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Total generalized variation regularization in data assimilation for Burgers' equation

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

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  • 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.

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

    Citation:

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  • 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
     | Show Table
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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
     | Show Table
    DownLoad: CSV
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