Geodesic
On reconstruction of Riemannian manifold via boundary data: theory and plan of numerical testing
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 40, Tome 380 (2010), pp. 8-30 
M. I. Belishev. On reconstruction of Riemannian manifold via boundary data: theory and plan of numerical testing. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 40, Tome 380 (2010), pp. 8-30. http://geodesic.mathdoc.fr/item/ZNSL_2010_380_a1/
@article{ZNSL_2010_380_a1,
     author = {M. I. Belishev},
     title = {On reconstruction of {Riemannian} manifold via boundary data: theory and plan of numerical testing},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {8--30},
     year = {2010},
     volume = {380},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_380_a1/}
}
TY  - JOUR
AU  - M. I. Belishev
TI  - On reconstruction of Riemannian manifold via boundary data: theory and plan of numerical testing
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2010
SP  - 8
EP  - 30
VL  - 380
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2010_380_a1/
LA  - ru
ID  - ZNSL_2010_380_a1
ER  - 
%0 Journal Article
%A M. I. Belishev
%T On reconstruction of Riemannian manifold via boundary data: theory and plan of numerical testing
%J Zapiski Nauchnykh Seminarov POMI
%D 2010
%P 8-30
%V 380
%U http://geodesic.mathdoc.fr/item/ZNSL_2010_380_a1/
%G ru
%F ZNSL_2010_380_a1

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The paper deals with an inverse problem of reconstruction of a Riemannian manifold via its boundary data. This problem has been solved by the boundary control method, whereas at the moment there are a few variants of solving it. In the paper, one more version of the procedure, which recovers the manifold via the scalar spectral or dynamical data, is proposed. This version is a simplest one in regard to the devices in use: we do not enlist geometrical optics, polar representation of operators, etc, but get by with a controllability property of the relevant dynamical system. With no substantial changes, this version is applicable to more the complicated (vector) problem of electrodynamics for the Maxwell system. Simplicity of the proposed procedure provides additional chances for its numerical realizability. At the end of the paper, a plan of numerical experiment is discussed. To draw attention to such new options is one of the main aims of the paper. Bibl. 9 titles.

[1] S. A. Avdonin, M. I. Belishev, S. A. Ivanov, “Upravlyaemost v zapolnennoi oblasti dlya mnogomernogo volnovogo uravneniya s singulyarnym upravleniem”, Zap. nauch. semin. POMI, 210, 1993, 3–14

[2] M. I. Belishev, “K obosnovaniyu pravila Gyuigensa”, Zap. nauch. semin. POMI, 218, 1994, 17–24 | MR | Zbl

[3] M. I. Belishev, “Boundary control in reconstruction of manifolds and metrics (the BC method)”, Inverse Problems, 13:5 (1997), R1–R45 | DOI | MR | Zbl

[4] M. I. Belishev, “On relations between spectral and dynamical inverse data”, J. Inverse and Ill-Posed Problems, 9:6 (2001), 547–565 | DOI | MR | Zbl

[5] M. I. Belishev, “O svyazi dannykh dinamicheskikh i spektralnykh obratnykh zadach”, Zap. nauch. semin. POMI, 297, 2003, 30–48 | MR | Zbl

[6] M. I. Belishev, “Recent progress in the boundary control method”, Inverse Problems, 23:5 (2007), R1–R67 | DOI | MR | Zbl

[7] M. I. Belishev, M. N. Demchenko, “Time-optimal reconstruction of Riemannian manifold via boundary electromagnetic measurements”, Inverse Problems, 26 (2010) (to appear)

[8] D. Gromol, W. Klingenberg, W. Meyer, Riemannische Geometrie im Grossen, Springer, Berlin, 1968 | MR | Zbl

[9] W. Klingenberg, Riemannian geometry, de Gruyer Studies in Mathematics, 1, Walter de Gruyter, 1982 | MR | Zbl