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May  2022, 15(5): 1143-1164. doi: 10.3934/dcdss.2021087

Recovering the initial condition in the one-phase Stefan problem

1. 

Faculty of Sciences of Tunis 2092, University of Tunis El Manar, Tunis, Tunisia

2. 

Laboratoire Jean Kuntzmann, UMR CNRS 5224, Université Grenoble-Alpes, 700 Avenue Centrale, 38401 Saint-Martin-d'Hères, France

* Corresponding author: C. Ghanmi

Received  March 2021 Revised  May 2021 Published  May 2022 Early access  July 2021

Fund Project: 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)

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.

Citation: Chifaa Ghanmi, Saloua Mani Aouadi, Faouzi Triki. Recovering the initial condition in the one-phase Stefan problem. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1143-1164. doi: 10.3934/dcdss.2021087
References:
[1]

K. AmmariA. Bchatnia and K. El Mufti, A remark on observability of the wave equation with moving boundary, J. Appl. Anal., 23 (2017), 43-51.  doi: 10.1515/jaa-2017-0007.

[2]

K. Ammari and F. Triki, On weak observability for evolution systems with skew-adjoint generators, SIAM J. Math. Anal., 52 (2020), 1884-1902.  doi: 10.1137/19M1241830.

[3]

G. Bruckner and J. Cheng, Tikhonov regularization for an integral equation of the first kind with logarithmic kernel, J. Inverse Ill-Posed Probl., 8 (2000), 665-675.  doi: 10.1515/jiip.2000.8.6.665.

[4]

J. R. Cannon and J. Douglas Jr., The Cauchy problem for the heat equation, SIAM J. Numer. Anal., 4 (1967), 317-336.  doi: 10.1137/0704028.

[5]

J. R. Cannon and J. Douglas Jr., The stability of the boundary in a Stefan problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 21 (1967), 83-91. 

[6]

J. R. Cannon and C. D. Hill, Existence, uniqueness, stability, and monotone dependence in a Stefan problem for the heat equation, J. Math. Mech., 17 (1967), 1-19.  doi: 10.1512/iumj.1968.17.17001.

[7]

J. R. Cannon and M. Primicerio, Remarks on the one-phase Stefan problem for the heat equation with the flux prescribed on the fixed boundary, J. Math. Anal. Appl., 35 (1971), 361-373.  doi: 10.1016/0022-247X(71)90223-X.

[8]

M. Choulli, Various stability estimates for the problem of determining an initial heat distribution from a single measurement, Riv. Math. Univ. Parma (N.S.), 7 (2016), 279-307. 

[9]

M. Choulli and M. Yamamoto, Logarithmic stability of parabolic Cauchy problems, preprint, arXiv: 1702.06299v4.

[10]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.

[11]

E. Fernández-CaraF. Hernández and J. Límaco, Local null controllability of a 1D Stefan problem, Bull. Braz. Math. Soc. (N.S.), 50 (2019), 745-769.  doi: 10.1007/s00574-018-0093-9.

[12]

E. Fernández-CaraJ. Limaco and S. B. de Menezes, On the controllability of a free-boundary problem for the 1D heat equation, Systems Control Lett., 87 (2016), 29-35.  doi: 10.1016/j.sysconle.2015.10.011.

[13]

A. Friedman, Variational Principles and Free Boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982.

[14]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.

[15]

G. C. GarciaA. Osses and M. Tapia, A heat source reconstruction formula from single internal measurements using a family of null controls, J. Inverse Ill-Posed Probl., 21 (2013), 755-779.  doi: 10.1515/jip-2011-0001.

[16]

B. Geshkovski and E. Zuazua, Controllability of one-dimensional viscous free boundary flows, SIAM J. Control Optim., 59 (2021), 1830-1850.  doi: 10.1137/19M1285354.

[17]

C. Ghanmi, S. Mani-Aouadi and F. Triki, Identification of a Boundary Influx Condition in A One-Phase Stefan Problem, Appl. Anal., to appear.

[18]

N. L Gol'dman, Inverse Stefan Problems, Springer Science & Business Media, 2012.

[19]

A. HajiollowY. LotfiK. ParandA. H. HadianK. Rashedi and J. A. Rad, Recovering a moving boundary from Cauchy data in an inverse problem which arises in modeling brain tumor treatment: The (quasi) linearization idea combined with radial basis functions (RBFs) approximation, Engineering with Computers, 37 (2021), 1735-1749.  doi: 10.1007/s00366-019-00909-8.

[20]

M. HankeA. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.  doi: 10.1007/s002110050158.

[21]

P. Jochum, The numerical solution of the inverse Stefan problem, Numer. Math., 34 (1980), 411-429.  doi: 10.1007/BF01403678.

[22]

B. T. JohanssonD. Lesnic and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan problem, Appl. Math. Model., 35 (2011), 4367-4378.  doi: 10.1016/j.apm.2011.03.005.

[23]

P. Knabner, Control of Stefan problems by means of linear-quadratic defect minimization, Numer. Math., 46 (1985), 429-442.  doi: 10.1007/BF01389495.

[24]

W. T. Kyner, An existence and uniqueness theorem for a nonlinear Stefan problem, J. Math. Mech., 8 (1959), 483-498.  doi: 10.1512/iumj.1959.8.58035.

[25]

O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Soc., 1968.

[26]

L. Landweber, An iteration formula for Fredholm integral equations of the first kind, Amer. J. Math., 73 (1951), 615-624.  doi: 10.2307/2372313.

[27]

J. LiM. Yamamoto and J. Zou, Conditional stability and numerical reconstruction of initial temperature, Commun. Pure Appl. Anal., 8 (2009), 361-382.  doi: 10.3934/cpaa.2009.8.361.

[28]

R. Nevanlinna, H. Behnke, L. V. Grauert, H. Ahlfors, D. C. Spencer, L. Bers, K. Kodaira, M. Heins and J. A. Jenkins, Analytic Functions, Berlin, Springer, 1970.

[29]

R. Reemtsen and A. Kirsch, A method for the numerical solution of the one-dimensional inverse Stefan problem, Numer. Math., 45 (1984), 253-273.  doi: 10.1007/BF01389470.

[30]

L. I. Rubenšteǐn, The Stefan Problem, Translations of Mathematical Monographs, 27, American Mathematical Society, Providence, RI, 1971.

[31]

W. Rudin, Real and Complex Analysis, 2$^{nd}$ edition, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974.

[32]

T. Wei and M. Yamamoto, Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation, Inverse Probl. Sci. Eng., 17 (2009), 551-567.  doi: 10.1080/17415970802231610.

[33]

L. C. Wrobel, A boundary element solution to Stefan's problem, Boundary Elements V, (1983).

show all references

References:
[1]

K. AmmariA. Bchatnia and K. El Mufti, A remark on observability of the wave equation with moving boundary, J. Appl. Anal., 23 (2017), 43-51.  doi: 10.1515/jaa-2017-0007.

[2]

K. Ammari and F. Triki, On weak observability for evolution systems with skew-adjoint generators, SIAM J. Math. Anal., 52 (2020), 1884-1902.  doi: 10.1137/19M1241830.

[3]

G. Bruckner and J. Cheng, Tikhonov regularization for an integral equation of the first kind with logarithmic kernel, J. Inverse Ill-Posed Probl., 8 (2000), 665-675.  doi: 10.1515/jiip.2000.8.6.665.

[4]

J. R. Cannon and J. Douglas Jr., The Cauchy problem for the heat equation, SIAM J. Numer. Anal., 4 (1967), 317-336.  doi: 10.1137/0704028.

[5]

J. R. Cannon and J. Douglas Jr., The stability of the boundary in a Stefan problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 21 (1967), 83-91. 

[6]

J. R. Cannon and C. D. Hill, Existence, uniqueness, stability, and monotone dependence in a Stefan problem for the heat equation, J. Math. Mech., 17 (1967), 1-19.  doi: 10.1512/iumj.1968.17.17001.

[7]

J. R. Cannon and M. Primicerio, Remarks on the one-phase Stefan problem for the heat equation with the flux prescribed on the fixed boundary, J. Math. Anal. Appl., 35 (1971), 361-373.  doi: 10.1016/0022-247X(71)90223-X.

[8]

M. Choulli, Various stability estimates for the problem of determining an initial heat distribution from a single measurement, Riv. Math. Univ. Parma (N.S.), 7 (2016), 279-307. 

[9]

M. Choulli and M. Yamamoto, Logarithmic stability of parabolic Cauchy problems, preprint, arXiv: 1702.06299v4.

[10]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.

[11]

E. Fernández-CaraF. Hernández and J. Límaco, Local null controllability of a 1D Stefan problem, Bull. Braz. Math. Soc. (N.S.), 50 (2019), 745-769.  doi: 10.1007/s00574-018-0093-9.

[12]

E. Fernández-CaraJ. Limaco and S. B. de Menezes, On the controllability of a free-boundary problem for the 1D heat equation, Systems Control Lett., 87 (2016), 29-35.  doi: 10.1016/j.sysconle.2015.10.011.

[13]

A. Friedman, Variational Principles and Free Boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982.

[14]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.

[15]

G. C. GarciaA. Osses and M. Tapia, A heat source reconstruction formula from single internal measurements using a family of null controls, J. Inverse Ill-Posed Probl., 21 (2013), 755-779.  doi: 10.1515/jip-2011-0001.

[16]

B. Geshkovski and E. Zuazua, Controllability of one-dimensional viscous free boundary flows, SIAM J. Control Optim., 59 (2021), 1830-1850.  doi: 10.1137/19M1285354.

[17]

C. Ghanmi, S. Mani-Aouadi and F. Triki, Identification of a Boundary Influx Condition in A One-Phase Stefan Problem, Appl. Anal., to appear.

[18]

N. L Gol'dman, Inverse Stefan Problems, Springer Science & Business Media, 2012.

[19]

A. HajiollowY. LotfiK. ParandA. H. HadianK. Rashedi and J. A. Rad, Recovering a moving boundary from Cauchy data in an inverse problem which arises in modeling brain tumor treatment: The (quasi) linearization idea combined with radial basis functions (RBFs) approximation, Engineering with Computers, 37 (2021), 1735-1749.  doi: 10.1007/s00366-019-00909-8.

[20]

M. HankeA. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.  doi: 10.1007/s002110050158.

[21]

P. Jochum, The numerical solution of the inverse Stefan problem, Numer. Math., 34 (1980), 411-429.  doi: 10.1007/BF01403678.

[22]

B. T. JohanssonD. Lesnic and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan problem, Appl. Math. Model., 35 (2011), 4367-4378.  doi: 10.1016/j.apm.2011.03.005.

[23]

P. Knabner, Control of Stefan problems by means of linear-quadratic defect minimization, Numer. Math., 46 (1985), 429-442.  doi: 10.1007/BF01389495.

[24]

W. T. Kyner, An existence and uniqueness theorem for a nonlinear Stefan problem, J. Math. Mech., 8 (1959), 483-498.  doi: 10.1512/iumj.1959.8.58035.

[25]

O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Soc., 1968.

[26]

L. Landweber, An iteration formula for Fredholm integral equations of the first kind, Amer. J. Math., 73 (1951), 615-624.  doi: 10.2307/2372313.

[27]

J. LiM. Yamamoto and J. Zou, Conditional stability and numerical reconstruction of initial temperature, Commun. Pure Appl. Anal., 8 (2009), 361-382.  doi: 10.3934/cpaa.2009.8.361.

[28]

R. Nevanlinna, H. Behnke, L. V. Grauert, H. Ahlfors, D. C. Spencer, L. Bers, K. Kodaira, M. Heins and J. A. Jenkins, Analytic Functions, Berlin, Springer, 1970.

[29]

R. Reemtsen and A. Kirsch, A method for the numerical solution of the one-dimensional inverse Stefan problem, Numer. Math., 45 (1984), 253-273.  doi: 10.1007/BF01389470.

[30]

L. I. Rubenšteǐn, The Stefan Problem, Translations of Mathematical Monographs, 27, American Mathematical Society, Providence, RI, 1971.

[31]

W. Rudin, Real and Complex Analysis, 2$^{nd}$ edition, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974.

[32]

T. Wei and M. Yamamoto, Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation, Inverse Probl. Sci. Eng., 17 (2009), 551-567.  doi: 10.1080/17415970802231610.

[33]

L. C. Wrobel, A boundary element solution to Stefan's problem, Boundary Elements V, (1983).

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
$ \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
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
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
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
$ \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
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
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
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
$ \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
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
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
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