• Previous Article
    On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method
  • IPI Home
  • This Issue
  • Next Article
    Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast
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.  Google Scholar

[2]

M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, No. 33. Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[6]

M. V. de HoopS. F. HolmanE. IversenM. 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.  Google Scholar

[7]

M. V. de HoopS. F. HolmanE. IversenM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

I. KupkaM. 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.  Google Scholar

[11]

Y. Kurylev, Multidimensional Gelfand inverse problem and boundary distance map, Inverse Problems Related with Geometry (ed. H. Soga), Ibaraki, 1-15. Google Scholar

[12]

Y. KurylevM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[16]

M. LassasV. Sharafutdinov and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Mathematische Annalen, 325 (2003), 767-793.  doi: 10.1007/s00208-002-0407-4.  Google Scholar

[17]

J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, vol. 176, Springer-Verlag, New York, 1997. doi: 10.1007/b98852.  Google Scholar

[18]

R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Inventiones mathematicae, 65 (1981/82), 71-83.  doi: 10.1007/BF01389295.  Google Scholar

[19]

T. Milne, Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

V. Sharafutdinov, Ray transform on riemannian manifolds. eight lectures on integral geometry, preprint. Google Scholar

[23]

P. Stefanov, Microlocal approach to tensor tomography and boundary and lens rigidity, Serdica Math. J, 34 (2008), 67-112.   Google Scholar

[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.  Google Scholar

[25]

P. StefanovG. Uhlmann and A. Vasy, Boundary rigidity with partial data, Journal of the American Mathematical Society, 29 (2016), 299-332.  doi: 10.1090/jams/846.  Google Scholar

[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. Google Scholar

[27]

P. Topalov and V. S. Matveev, Geodesic equivalence via integrability, Geometriae Dedicata, 96 (2003), 91-115.  doi: 10.1023/A:1022166218282.  Google Scholar

[28]

G. Uhlmann and H. Zhou, Journey to the Center of the Earth, arXiv: 1604.00630. Google Scholar

show all references

References:
[1]

L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces, 25, Oxford University Press on Demand, 2004.  Google Scholar

[2]

M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, No. 33. Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[6]

M. V. de HoopS. F. HolmanE. IversenM. 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.  Google Scholar

[7]

M. V. de HoopS. F. HolmanE. IversenM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

I. KupkaM. 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.  Google Scholar

[11]

Y. Kurylev, Multidimensional Gelfand inverse problem and boundary distance map, Inverse Problems Related with Geometry (ed. H. Soga), Ibaraki, 1-15. Google Scholar

[12]

Y. KurylevM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[16]

M. LassasV. Sharafutdinov and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Mathematische Annalen, 325 (2003), 767-793.  doi: 10.1007/s00208-002-0407-4.  Google Scholar

[17]

J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, vol. 176, Springer-Verlag, New York, 1997. doi: 10.1007/b98852.  Google Scholar

[18]

R. Michel, Sur la rigidité imposée par la longueur des géodésiques, Inventiones mathematicae, 65 (1981/82), 71-83.  doi: 10.1007/BF01389295.  Google Scholar

[19]

T. Milne, Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

V. Sharafutdinov, Ray transform on riemannian manifolds. eight lectures on integral geometry, preprint. Google Scholar

[23]

P. Stefanov, Microlocal approach to tensor tomography and boundary and lens rigidity, Serdica Math. J, 34 (2008), 67-112.   Google Scholar

[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.  Google Scholar

[25]

P. StefanovG. Uhlmann and A. Vasy, Boundary rigidity with partial data, Journal of the American Mathematical Society, 29 (2016), 299-332.  doi: 10.1090/jams/846.  Google Scholar

[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. Google Scholar

[27]

P. Topalov and V. S. Matveev, Geodesic equivalence via integrability, Geometriae Dedicata, 96 (2003), 91-115.  doi: 10.1023/A:1022166218282.  Google Scholar

[28]

G. Uhlmann and H. Zhou, Journey to the Center of the Earth, arXiv: 1604.00630. Google Scholar

Figure 1.  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
Figure 2.  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
Figure 3.  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 $(w, \eta)\in K(p)$
Figure 4.  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 $v \in T_{\tilde q}N$
Figure 5.  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 $p = \Pi_W(w)$
Figure 6.  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]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[2]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[3]

Xinlin Cao, Huaian Diao, Jinhong Li. Some recent progress on inverse scattering problems within general polyhedral geometry. Electronic Research Archive, 2021, 29 (1) : 1753-1782. doi: 10.3934/era.2020090

[4]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[5]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[6]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108

[7]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398

[8]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[9]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[10]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004

[11]

Shahede Omidi, Jafar Fathali. Inverse single facility location problem on a tree with balancing on the distance of server to clients. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021017

[12]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005

[13]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006

[14]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050

[15]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124

[16]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[17]

Weihong Guo, Yifei Lou, Jing Qin, Ming Yan. IPI special issue on "mathematical/statistical approaches in data science" in the Inverse Problem and Imaging. Inverse Problems & Imaging, 2021, 15 (1) : I-I. doi: 10.3934/ipi.2021007

[18]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[19]

John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044

[20]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042

2019 Impact Factor: 1.373

Metrics

  • PDF downloads (112)
  • HTML views (109)
  • Cited by (1)

Other articles
by authors

[Back to Top]