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doi: 10.3934/dcdss.2021087
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## 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 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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021087
##### References:
 [1] K. Ammari, A. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [9] M. Choulli and M. Yamamoto, Logarithmic stability of parabolic Cauchy problems, preprint, arXiv: 1702.06299v4. Google Scholar [10] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.  Google Scholar [11] E. Fernández-Cara, F. 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.  Google Scholar [12] E. Fernández-Cara, J. 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.  Google Scholar [13] A. Friedman, Variational Principles and Free Boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982.  Google Scholar [14] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.  Google Scholar [15] G. C. Garcia, A. 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.  Google Scholar [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.  Google Scholar [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. Google Scholar [18] N. L Gol'dman, Inverse Stefan Problems, Springer Science & Business Media, 2012. Google Scholar [19] A. Hajiollow, Y. Lotfi, K. Parand, A. H. Hadian, K. 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.  Google Scholar [20] M. Hanke, A. 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.  Google Scholar [21] P. Jochum, The numerical solution of the inverse Stefan problem, Numer. Math., 34 (1980), 411-429.  doi: 10.1007/BF01403678.  Google Scholar [22] B. T. Johansson, D. 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.  Google Scholar [23] P. Knabner, Control of Stefan problems by means of linear-quadratic defect minimization, Numer. Math., 46 (1985), 429-442.  doi: 10.1007/BF01389495.  Google Scholar [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.  Google Scholar [25] O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Soc., 1968. Google Scholar [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.  Google Scholar [27] J. Li, M. 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.  Google Scholar [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. Google Scholar [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.  Google Scholar [30] L. I. Rubenšteǐn, The Stefan Problem, Translations of Mathematical Monographs, 27, American Mathematical Society, Providence, RI, 1971.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] L. C. Wrobel, A boundary element solution to Stefan's problem, Boundary Elements V, (1983). Google Scholar

show all references

##### References:
 [1] K. Ammari, A. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [9] M. Choulli and M. Yamamoto, Logarithmic stability of parabolic Cauchy problems, preprint, arXiv: 1702.06299v4. Google Scholar [10] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.  Google Scholar [11] E. Fernández-Cara, F. 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.  Google Scholar [12] E. Fernández-Cara, J. 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.  Google Scholar [13] A. Friedman, Variational Principles and Free Boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982.  Google Scholar [14] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.  Google Scholar [15] G. C. Garcia, A. 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.  Google Scholar [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.  Google Scholar [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. Google Scholar [18] N. L Gol'dman, Inverse Stefan Problems, Springer Science & Business Media, 2012. Google Scholar [19] A. Hajiollow, Y. Lotfi, K. Parand, A. H. Hadian, K. 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.  Google Scholar [20] M. Hanke, A. 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.  Google Scholar [21] P. Jochum, The numerical solution of the inverse Stefan problem, Numer. Math., 34 (1980), 411-429.  doi: 10.1007/BF01403678.  Google Scholar [22] B. T. Johansson, D. 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.  Google Scholar [23] P. Knabner, Control of Stefan problems by means of linear-quadratic defect minimization, Numer. Math., 46 (1985), 429-442.  doi: 10.1007/BF01389495.  Google Scholar [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.  Google Scholar [25] O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Soc., 1968. Google Scholar [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.  Google Scholar [27] J. Li, M. 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.  Google Scholar [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. Google Scholar [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.  Google Scholar [30] L. I. Rubenšteǐn, The Stefan Problem, Translations of Mathematical Monographs, 27, American Mathematical Society, Providence, RI, 1971.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] L. C. Wrobel, A boundary element solution to Stefan's problem, Boundary Elements V, (1983). Google Scholar
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
The exact initial condition $u_0(x)$ and approximate solution with different Gaussian noise levels obtained with $M = 250$ using Landweber method
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
The exact initial condition $u_0(x)$ and approximate solution with different Gaussian noise levels obtained with $M = 250$ using Landweber method
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
The initial condition $u_0(x)$ and approximate solution with different Gaussian noise levels obtained with $M = 250$ using Landweber method
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
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
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
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
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
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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