Multidimensional Analogues of Gelfand–Levitan, Marchenko and Krein Equations. Theory, Numerics and Applications
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Keywords:
Gelfand-Levitan equation, Krein equation, Marchenko equation, inverse coefficient problem, inverse scattering problemAbstract
We consider the method of regularization of two dimensional (2D) inverse coefficient
problems based on the projection method and the approach of I.M. Gelfand, B.M. Levitan, M.G.
Krein and V.A. Marchenko. We propose a method of reconstruction of the potential, density and
velocity in 2D inverse coefficient problems. The 2D analogies of the I.M. Gelfand, B.M. Levitan and
M.G. Krein method are established. The 2D analog of the V.A. Marchenko equation is considered
for the Kadomtsev-Petviashvili equation. This approach can be easily applied to corresponding
multidimensional inverse problems. The results of numerical calculations are presented.
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projection method. Doklady Akademii Nauk. 292 (3), 534–537 (1987)
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19. Kabanikhin, S.I., Bakanov, G.B.: Discrete analogy of Gelfand-Levitan method Doklady Akademii Nauk. 356
(2) 157–160 (1997).
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inverse problem for a hyperbolic equation. Siberian Mathematical Journal. 40 (2), 262–280 (1999)
21. Kabanikhin, S.I., Satybaev, A.D., Shishlenin, M.A.: Direct Methods of Solving Inverse Hyperbolic Problems
VSP, The Netherlands (2005).
22. Kabanikhin, S.I., Shishlenin, M.A.: Quasi-solution in inverse coefficient problems. J. of Inverse and Ill-Posed
Problems. 16 (7), 705–713 (2008).
23. Kabanikhin, S.I., Shishlenin, M.A.: Numerical algorithm for two-dimensional inverse acoustic problem based
on Gel’fand-Levitan-Krein equation. J. of Inverse and Ill-Posed Problems. 18 (9), 979–995 (2011).
24. Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersive media. Sov. Phys.
Dokl. 15, 539–541 (1970).
25. Natterer, F.: A discrete Gelfand-Levitan theory. Technical report, Institut fuer Numerische und instrumentelle
Mathematik Universitaet Muenster Germany (1994).
26. Santosa, F.: Numerical scheme for the inversion of acoustical impedance profile based on the Gelfand-Levitan
method. Geophys. J. Roy. Astr. Soc. 70, 229–244 (1982).
27. Symes, W.W.: Inverse boundary value problems and a theorem of Gel’fand and Levitan. J. Math. Anal.
Appl. 71 378–402 (1979).
28. Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. of
Math. 125, 153–169 (1987).
29. Rakesh: An inverse problem for the wave equation in the half plane. Inverse Problems. 9, 433–441 (1993).
2. Belishev, M.I.: Boundary control in reconstruction of manifolds and metrics (the BC method). Inverse
Problems. 13 (5) R1–R45 (1997).
3. Belishev, M.I.: Recent progress in the boundary control method. Inverse Problems. 23 (5), R1–R67 (2007).
4. Beilina, L., Klibanov, M.V.: A globally convergent numerical method for a coefficient inverse problem. SIAM
J. Sci. Comp. 31, 478–509 (2008).
5. Beilina, L., Klibanov, M.V., Kokurin, M.Yu.: Adaptivity with relaxation for ill-posed problems and global
convergence for a coefficient inverse problem. J. of Mathematical Sciences. 167, 279–325 (2010).
6. Burridge, R.: The Gelfand-Levitan, the Marchenko and the Gopinath-Sondhi integral equation of inverse
scattering theory, regarded in the context of inverse impulse-response problems. Wave Motion. 2, 305–323
(1980).
7. Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for Solving the Korteweg-deVries Equation.
Physical Review Letters. 19(19), 1095–1097 (1967).
8. Gelfand, I.M., Levitan, B.M.: On the determination of a differential equation from its spectral function. Izv.
Akad. Nauk SSSR. Ser. Mat. 15, 309–360 (1951) (in Russian).
