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doi: 10.3934/ipi.2021062
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## Uniqueness and numerical reconstruction for inverse problems dealing with interval size search

 1 Universidad del País Vasco, Dpto. Matemáticas, , Barrio Sarriena s/n, 48940 Leioa (Bizkaia), Spain 2 Fudan University, School of Mathematical Sciences, Shanghai 200433, China 3 Universidad de Sevilla, Dpto. EDAN e IMUS, Campus Reina Mercedes, 41012 Sevilla, Spain 4 The University of Tokyo, Department of Mathematical Sciences, 3-8-1 Komaba, Meguro, Tokyo 153, Japan

*Corresponding author: Jone Apraiz

Received  January 2021 Revised  July 2021 Early access October 2021

We consider a heat equation and a wave equation in one spatial dimension. This article deals with the inverse problem of determining the size of the spatial interval from some extra boundary information on the solution. Under several different circumstances, we prove uniqueness, non-uniqueness and some size estimates. Moreover, we numerically solve the inverse problems and compute accurate approximations of the size. This is illustrated with several satisfactory numerical experiments.

Citation: Jone Apraiz, Jin Cheng, Anna Doubova, Enrique Fernández-Cara, Masahiro Yamamoto. Uniqueness and numerical reconstruction for inverse problems dealing with interval size search. Inverse Problems & Imaging, doi: 10.3934/ipi.2021062
##### References:
 [1] H. T. Banks, F. Kojima and W. P. Winfree, Boundary estimation problems arising in thermal tomography, Inverse Problems, 6 (1990), 897-921.  doi: 10.1088/0266-5611/6/6/003.  Google Scholar [2] M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.  Google Scholar [3] L. Borcea, H. Kang, H. Liu and G. Uhlmann, Inverse Problems and Imaging, Lectures from the Workshop held at the Institut Henri Poincaré, Paris, February 20–22, 2013. Edited by H. Ammari and J. Garnier. Panoramas et Synthèses, 44. Société Mathématique de France, Paris, 2015.  Google Scholar [4] K. Bryan and L. Caudill, Reconstruction of an unknown boundary portion from Cauchy data in $n$ dimensions, Inverse Problems, 21 (2005), 239-255.  doi: 10.1088/0266-5611/21/1/015.  Google Scholar [5] K. Bryan and L. F. Caudill Jr., An inverse problem in thermal imaging, SIAM J. Appl. Math., 56 (1996), 715-735.  doi: 10.1137/S0036139994277828.  Google Scholar [6] K. Bryan and L. F. Caudill Jr., Stability and reconstruction for an inverse problem for the heat equation, Inverse Problems, 14 (1998), 1429-1453.  doi: 10.1088/0266-5611/14/6/005.  Google Scholar [7] R. Chapko, R. Kress and J. R. Yoon, On the numerical solution of an inverse boundary value problem for the heat equation, Inverse Problems, 14 (1998), 853-867.  doi: 10.1088/0266-5611/14/4/006.  Google Scholar [8] R. Chapko, R. Kress and J. R. Yoon, An inverse boundary value problem for the heat equation: The Neumann condition, Inverse Problems, 15 (1999), 1033-1046.  doi: 10.1088/0266-5611/15/4/313.  Google Scholar [9] P. P. Carvalho, A. Doubova, E. Fernández-Cara and J. Rocha, Some new results for geometric inverse problems with the method of fundamental solutions, Inverse Probl. Sci. Eng., 29 (2021), 131-152.  