An iterative approach to non-overdetermined inverse scattering at fixed energy
Sbornik. Mathematics, Tome 206 (2015) no. 1, pp. 120-134 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We present an iterative approximate reconstruction algorithm for non-overdetermined inverse scattering at fixed energy $E$ with incomplete data, where the dimension $d\geqslant 2$. In particular, we obtain rapidly converging approximate reconstructions for this inverse scattering for $E\to +\infty$. Bibliography: 38 titles.
Keywords: inverse scattering problem, non-overdetermined monochromatic data, iterative approximate reconstruction, rapid high energy convergence.
@article{SM_2015_206_1_a7,
     author = {R. G. Novikov},
     title = {An iterative approach to non-overdetermined inverse scattering at fixed energy},
     journal = {Sbornik. Mathematics},
     pages = {120--134},
     year = {2015},
     volume = {206},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2015_206_1_a7/}
}
TY  - JOUR
AU  - R. G. Novikov
TI  - An iterative approach to non-overdetermined inverse scattering at fixed energy
JO  - Sbornik. Mathematics
PY  - 2015
SP  - 120
EP  - 134
VL  - 206
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_2015_206_1_a7/
LA  - en
ID  - SM_2015_206_1_a7
ER  - 
%0 Journal Article
%A R. G. Novikov
%T An iterative approach to non-overdetermined inverse scattering at fixed energy
%J Sbornik. Mathematics
%D 2015
%P 120-134
%V 206
%N 1
%U http://geodesic.mathdoc.fr/item/SM_2015_206_1_a7/
%G en
%F SM_2015_206_1_a7
R. G. Novikov. An iterative approach to non-overdetermined inverse scattering at fixed energy. Sbornik. Mathematics, Tome 206 (2015) no. 1, pp. 120-134. http://geodesic.mathdoc.fr/item/SM_2015_206_1_a7/

[1] F. A. Berezin, M. A. Shubin, The Schrödinger equation, Math. Appl. (Soviet Ser.), 66, Kluwer Academic Publishers Group, Dordrecht, 1991, xviii+555 pp. | DOI | MR | MR | Zbl | Zbl

[2] L. D. Faddeev, “The inverse problem in the quantum theory of scattering. II”, J. Soviet Math., 5:3 (1976), 334–396 | DOI | MR | Zbl

[3] R. G. Novikov, “Approximate Lipschitz stability for non-overdetermined inverse scattering at fixed energy”, J. Inverse Ill-Posed Probl., 21:6 (2013), 813–823 | DOI | MR | Zbl

[4] N. V. Alexeenko, V. A. Burov, O. D. Rumyantseva, “Solution of the three-dimensional acoustical inverse scattering problem. The modified Novikov algorithm”, Acoust. Phys., 54:3 (2008), 407–419 | DOI

[5] A. L. Bukhgeim, “Recovering a potential from Cauchy data in the two-dimensional case”, J. Inverse Ill-Posed Probl., 16:1 (2008), 19–33 | DOI | MR | Zbl

[6] V. A. Burov, N. V. Alekseenko, O. D. Rumyantseva, “Multifrequency generalization of the Novikov algorithm for the two-dimensional inverse scattering problem”, Acoust. Phys., 55:6 (2009), 843–856 | DOI

[7] K. Chadan, P. C. Sabatier, Inverse problems in quantum scattering theory, With a foreword by R. G. Newton, Texts Monogr. Phys., 2nd ed., Springer-Verlag, New York, 1989, xxxii+499 pp. | DOI | MR | Zbl

[8] G. Eskin, Lectures on linear partial differential equations, Grad. Stud. Math., 123, Amer. Math. Soc., Providence, RI, 2011, xviii+410 pp. | MR | Zbl

[9] G. Eskin, J. Ralston, “Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy”, Comm. Math. Phys., 173:1 (1995), 199–224 | DOI | MR | Zbl

[10] L. D. Faddeev, “Edinstvennost resheniya obratnoi zadachi rasseyaniya”, Vestn. LGU, 11:7 (1956), 126–130 | MR

[11] A. A. Gonchar, N. N. Novikova, G. M. Henkin, “Multipoint Padé approximants in the inverse Sturm–Liouville problem”, Math. USSR-Sb., 73:2 (1992), 479–489 | DOI | MR | Zbl

[12] P. G. Grinevich, “Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity”, Russian Math. Surveys, 55:6 (2000), 1015–1083 | DOI | DOI | MR | Zbl

[13] P. Hähner, T. Hohage, “New stability estimates for the inverse acoustic inhomogeneous medium problem and applications”, SIAM J. Math. Anal., 33:3 (2001), 670–685 | DOI | MR | Zbl

[14] M. I. Isaev, “Exponential instability in the inverse scattering problem on the energy interval”, Funct. Anal. Appl., 47:3 (2013), 187–194 | DOI | DOI | MR