9. Gladwell, G.M.L., Willms, N.B.: A discrete Gelfand-Levitan method for band-matrix inverse eigenvalue
problems. Inverse Problems. 5, 165–179 (1989).
10. Gudkova, E.V., Habibullin, I.T.: Kadomtsev–Petviashvili Equation on the HalfPlane. Teoret. Mat. Fiz.
140(2), 230–240 (2004).
11. Krein, M.G.: Solution of the inverse Sturm-Liouville problem. Dokl. Akad. Nauk SSSR. 76, 21–24 (1951)
(in Russian).
12. Krein, M.G.: On a method of effective solution of an inverse boundary problem. Dokl. Akad. Nauk SSSR.
94, 987–990 (1954) (in Russian).
13. Marchenko, V.A.: Several quesions of the theory of the differential operator of the second order. Doklady
Akademii Nauk. 72 457–460 (1950) (in Russian).
14. Marchenko, V.A.: Several questions about the theory of one dimensional linear differential operator of the
second order I. Proceedings of Moscow Mathematical Society. 1, 327–420 (1952) (in Russian).
15. Pelinovsky, D.E., Stepanyants, Yu.A., Kivshar, Yu.A.: Self-focusing of plane dark solitons in nonlinear
defocusing media. Phys. Rev. 51, 5016–5026 (1995).
16. Kabanikhin, S.I., Shishlenin, M.A.: Boundary control and Gel’fand-Levitan-Krein methods in inverse acoustic
problem. J. Inv. Ill-Posed Problems. 12 (2), 125–144 (2004).
17. Kabanikhin, S.I.: Regularization of multidimensional inverse problems for hyperbolic equations based on a
projection method. Doklady Akademii Nauk. 292 (3), 534–537 (1987)
18. Kabanikhin, S.I.: On linear regularization of multidimensional inverse problems for hyperbolic equations. Sov.
Math. Dokl. 40(3), 579–583 (1990).
19. Kabanikhin, S.I., Bakanov, G.B.: Discrete analogy of Gelfand-Levitan method Doklady Akademii Nauk. 356
(2) 157–160 (1997).
20. Kabanikhin, S.I., Bakanov, G.B.: A discrete analog of the Gelfand-Levitan method in a two-dimensional
inverse problem for a hyperbolic equation. Siberian Mathematical Journal. 40 (2), 262–280 (1999)
21. Kabanikhin, S.I., Satybaev, A.D., Shishlenin, M.A.: Direct Methods of Solving Inverse Hyperbolic Problems
VSP, The Netherlands (2005).
22. Kabanikhin, S.I., Shishlenin, M.A.: Quasi-solution in inverse coefficient problems. J. of Inverse and Ill-Posed
Problems. 16 (7), 705–713 (2008).
23. Kabanikhin, S.I., Shishlenin, M.A.: Numerical algorithm for two-dimensional inverse acoustic problem based
on Gel’fand-Levitan-Krein equation. J. of Inverse and Ill-Posed Problems. 18 (9), 979–995 (2011).
24. Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersive media. Sov. Phys.
Dokl. 15, 539–541 (1970).
25. Natterer, F.: A discrete Gelfand-Levitan theory. Technical report, Institut fuer Numerische und instrumentelle
Mathematik Universitaet Muenster Germany (1994).
26. Santosa, F.: Numerical scheme for the inversion of acoustical impedance profile based on the Gelfand-Levitan
method. Geophys. J. Roy. Astr. Soc. 70, 229–244 (1982).
27. Symes, W.W.: Inverse boundary value problems and a theorem of Gel’fand and Levitan. J. Math. Anal.
Appl. 71 378–402 (1979).
28. Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. of
Math. 125, 153–169 (1987).
29. Rakesh: An inverse problem for the wave equation in the half plane. Inverse Problems. 9, 433–441 (1993).
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How to Cite
Kabanikhin, S., & Shishlenin, M. (2018). Multidimensional Analogues of Gelfand–Levitan, Marchenko and Krein Equations. Theory, Numerics and Applications. Journal of Mathematics, Mechanics and Computer Science, 86(3), 63–69. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/498
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