doi: 10.1080/17415977.2020.1782398.  Google Scholar [10] A. Doubova and E. Fernández-Cara, Some geometric inverse problems for the linear wave equation, Inverse Probl. Imaging, 9 (2015), 371-393.  doi: 10.3934/ipi.2015.9.371.  Google Scholar [11] A. Doubova and E. Fernández-Cara, Some geometric inverse problems for the Lamé system with applications in elastography, Appl. Math. Optim., 82 (2020), 1-21.  doi: 10.1007/s00245-018-9487-8.  Google Scholar [12] T. P. Fredman, A boundary identification method for an inverse heat conduction problem with an application in ironmaking, Heat Mass Transfer, 41 (2004), 95-103.  doi: 10.1007/s00231-004-0543-3.  Google Scholar [13] J. Hadamard, Sur les problèmes aux dérivées partielles et leur signification physique, Princeton University Bulletin, (1902), 49–52. Google Scholar [14] M. Hanke, A Taste of Inverse Problems — Basic Theory and Examples, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2017. doi: 10.1137/1.9781611974942.ch1.  Google Scholar [15] A. Hasanov and V. G. Romanov, Introduction to Inverse Problems for Differential Equations, Springer, Cham, 2017. doi: 10.1007/978-3-319-62797-7.  Google Scholar [16] V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ edition, Springer, New York, 2006.  Google Scholar [17] V. Komornik, Exact Controllability and Stabilization the Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar [18] M. M. Lavrentiev, A. V. Avdeev, M. M. Lavrentiev Jr. and V. I. Priimenko, Inverse Problems of Mathematical Physics, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2003. doi: 10.1515/9783110915525.  Google Scholar [19] J. Nocedal and S. J. Wright, Numerical Optimization, 2$^{nd}$ edition, Springer-Verlag, New York, 2006.  Google Scholar [20] M. Richter, Inverse Problems — Basics, Theory and Applications in Geophysics, Lecture Notes in Geosystems Mathematics and Computing, Birkhäuser/Springer, Cham, 2016.  Google Scholar [21] V. G. Romanov, Investigation Methods for Inverse Problems, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002. doi: 10.1515/9783110943849.  Google Scholar [22] A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Inverse and Ill-posed Problems Series, 52. Walter de Gruyter GmbH & Co. KG, Berlin, 2007. doi: 10.1515/9783110205794.  Google Scholar [23] S. Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates, Inverse Problems, 24 (2008), 81pp. doi: 10.1088/0266-5611/24/2/023001.  Google Scholar [24] C. R. Vogel, Computational Methods for Inverse Problems, Frontiers in Applied Mathematics, 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717570.  Google Scholar [25] Y. Wang, J. Cheng, J. Nakagawa and M. Yamamoto, A numerical method for solving the inverse heat conduction problem without initial value, Inverse Probl. Sci. Eng., 18 (2010), 655-671.  doi: 10.1080/17415971003698615.  Google Scholar [26] 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 [27] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 75pp. doi: 10.1088/0266-5611/25/12/123013.  Google Scholar

