# American Institute of Mathematical Sciences

• Previous Article
Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast
• IPI Home
• This Issue
• Next Article
On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method
August 2018, 12(4): 993-1031. doi: 10.3934/ipi.2018042

## Reconstruction of a compact manifold from the scattering data of internal sources

 1 Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland 2 Department of computational and applied mathematics, Rice University, Houston, Texas, USA 3 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK 4 Department of Mathematics, University of California Santa Barbara, Santa Barbara, California, USA

Received  August 2017 Revised  March 2018 Published  June 2018

Given a smooth non-trapping compact manifold with strictly convex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. These data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of the metric, the scattering data measured on the boundary determine the Riemannian manifold up to isometry.

Citation: Matti Lassas, Teemu Saksala, Hanming Zhou. Reconstruction of a compact manifold from the scattering data of internal sources. Inverse Problems & Imaging, 2018, 12 (4) : 993-1031. doi: 10.3934/ipi.2018042
##### References:
 [1] L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces, 25, Oxford University Press on Demand, 2004. [2] M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527. [3] M. I. Belishev and Y. V. Kuryiev, To the reconstruction of a riemannian manifold via its spectral data (bc-method), Communications in Partial Differential Equations, 17 (1992), 767-804. doi: 10.1080/03605309208820863. [4] C. B. Croke, Rigidity theorems in Riemannian geometry, in Geometric Methods in Inverse Problems and PDE Control, Springer, 137 (2004), 47-72. doi: 10.1007/978-1-4684-9375-7_4. [5] M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, No. 33. Springer-Verlag, New York-Heidelberg, 1976. [6] M. V. de Hoop, S. F. Holman, E. Iversen, M. Lassas and B. Ursin, Reconstruction of a conformally euclidean metric from local boundary diffraction travel times, SIAM Journal on Mathematical Analysis, 46 (2014), 3705-3726. doi: 10.1137/130931291. [7] M. V. de Hoop, S. F. Holman, E. Iversen, M. Lassas and B. Ursin, Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times, Journal de Mathématiques Pures et Appliquées, 103 (2015), 830-848. doi: 10.1016/j.matpur.2014.09.003. [8] A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data, Communications in Partial Differential Equations, 23 (1998), 27-95. doi: 10.1080/03605309808821338. [9] A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, vol. 123 of Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220. [10] I. Kupka, M. Peixoto and C. Pugh, Focal stability of Riemann metrics, Journal fur die reine und angewandte Mathematik (Crelles Journal), 593 (2006), 31-72. doi: 10.1515/CRELLE.2006.029. [11] Y. Kurylev, Multidimensional Gelfand inverse problem and boundary distance map, Inverse Problems Related with Geometry (ed. H. Soga), Ibaraki, 1-15. [12] Y. Kurylev, M. Lassas and G. Uhlmann, Rigidity of broken geodesic flow and inverse problems, American Journal of Mathematics, 132 (2010), 529-562. doi: 10.1353/ajm.0.0103. [13] M. Lassas and L. Oksanen, An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Problems, 26 (2010), 085012, 19pp. doi: 10.1088/0266-5611/26/8/085012. [14] M. Lassas and L. Oksanen, Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets, Duke Mathematical Journal, 163 (2014), 1071-1103. doi: 10.1215/00127094-2649534. [15] M. Lassas and T. Saksala, Determination of a Riemannian manifold from the distance difference functions, Asian journal of mathematics (to appear), arXiv preprint arXiv: 1510.06157. [16] M. Lassas, V. Sharafutdinov and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Mathematische Annalen, 325 (2003), 767-793. doi: 10.1007/s00208-002-0407-4. [17] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, vol. 176, Springer-Verlag, New York, 1997. doi: 10.1007/b98852. [18] R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Inventiones mathematicae, 65 (1981/82), 71-83. doi: 10.1007/BF01389295. [19] T. Milne, Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation. [20] L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, 161 (2015), 1093-1110. doi: 10.4007/annals.2005.161.1093. [21] L. Pestov, G. Uhlmann and H. Zhou, An inverse kinematic problem with internal sources, Inverse Problems, 31 (2015), 055006, 6pp. doi: 10.1088/0266-5611/31/5/055006. [22] V. Sharafutdinov, Ray transform on riemannian manifolds. eight lectures on integral geometry, preprint. [23] P. Stefanov, Microlocal approach to tensor tomography and boundary and lens rigidity, Serdica Math. J, 34 (2008), 67-112. [24] P. Stefanov and G. Uhlmann, Boundary and lens rigidity, tensor tomography and analytic microlocal analysis, in Algebraic Analysis of Differential Equations, Springer, 2008,275-293. doi: 10.1007/978-4-431-73240-2_23. [25] P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data, Journal of the American Mathematical Society, 29 (2016), 299-332. doi: 10.1090/jams/846. [26] P. Stefanov, G. Uhlmann and A. Vasy, Local and global boundary rigidity and the geodesic x-ray transform in the normal gauge, arXiv: 1702.03638. [27] P. Topalov and V. S. Matveev, Geodesic equivalence via integrability, Geometriae Dedicata, 96 (2003), 91-115. doi: 10.1023/A:1022166218282. [28] G. Uhlmann and H. Zhou, Journey to the Center of the Earth, arXiv: 1604.00630.

