# American Institute of Mathematical Sciences

June  2022, 16(3): 569-594. doi: 10.3934/ipi.2021062

## 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 Published  June 2022 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 and Imaging, 2022, 16 (3) : 569-594. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [13] J. Hadamard, Sur les problèmes aux dérivées partielles et leur signification physique, Princeton University Bulletin, (1902), 49–52. [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. [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. [16] V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ edition, Springer, New York, 2006. [17] V. Komornik, Exact Controllability and Stabilization the Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. [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. [19] J. Nocedal and S. J. Wright, Numerical Optimization, 2$^{nd}$ edition, Springer-Verlag, New York, 2006. [20] M. Richter, Inverse Problems — Basics, Theory and Applications in Geophysics, Lecture Notes in Geosystems Mathematics and Computing, Birkhäuser/Springer, Cham, 2016. [21] V. G. Romanov, Investigation Methods for Inverse Problems, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002. doi: 10.1515/9783110943849. [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. [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. [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. [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. [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. [27] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 75pp. doi: 10.1088/0266-5611/25/12/123013.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [13] J. Hadamard, Sur les problèmes aux dérivées partielles et leur signification physique, Princeton University Bulletin, (1902), 49–52. [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. [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. [16] V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ edition, Springer, New York, 2006. [17] V. Komornik, Exact Controllability and Stabilization the Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. [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. [19] J. Nocedal and S. J. Wright, Numerical Optimization, 2$^{nd}$ edition, Springer-Verlag, New York, 2006. [20] M. Richter, Inverse Problems — Basics, Theory and Applications in Geophysics, Lecture Notes in Geosystems Mathematics and Computing, Birkhäuser/Springer, Cham, 2016. [21] V. G. Romanov, Investigation Methods for Inverse Problems, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002. doi: 10.1515/9783110943849. [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. [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. [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. [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. [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. [27] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 75pp. doi: 10.1088/0266-5611/25/12/123013.
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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