni Di Pillo, Giampaolo Liuzzi, Stefano Lucidi. A primal-dual algorithm for nonlinear programming exploiting negative curvature directions. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 509-528. doi: 10.3934/naco.2011.1.509
[5]

Panagiotis Stinis. A hybrid method for the inviscid Burgers equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 793-799. doi: 10.3934/dcds.2003.9.793
[6]

Fengmin Wang, Dachuan Xu, Donglei Du, Chenchen Wu. Primal-dual approximation algorithms for submodular cost set cover problems with linear/submodular penalties. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 91-100. doi: 10.3934/naco.2015.5.91
[7]

Xiaojing Ye, Haomin Zhou. Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm. Inverse Problems & Imaging, 2013, 7 (3) : 1031-1050. doi: 10.3934/ipi.2013.7.1031
[8]

Yanqin Bai, Xuerui Gao, Guoqiang Wang. Primal-dual interior-point algorithms for convex quadratic circular cone optimization. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 211-231. doi: 10.3934/naco.2015.5.211
[9]

Siqi Li, Weiyi Qian. Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 37-46. doi: 10.3934/naco.2015.5.37
[10]

Yu-Hong Dai, Xin-Wei Liu, Jie Sun. A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-27. doi: 10.3934/jimo.2018190
[11]

Jie Shen, Li-Lian Wang. Laguerre and composite Legendre-Laguerre Dual-Petrov-Galerkin methods for third-order equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1381-1402. doi: 10.3934/dcdsb.2006.6.1381
[12]

José F. Cariñena, Javier de Lucas Araujo. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics, 2011, 3 (1) : 1-22. doi: 10.3934/jgm.2011.3.1
[13]

Eugenii Shustin, Emilia Fridman, Leonid Fridman. Oscillations in a second-order discontinuous system with delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 339-358. doi: 10.3934/dcds.2003.9.339
[14]

Jiyu Zhong, Shengfu Deng. Two codimension-two bifurcations of a second-order difference equation from macroeconomics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1581-1600. doi: 10.3934/dcdsb.2018062
[15]

Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945
[16]

Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-20. doi: 10.3934/dcdsb.2019146
[17]

Naoki Fujino, Mitsuru Yamazaki. Burgers' type equation with vanishing higher order. Communications on Pure & Applied Analysis, 2007, 6 (2) : 505-520. doi: 10.3934/cpaa.2007.6.505
[18]

Engu Satynarayana, Manas R. Sahoo, Manasa M. Higher order asymptotic for Burgers equation and Adhesion model. Communications on Pure & Applied Analysis, 2017, 16 (1) : 253-272. doi: 10.3934/cpaa.2017012
[19]

Alexandre J. Chorin, Fei Lu, Robert N. Miller, Matthias Morzfeld, Xuemin Tu. Sampling, feasibility, and priors in data assimilation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4227-4246. doi: 10.3934/dcds.2016.36.4227
[20]

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (68)
  • HTML views (216)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences