# American Institute of Mathematical Sciences

April  2021, 15(2): 367-386. doi: 10.3934/ipi.2020072

## Duality between range and no-response tests and its application for inverse problems

 1 Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan 2 Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan 3 Department of Mathematics and Statistics, University of Reading, RG6 6AH, UK 4 School of Mathematics, Southeast University, Nanjing 210096, China, Nanjing Center for Applied Mathematics, Nanjing 211135, China

* Corresponding author: Gen Nakamura

Received  April 2020 Revised  September 2020 Published  April 2021 Early access  November 2020

In this paper we will show the duality between the range test (RT) and no-response test (NRT) for the inverse boundary value problem for the Laplace equation in $\Omega\setminus\overline D$ with an unknown obstacle $D\Subset\Omega$ whose boundary $\partial D$ is visible from the boundary $\partial\Omega$ of $\Omega$ and a measurement is given as a set of Cauchy data on $\partial\Omega$. Here the Cauchy data is given by a unique solution $u$ of the boundary value problem for the Laplace equation in $\Omega\setminus\overline D$ with homogeneous and inhomogeneous Dirichlet boundary condition on $\partial D$ and $\partial\Omega$, respectively. These testing methods are domain sampling methods to estimate the location of the obstacle using test domains and the associated indicator functions. Also both of these testing methods can test the analytic extension of $u$ to the exterior of a test domain. Since these methods are defined via some operators which are dual to each other, we could expect that there is a duality between the two methods. We will give this duality in terms of the equivalence of the pre-indicator functions associated to their indicator functions. As an application of the duality, the reconstruction of $D$ using the RT gives the reconstruction of $D$ using the NRT and vice versa. We will also give each of these reconstructions without using the duality if the Dirichlet data of the Cauchy data on $\partial\Omega$ is not identically zero and the solution to the associated forward problem does not have any analytic extension across $\partial D$. Moreover, we will show that these methods can still give the reconstruction of $D$ if we a priori knows that $D$ is a convex polygon and it satisfies one of the following two properties: all of its corner angles are irrational and its diameter is less than its distance to $\partial\Omega$.

Citation: Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems and Imaging, 2021, 15 (2) : 367-386. doi: 10.3934/ipi.2020072
##### References:
 [1] G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 755-806. [2] H. Ammari and H. Kang, Polarization and Moment Tensors, With Applications to Inverse Problems and Effective Medium Theory, Springer-Verlag, Berlin, 2007. [3] D. D. Ang, D. D. Trong and M. Yamamoto, Unique continuation and identification of boundary of an elastic body, J. Inverse Ill-Posed Probl., 3 (1996), 417-428.  doi: 10.1515/jiip.1995.3.6.417. [4] M. Bonnet, Inverse acoustic scattering by small-obstacle expansion of a misfit function, Inverse Problems, 24 (2008), 035022, 27pp. doi: 10.1088/0266-5611/24/3/035022. [5] L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Probl. Imaging, 4 (2010), 351-377.  doi: 10.3934/ipi.2010.4.351. [6] M. Burger, A level set method for inverse problems, Inverse Problems, 17 (2001), 1327-1355.  doi: 10.1088/0266-5611/17/5/307. [7] T. Chow, K. Ito and J. Zou, A direct sampling method for electrical impedance tomography, Inverse Problems, 30 (2014), 095003, 25pp. doi: 10.1088/0266-5611/30/9/095003. [8] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^rd$ edition, Springer-Verlag, Berlin, 2013. doi: 10.1007/978-1-4614-4942-3. [9] A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement, Indiana University Mathematics Journal, 38 (1989), 563-579.  doi: 10.1512/iumj.1989.38.38027. [10] N. Higashimori, A conditional stability estimate for determining a cavity in an elastic material, Proc. Japan Acad. Ser. A Math. Sci., 78 (2002), 15-17. [11] N. Honda, G. Nakamura and M. Sini, Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators, Math. Ann., 355 (2013), 401-427.  doi: 10.1007/s00208-012-0786-0. [12] N. Honda, G. Nakamura, R. Potthast and M. Sini, The no-response approach and its relation to non-iterative methods for the inverse scattering, Annali di Matematica, 187 (2008), 7-37.  doi: 10.1007/s10231-006-0030-1. [13] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241.  doi: 10.1088/0266-5611/15/5/308. [14] M. Ikehata and T. Ohe, A numerical method for finding the convex hull of polygonal cavities using the enclosure method, Inverse Problems, 18 (2002), 111-124.  doi: 10.1088/0266-5611/18/1/308. [15] V. Isakov, Inverse Problems for Partial Differential Equations, 3$^rd$ edition, Springer-Verlag, Berlin, 2017. doi: 10.1007/978-3-319-51658-5. [16] R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 1207-1223.  doi: 10.1088/0266-5611/21/4/002. [17] D. R. Luke and R. Potthast, The no response test-a sampling method for inverse scattering problems, SIAM J. Appl. Math., 63 (2003), 1292-1312.  doi: 10.1137/S0036139902406887. [18] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. [19] S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, Cambridge, 1973. [20] A. Morrassi and E. Rosset, Stable determination of cavities in elastic bodies, Inverse Problems, 20 (2004), 453-480.  doi: 10.1088/0266-5611/20/2/010. [21] G. Nakamura and R. Potthast, Inverse Modelling, IOP Publishing, Bristol, 2015. [22] R. Potthast, On the convergence of the no response test, SIAM J. Math. Anal., 38 (2007), 1808-1824.  doi: 10.1137/S0036141004441003. [23] R. Potthast and M. Sini, The no response test for the reconstruction of polyhedral objects in electromagnetics, J. Comput. Appl. Math., 234 (2010), 1739-1746.  doi: 10.1016/j.cam.2009.08.023. [24] R. Potthast, J. Sylvester and S. Kusiak, A 'range test' for determining scatteres with unknown physical properties, Inverse Problems, 19 (2003), 533-547.  doi: 10.1088/0266-5611/19/3/304. [25] Q. Zia and R. Potthast, The range test and the no response test for Oseen problems: Theoretical foundation, J. Comput. Appl. Math., 304 (2016), 201-211.  doi: 10.1016/j.cam.2015.11.029.