show all references

##### References:
 [1] L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces, 25, Oxford University Press on Demand, 2004. [2] M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527. [3] M. I. Belishev and Y. V. Kuryiev, To the reconstruction of a riemannian manifold via its spectral data (bc-method), Communications in Partial Differential Equations, 17 (1992), 767-804. doi: 10.1080/03605309208820863. [4] C. B. Croke, Rigidity theorems in Riemannian geometry, in Geometric Methods in Inverse Problems and PDE Control, Springer, 137 (2004), 47-72. doi: 10.1007/978-1-4684-9375-7_4. [5] M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, No. 33. Springer-Verlag, New York-Heidelberg, 1976. [6] M. V. de Hoop, S. F. Holman, E. Iversen, M. Lassas and B. Ursin, Reconstruction of a conformally euclidean metric from local boundary diffraction travel times, SIAM Journal on Mathematical Analysis, 46 (2014), 3705-3726. doi: 10.1137/130931291. [7] M. V. de Hoop, S. F. Holman, E. Iversen, M. Lassas and B. Ursin, Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times, Journal de Mathématiques Pures et Appliquées, 103 (2015), 830-848. doi: 10.1016/j.matpur.2014.09.003. [8] A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data, Communications in Partial Differential Equations, 23 (1998), 27-95. doi: 10.1080/03605309808821338. [9] A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, vol. 123 of Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220. [10] I. Kupka, M. Peixoto and C. Pugh, Focal stability of Riemann metrics, Journal fur die reine und angewandte Mathematik (Crelles Journal), 593 (2006), 31-72. doi: 10.1515/CRELLE.2006.029. [11] Y. Kurylev, Multidimensional Gelfand inverse problem and boundary distance map, Inverse Problems Related with Geometry (ed. H. Soga), Ibaraki, 1-15. [12] Y. Kurylev, M. Lassas and G. Uhlmann, Rigidity of broken geodesic flow and inverse problems, American Journal of Mathematics, 132 (2010), 529-562. doi: 10.1353/ajm.0.0103. [13] M. Lassas and L. Oksanen, An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Problems, 26 (2010), 085012, 19pp. doi: 10.1088/0266-5611/26/8/085012. [14] M. Lassas and L. Oksanen, Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets, Duke Mathematical Journal, 163 (2014), 1071-1103. doi: 10.1215/00127094-2649534. [15] M. Lassas and T. Saksala, Determination of a Riemannian manifold from the distance difference functions, Asian journal of mathematics (to appear), arXiv preprint arXiv: 1510.06157. [16] M. Lassas, V. Sharafutdinov and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Mathematische Annalen, 325 (2003), 767-793. doi: 10.1007/s00208-002-0407-4. [17] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, vol. 176, Springer-Verlag, New York, 1997. doi: 10.1007/b98852. [18] R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Inventiones mathematicae, 65 (1981/82), 71-83. doi: 10.1007/BF01389295. [19] T. Milne, Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation. [20] L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, 161 (2015), 1093-1110. doi: 10.4007/annals.2005.161.1093. [21] L. Pestov, G. Uhlmann and H. Zhou, An inverse kinematic problem with internal sources, Inverse Problems, 31 (2015), 055006, 6pp. doi: 10.1088/0266-5611/31/5/055006. [22] V. Sharafutdinov, Ray transform on riemannian manifolds. eight lectures on integral geometry, preprint. [23] P. Stefanov, Microlocal approach to tensor tomography and boundary and lens rigidity, Serdica Math. J, 34 (2008), 67-112. [24] P. Stefanov and G. Uhlmann, Boundary and lens rigidity, tensor tomography and analytic microlocal analysis, in Algebraic Analysis of Differential Equations, Springer, 2008,275-293. doi: 10.1007/978-4-431-73240-2_23. [25] P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data, Journal of the American Mathematical Society, 29 (2016), 299-332. doi: 10.1090/jams/846. [26] P. Stefanov, G. Uhlmann and A. Vasy, Local and global boundary rigidity and the geodesic x-ray transform in the normal gauge, arXiv: 1702.03638. [27] P. Topalov and V. S. Matveev, Geodesic equivalence via integrability, Geometriae Dedicata, 96 (2003), 91-115. doi: 10.1023/A:1022166218282. [28] G. Uhlmann and H. Zhou, Journey to the Center of the Earth, arXiv: 1604.00630.
