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Recovering the initial condition in the one-phase Stefan problem

  • * Corresponding author: C. Ghanmi

    * Corresponding author: C. Ghanmi 

The work of F. Triki is supported in part by the grant ANR-17-CE40-0029 of the French National Research Agency ANR (project MultiOnde)

Abstract / Introduction Full Text(HTML) Figure(6) / Table(6) Related Papers Cited by
  • We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data.

    Mathematics Subject Classification: Primary: 35R30, 80A22, 45Q05, 35B35; Secondary: 65M32.

    Citation:

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  • Figure 1.  The exact initial condition $ u_0(x) $ and approximate solution with different Gaussian noise levels obtained with $ \lambda = 10^{-3} $, $ M = 250 $ using Tikhonov Regularization method

    Figure 2.  The exact initial condition $ u_0(x) $ and approximate solution with different Gaussian noise levels obtained with $ M = 250 $ using Landweber method

    Figure 3.  The exact initial condition $ u_0(x) $ and the approximate solution with different Gaussian noise levels obtained with $ \lambda = 10^{-2} $, $ M = 250 $ using Tikhonov method

    Figure 4.  The exact initial condition $ u_0(x) $ and approximate solution with different Gaussian noise levels obtained with $ M = 250 $ using Landweber method

    Figure 5.  The initial condition $ u_0(x) $ and approximate solution with different Gaussian noise levels obtained with $ \lambda = 10^{-3} $ and $ M = 250 $ using Tikhonov method

    Figure 6.  The initial condition $ u_0(x) $ and approximate solution with different Gaussian noise levels obtained with $ M = 250 $ using Landweber method

    Table 1.  Relative errors using Tikhonov method

    $ \lambda $ Noise on $ s(t) $ ($ \% $) $ \frac{||{u_0-U_0}||_2}{||{u_0}||_2} $
    $ 10^{-3} $ 0 $ \% $ 0.0425
    $ 10^{-3} $ 1 $ \% $ 0.0472
    $ 10^{-3} $ 2 $ \% $ 0.0571
    $ 10^{-3} $ 3 $ \% $ 0.0669
     | Show Table
    DownLoad: CSV

    Table 2.  Relative errors using Landweber method

    Noise on $ s(t) $ ($ \% $) $ \frac{||{u_0-U_0}||_2}{||{u_0}||_2} $
    0 $ \% $ 0.0846
    1 $ \% $ 0.0917
    2 $ \% $ 0.1026
    3 $ \% $ 0.1115
     | Show Table
    DownLoad: CSV

    Table 3.  Relative errors using Tikhonov method

    $ \lambda $ Noise on $ s(t) $ ($ \% $) $ \frac{||{u_0-U_0}||_2}{||{u_0}||_2} $
    $ 10^{-2} $ 0 $ \% $ 0.0953
    $ 10^{-2} $ 1 $ \% $ 0.0997
    $ 10^{-2} $ 2 $ \% $ 0.1082
    $ 10^{-2} $ 3 $ \% $ 0.1465
     | Show Table
    DownLoad: CSV

    Table 4.  Relative errors using Landweber method

    Noise on $ s(t) $ ($ \% $) $ \frac{||{u_0-U_0}||_2}{||{u_0}||_2} $
    0 $ \% $ 0.1017
    1 $ \% $ 0.1188
    2 $ \% $ 0.1321
    3 $ \% $ 0.1520
     | Show Table
    DownLoad: CSV

    Table 5.  Relative errors using Tikhonov method

    $ \lambda $ Noise on $ s(t) $ ($ \% $) $ \frac{||{u_0-U_0}||_2}{||{u_0}||_2} $
    $ 10^{-3} $ 0 $ \% $ 0.0714
    $ 10^{-3} $ 1 $ \% $ 0.0866
    $ 10^{-3} $ 2 $ \% $ 0.0916
    $ 10^{-3} $ 3 $ \% $ 0.1002
     | Show Table
    DownLoad: CSV

    Table 6.  Relative errors using Landweber method

    Noise on $ s(t) $ ($ \% $) $ \frac{||{u_0-U_0}||_2}{||{u_0}||_2} $
    0 $ \% $ 0.0690
    1 $ \% $ 0.0755
    2 $ \% $ 0.0970
    3 $ \% $ 0.1132
     | Show Table
    DownLoad: CSV
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