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##### References:
 [1] G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 755-806. [2] H. Ammari and H. Kang, Polarization and Moment Tensors, With Applications to Inverse Problems and Effective Medium Theory, Springer-Verlag, Berlin, 2007. [3] D. D. Ang, D. D. Trong and M. Yamamoto, Unique continuation and identification of boundary of an elastic body, J. Inverse Ill-Posed Probl., 3 (1996), 417-428.  doi: 10.1515/jiip.1995.3.6.417. [4] M. Bonnet, Inverse acoustic scattering by small-obstacle expansion of a misfit function, Inverse Problems, 24 (2008), 035022, 27pp. doi: 10.1088/0266-5611/24/3/035022. [5] L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Probl. Imaging, 4 (2010), 351-377.  doi: 10.3934/ipi.2010.4.351. [6] M. Burger, A level set method for inverse problems, Inverse Problems, 17 (2001), 1327-1355.  doi: 10.1088/0266-5611/17/5/307. [7] T. Chow, K. Ito and J. Zou, A direct sampling method for electrical impedance tomography, Inverse Problems, 30 (2014), 095003, 25pp. doi: 10.1088/0266-5611/30/9/095003. [8] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3$^rd$ edition, Springer-Verlag, Berlin, 2013. doi: 10.1007/978-1-4614-4942-3. [9] A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement, Indiana University Mathematics Journal, 38 (1989), 563-579.  doi: 10.1512/iumj.1989.38.38027. [10] N. Higashimori, A conditional stability estimate for determining a cavity in an elastic material, Proc. Japan Acad. Ser. A Math. Sci., 78 (2002), 15-17. [11] N. Honda, G. Nakamura and M. Sini, Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators, Math. Ann., 355 (2013), 401-427.  doi: 10.1007/s00208-012-0786-0. [12] N. Honda, G. Nakamura, R. Potthast and M. Sini, The no-response approach and its relation to non-iterative methods for the inverse scattering, Annali di Matematica, 187 (2008), 7-37.  doi: 10.1007/s10231-006-0030-1. [13] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241.  doi: 10.1088/0266-5611/15/5/308. [14] M. Ikehata and T. Ohe, A numerical method for finding the convex hull of polygonal cavities using the enclosure method, Inverse Problems, 18 (2002), 111-124.  doi: 10.1088/0266-5611/18/1/308. [15] V. Isakov, Inverse Problems for Partial Differential Equations, 3$^rd$ edition, Springer-Verlag, Berlin, 2017. doi: 10.1007/978-3-319-51658-5. [16] R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 1207-1223.  doi: 10.1088/0266-5611/21/4/002. [17] D. R. Luke and R. Potthast, The no response test-a sampling method for inverse scattering problems, SIAM J. Appl. Math., 63 (2003), 1292-1312.  doi: 10.1137/S0036139902406887. [18] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. [19] S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, Cambridge, 1973. [20] A. Morrassi and E. Rosset, Stable determination of cavities in elastic bodies, Inverse Problems, 20 (2004), 453-480.  doi: 10.1088/0266-5611/20/2/010. [21] G. Nakamura and R. Potthast, Inverse Modelling, IOP Publishing, Bristol, 2015. [22] R. Potthast, On the convergence of the no response test, SIAM J. Math. Anal., 38 (2007), 1808-1824.  doi: 10.1137/S0036141004441003. [23] R. Potthast and M. Sini, The no response test for the reconstruction of polyhedral objects in electromagnetics, J. Comput. Appl. Math., 234 (2010), 1739-1746.  doi: 10.1016/j.cam.2009.08.023. [24] R. Potthast, J. Sylvester and S. Kusiak, A 'range test' for determining scatteres with unknown physical properties, Inverse Problems, 19 (2003), 533-547.  doi: 10.1088/0266-5611/19/3/304. [25] Q. Zia and R. Potthast, The range test and the no response test for Oseen problems: Theoretical foundation, J. Comput. Appl. Math., 304 (2016), 201-211.  doi: 10.1016/j.cam.2015.11.029.
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