show all references

##### References:
 [1] H. T. Banks, F. Kojima and W. P. Winfree, Boundary estimation problems arising in thermal tomography, Inverse Problems, 6 (1990), 897-921.  doi: 10.1088/0266-5611/6/6/003.  Google Scholar [2] M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.  Google Scholar [3] L. Borcea, H. Kang, H. Liu and G. Uhlmann, Inverse Problems and Imaging, Lectures from the Workshop held at the Institut Henri Poincaré, Paris, February 20–22, 2013. Edited by H. Ammari and J. Garnier. Panoramas et Synthèses, 44. Société Mathématique de France, Paris, 2015.  Google Scholar [4] K. Bryan and L. Caudill, Reconstruction of an unknown boundary portion from Cauchy data in $n$ dimensions, Inverse Problems, 21 (2005), 239-255.  doi: 10.1088/0266-5611/21/1/015.  Google Scholar [5] K. Bryan and L. F. Caudill Jr., An inverse problem in thermal imaging, SIAM J. Appl. Math., 56 (1996), 715-735.  doi: 10.1137/S0036139994277828.  Google Scholar [6] K. Bryan and L. F. Caudill Jr., Stability and reconstruction for an inverse problem for the heat equation, Inverse Problems, 14 (1998), 1429-1453.  doi: 10.1088/0266-5611/14/6/005.  Google Scholar [7] R. Chapko, R. Kress and J. R. Yoon, On the numerical solution of an inverse boundary value problem for the heat equation, Inverse Problems, 14 (1998), 853-867.  doi: 10.1088/0266-5611/14/4/006.  Google Scholar [8] R. Chapko, R. Kress and J. R. Yoon, An inverse boundary value problem for the heat equation: The Neumann condition, Inverse Problems, 15 (1999), 1033-1046.  doi: 10.1088/0266-5611/15/4/313.  Google Scholar [9] P. P. Carvalho, A. Doubova, E. Fernández-Cara and J. Rocha, Some new results for geometric inverse problems with the method of fundamental solutions, Inverse Probl. Sci. Eng., 29 (2021), 131-152.  doi: 10.1080/17415977.2020.1782398.  Google Scholar [10] A. Doubova and E. Fernández-Cara, Some geometric inverse problems for the linear wave equation, Inverse Probl. Imaging, 9 (2015), 371-393.  doi: 10.3934/ipi.2015.9.371.  Google Scholar [11] A. Doubova and E. Fernández-Cara, Some geometric inverse problems for the Lamé system with applications in elastography, Appl. Math. Optim., 82 (2020), 1-21.  doi: 10.1007/s00245-018-9487-8.  Google Scholar [12] T. P. Fredman, A boundary identification method for an inverse heat conduction problem with an application in ironmaking, Heat Mass Transfer, 41 (2004), 95-103.  doi: 10.1007/s00231-004-0543-3.  Google Scholar [13] J. Hadamard, Sur les problèmes aux dérivées partielles et leur signification physique, Princeton University Bulletin, (1902), 49–52. Google Scholar [14] M. Hanke, A Taste of Inverse Problems — Basic Theory and Examples, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2017. doi: 10.1137/1.9781611974942.ch1.  Google Scholar [15] A. Hasanov and V. G. Romanov, Introduction to Inverse Problems for Differential Equations, Springer, Cham, 2017. doi: 10.1007/978-3-319-62797-7.  Google Scholar [16] V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ edition, Springer, New York, 2006.  Google Scholar [17] V. Komornik, Exact Controllability and Stabilization the Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.  Google Scholar [18] M. M. Lavrentiev, A. V. Avdeev, M. M. Lavrentiev Jr. and V. I. Priimenko, Inverse Problems of Mathematical Physics, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2003. doi: 10.1515/9783110915525.  Google Scholar [19] J. Nocedal and S. J. Wright, Numerical Optimization, 2$^{nd}$ edition, Springer-Verlag, New York, 2006.  Google Scholar [20] M. Richter, Inverse Problems — Basics, Theory and Applications in Geophysics, Lecture Notes in Geosystems Mathematics and Computing, Birkhäuser/Springer, Cham, 2016.  