Here is a schematic picture about our data $R_{\partial M}(p)$, where the point $p$ is the blue dot. Here the black arrows are the exit directions of geodesics emitted from $p$ and the blue arrows are our data
Here is a visualization of the set up in the definition of the function $\varrho_k$ in Lemma 2.8. The blue dot is $p$ and the red&blue dot is $q$. The black curve is the geodesic $\gamma_{p, \eta}$. The red line is the hypersurface $\tilde S_1$ and the blue line is the hypersurface $\tilde S_2$. The small blue and red segments indicate the intervals $(s_k-\delta, s_k+\delta)$ where the function $\varrho_k(\cdot, \eta)$ is defined
Here is a schematic picture about $K(p)$, where the point $p \in M$ is the blue dot. The black curves represent the geodesics $\gamma_{z, \xi}$, and $\gamma_{w, \eta}$ respectively, where vectors $(z, \xi), (w, \eta) \in R_{\partial M}^E(p)$. Notice that only $(w, \eta)\in K(p)$
Here is a schematic picture about the map $\Theta_{q, \tilde q, v}$ evaluated at point $p\in M$, where the point $p \in M$ is the blue dot and $V_q\cap V_{\tilde q}$ is the blue ellipse. The higher red&blue dot is $q$ and the lower is $\tilde q$. The blue vector is the given direction $v \in T_{\tilde q}N$
Here is a schematic picture about the situation where the boundary normal geodesic $\gamma_{p, \nu}$ (the black curve) is self-intersecting at $p \in \partial M$ (red&blue dot). Here the blue curve is the geodesic $\gamma_{p, W(p)}$, where $W(p)\notin I_p$. For the point $w \in M$ (blue dot) the point $p$ satisfies $p = \Pi_W(w)$
Here is a schematic picture about the map $(\tilde Q_{q, v}, \Pi^E_{q, W})$ evaluated at a point $x\in M$ that is close to $\partial M$, where the point $x \in M$ is the blue dot. The right hand side red&blue dot is $\Pi^E_{q, W}(x)$ and the left hand side red&blue dot is $q$. The blue arrow is the given vector $v \in T_qN$
 [1] Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015 [2] Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098 [3] Alexander Nabutovsky and Regina Rotman. Lengths of geodesics between two points on a Riemannian manifold. Electronic Research Announcements, 2007, 13: 13-20. [4] Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems & Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034 [5] Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil. Opinion Dynamics on a General Compact Riemannian Manifold. Networks & Heterogeneous Media, 2017, 12 (3) : 489-523. doi: 10.3934/nhm.2017021 [6] Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5799-5825. doi: 10.3934/dcds.2015.35.5799 [7] Anna Maria Candela, J.L. Flores, M. Sánchez. A quadratic Bolza-type problem in a non-complete Riemannian manifold. Conference Publications, 2003, 2003 (Special) : 173-181. doi: 10.3934/proc.2003.2003.173 [8] Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203 [9] Guowei Dai, Ruyun Ma, Haiyan Wang, Feng Wang, Kuai Xu. Partial differential equations with Robin boundary condition in online social networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1609-1624. doi: 10.3934/dcdsb.2015.20.1609 [10] Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115 [11] Angelo Favini, Rabah Labbas, Stéphane Maingot, Maëlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4967-4990. doi: 10.3934/dcds.2013.33.4967 [12] Alfonso C. Casal, Jesús Ildefonso Díaz, José M. Vegas. Finite extinction time property for a delayed linear problem on a manifold without boundary. Conference Publications, 2011, 2011 (Special) : 265-271. doi: 10.3934/proc.2011.2011.265 [13] Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993 [14] Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems & Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959 [15] Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169 [16] Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469 [17] Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703 [18] Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053 [19] Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 [20] Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

2017 Impact Factor: 1.465