[15] M. I. Isaev, R. G. Novikov, “New global stability estimates for monochromatic inverse acoustic scattering”, SIAM J. Math. Anal., 45:3 (2013), 1495–1504 | DOI | MR | Zbl

[16] R. G. Novikov, “Multidimensional inverse spectral problem for the equation $-\Delta\psi+(v(x)-Eu(x))\psi=0$”, Funct. Anal. Appl., 22:4 (1988), 263–272 | DOI | MR | Zbl

[17] R. G. Novikov, “The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator”, J. Funct. Anal., 103:2 (1992), 409–463 | DOI | MR | Zbl

[18] R. G. Novikov, “The inverse scattering problem at fixed energy for the three-dimensional Schrödinger equation with an exponentially decreasing potential”, Comm. Math. Phys., 161:3 (1994), 569–595 | DOI | MR | Zbl

[19] R. G. Novikov, “Rapidly converging approximation in inverse quantum scattering in dimension $2$”, Phys. Lett. A, 238:2-3 (1998), 73–78 | DOI | MR | Zbl

[20] R. G. Novikov, “The $\bar\partial$-approach to approximate inverse scattering at fixed energy in three dimensions”, IMRP Int. Math. Res. Pap., 2005:6 (2005), 287–349 | DOI | MR | Zbl

[21] T. Regge, “Introduction to complex orbital moments”, Nuovo Cimento (10), 14:5 (1959), 951–976 | DOI | MR | Zbl

[22] P. Stefanov, “Stability of the inverse problem in potential scattering at fixed energy”, Ann. Inst. Fourier (Grenoble), 40:4 (1990), 867–884 | DOI | MR | Zbl

[23] A. Vasy, Xue-Ping Wang, “Inverse scattering with fixed energy for dilation-analytic potentials”, Inverse Problems, 20:4 (2004), 1349–1354 | DOI | MR | Zbl

[24] R. Weder, “Global uniqueness at fixed energy in multidimensional inverse scattering theory”, Inverse Problems, 7:6 (1991), 927–938 | DOI | MR | Zbl

[25] R. Weder, D. Yafaev, “On inverse scattering at a fixed energy for potentials with a regular behaviour at infinity”, Inverse Problems, 21:6 (2005), 1937–1952 | DOI | MR | Zbl

[26] L. Beilina, M. V. Klibanov, Approximate global convergence and adaptivity for coefficient inverse problems, Springer, New York, 2012, xv+407 pp. | Zbl

[27] B. A. Dubrovin, I. M. Krichever, S. P. Novikov, “The Schrödinger equation in a periodic field and Riemann surfaces”, Soviet Math. Dokl., 17 (1976), 947–951 | MR | Zbl

[28] G. Eskin, J. Ralston, “The inverse backscattering problem in three dimensions”, Comm. Math. Phys., 124:2 (1989), 169–215 | DOI | MR | Zbl

[29] G. M. Henkin, R. G. Novikov, “The $\bar\partial$-equation in the multidimensional inverse scattering problem”, Russian Math. Surveys, 42:3 (1987), 109–180 | DOI | MR | Zbl

[30] M. V. Klibanov, “Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems”, J. Inverse and Ill-Posed Probl., 21:4 (2013), 477–560 | DOI | MR | Zbl

[31] H. E. Moses, “Calculation of the scattering potential from reflection coefficients”, Phys. Rev. (2), 102 (1956), 559–567 | DOI | MR | Zbl

[32] R. G. Novikov, “On non-overdetermined inverse scattering at zero energy in three dimensions”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 5:3 (2006), 279–328 | MR | Zbl

[33] R. G. Novikov, M. Santacesaria, “Monochromatic reconstruction algorithms for two-dimensional multi-channel inverse problems”, Int. Math. Res. Not. IMRN, 2013, no. 6, 1205–1229 | MR

[34] P. D. Stefanov, “A uniqueness result for the inverse back-scattering problem”, Inverse Problems, 6:6 (1990), 1055–1064 | DOI | MR | Zbl

[35] M. M. Lavrent'ev, V. G. Romanov, S. P. Shishatskii, Ill-posed problems of mathematical physics and analysis, Transl. Math. Monogr., 64, Amer. Math. Soc., Providence, RI, 1986, vi+290 pp. | MR | MR | Zbl | Zbl

[36] R. G. Novikov, “On iterative reconstruction in the nonlinearized polarization tomography”, Inverse Problems, 25:11 (2009), 115010, 18 pp. | DOI | MR | Zbl

[37] L. Päivärinta, E. Somersalo, “Inversion of discontinuities for the Schrödinger equation in three dimensions”, SIAM J. Math. Anal., 22:2 (1991), 480–499 | DOI | MR | Zbl

[38] A. Jensen, “High energy resolvent estimates for generalized many-body Schrödinger operators”, Publ. Res. Inst. Math. Sci., 25:1 (1989), 155–167 | DOI | MR | Zbl