Google Scholar [21] V. G. Romanov, Investigation Methods for Inverse Problems, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002. doi: 10.1515/9783110943849.  Google Scholar [22] A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Inverse and Ill-posed Problems Series, 52. Walter de Gruyter GmbH & Co. KG, Berlin, 2007. doi: 10.1515/9783110205794.  Google Scholar [23] S. Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates, Inverse Problems, 24 (2008), 81pp. doi: 10.1088/0266-5611/24/2/023001.  Google Scholar [24] C. R. Vogel, Computational Methods for Inverse Problems, Frontiers in Applied Mathematics, 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717570.  Google Scholar [25] Y. Wang, J. Cheng, J. Nakagawa and M. Yamamoto, A numerical method for solving the inverse heat conduction problem without initial value, Inverse Probl. Sci. Eng., 18 (2010), 655-671.  doi: 10.1080/17415971003698615.  Google Scholar [26] 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 [27] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 75pp. doi: 10.1088/0266-5611/25/12/123013.  Google Scholar
Heat equation, $u_0 = 0$ and $\eta\neq 0$. The computed solution
Heat equation with $u_0 = 0$ and $\eta\neq 0$
Heat equation, $u_0\neq 0$ and large $\eta$. The computed solution
Heat equation, fixed $u_0$ and large $\eta$
Heat equation, $\eta = 0$ and fixed $u_0(x)$
Heat equation, $\eta = 0$, fixed $u_0(x)$
Heat equation, fixed $u_0$ and $\eta = 0$
Wave equation with $(u_0, u_1) = (0, 0)$ and $\eta \not = 0$. The computed solution
Wave equation with $(u_0, u_1) = (0, 0)$ and $\eta \not = 0$
Wave equation with $(u_0, u_1) \not = (0, 0)$ and large $\eta = 0$. The computed solution
Wave equation with $(u_0, u_1) \not = (0, 0)$ and large $\eta = 0$
Wave equation, $\eta = 0$, fixed $u_0(x)$
Wave equation, $\eta = 0$, fixed $u_0(x)$
Wave equation, fixed $u_0$, $\eta = 0$
Wave equation, fixed $u_0(x)$, $\eta = 0$
The non-uniqueness cases for the two equations
Heat equation, $u_0 = 0$ and $\eta\neq 0$. Results with random noise in the target (the desired length is $L_d = 2$)
 % noise Cost Iterates Computed $L_c$ 1% 1.e-4 10 1.997586488 0.1% 1.e-6 9 1.999864829 0.01% 1.e-9 8 2.000017283 0.001% 1.e-1 9 1.999998535 0% 1.e-16 9 1.999999991
 % noise Cost Iterates Computed $L_c$ 1% 1.e-4 10 1.997586488 0.1% 1.e-6 9 1.999864829 0.01% 1.e-9 8 2.000017283 0.001% 1.e-1 9 1.999998535 0% 1.e-16 9 1.999999991
Heat equation, fixed $u_0$ and large $\eta$. Results with random noise in the target (the desired length is $L_d = 2$)
 % noise Cost Iterates Computed $L_c$ 1% 1.e-3 8 1.999952948 0.1% 1.e-5 11 2.000008948 0.01% 1.e-8 9 2.000003174 0.001% 1.e-10 7 1.999999932 0% 1.e-12 13 1.999999964
 % noise Cost Iterates Computed $L_c$ 1% 1.e-3 8 1.999952948 0.1% 1.e-5 11 2.000008948 0.01% 1.e-8 9 2.000003174 0.001% 1.e-10 7 1.999999932 0% 1.e-12 13 1.999999964
Wave equation, $(u_0, u_1) = (0, 0)$ and $\eta \not = 0$. Results with random noise in the target {(the desired length is $L_d = 2$)
 % noise Cost Iterates Computed $L_c$ 1% 1.e-4 12 2.004287790 0.1% 1.e-6 8 2.000029532 0.01% 1.e-8 8 2.000018464 0.001% 1.e-10 9 1.999995613 0% 1.e-17 8 1.999999994
 % noise Cost Iterates Computed $L_c$ 1% 1.e-4 12 2.004287790 0.1% 1.e-6 8 2.000029532 0.01% 1.e-8 8 2.000018464 0.001% 1.e-10 9 1.999995613 0% 1.e-17 8 1.999999994
Wave equation, $(u_0, u_1) \not = (0, 0)$ and large $\eta = 0$. Cost and $L_c$ with random noise in the target {(desired length is $L_d = 2$)
 % noise Cost Iterates Computed $L_c$ 1% 1.e-1 12 2.324511735 0.1% 1.e-6 11 2.000008724 0.01% 1.e-7 9 1.999805367 0.001% 1.e-12 10 2.000000789 0% 1.e-17 14 1.999931083
 % noise Cost Iterates Computed $L_c$ 1% 1.e-1 12 2.324511735 0.1% 1.e-6 11 2.000008724 0.01% 1.e-7 9 1.999805367 0.001% 1.e-12 10 2.000000789 0% 1.e-17 14 1.999931083
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