Total generalized variation regularization in data assimilation for Burgers' equation

Juan Carlos De los Reyes; Estefanía Loayza-Romero

doi:10.3934/ipi.2019035

Article Contents

2019, Volume 13, Issue 4: 755-786. Doi: 10.3934/ipi.2019035

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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

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Abstract

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:

Mathematics Subject Classification: Primary: 76B75, 49M15; Secondary: 49K20.

Citation:

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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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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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