
-
Previous Article
On Tikhonov-type regularization with approximated penalty terms
- IPI Home
- This Issue
-
Next Article
A note on transmission eigenvalues in electromagnetic scattering theory
Two-dimensional inverse scattering for quasi-linear biharmonic operator
1. | Biomimetics and Intelligent Systems Group, P.O. BOX 8000, FIN-90014 University of Oulu, Finland |
2. | Research Unit of Mathematical Sciences, P.O. BOX 3000, FIN-90014 University of Oulu, Finland |
3. | Department of Mathematics and Statistics, P.O. BOX 68, FI-00014 University of Helsinki |
The subject of this work concerns the classical direct and inverse scattering problems for quasi-linear perturbations of the two-dimensional biharmonic operator. The quasi-linear perturbations of the first and zero order might be complex-valued and singular. We show the existence of the scattering solutions to the direct scattering problem in the Sobolev space $ W^1_{\infty}( \mathbb{{R}}^2) $. Then the inverse scattering problem can be formulated as follows: does the knowledge of the far field pattern uniquely determine the unknown coefficients for given differential operator? It turns out that the answer to this classical question is affirmative for quasi-linear perturbations of the biharmonic operator. Moreover, we present a numerical method for the reconstruction of unknown coefficients, which from the practical point of view can be thought of as recovery of the coefficients from fixed energy measurements.
References:
[1] |
S. Agmon,
Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (1975), 151-218.
|
[2] |
T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-Beam CT, Springer, Berlin Heidelberg, 2008. |
[3] |
F. Cakoni and D. Colton, A Qualitative Approach in Inverse Scattering Theory, Springer, New York, 2014.
doi: 10.1007/978-1-4614-8827-9. |
[4] |
G. Fotopoulos and M. Harju,
Inverse scattering with fixed observation angle data in 2D, Inv. Prob. Sci. Eng., 25 (2017), 1492-1507.
doi: 10.1080/17415977.2016.1267170. |
[5] |
G. Fotopoulos, M. Harju and V. Serov,
Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D, Inverse Problems ans Imaging, 7 (2013), 183-197.
doi: 10.3934/ipi.2013.7.183. |
[6] |
G. Fotopoulos and V. Serov,
Inverse fixed energy scattering problem for the two-dimensional nonlinear Schrödinger operator, Inv. Prob. Sci. Eng., 24 (2016), 692-710.
doi: 10.1080/17415977.2015.1055263. |
[7] |
F. Gazzola, H.-Ch. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin Heidelberg, 2010.
doi: 10.1007/978-3-642-12245-3. |
[8] |
L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, New Jersey, 2004. |
[9] |
M. Harju, On the Direct and Inverse Scattering Problems for a Nonlinear Three-Dimensional Schrödinger Equation, PhD-thesis, University of Oulu, 2010. |
[10] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Determining a first order perturbation of the biharmonic operator by partial boundary measurements, Journal of Functional Analysis, 262 (2012), 1781-1801.
doi: 10.1016/j.jfa.2011.11.021. |
[11] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Inverse boundary value problems for the perturbed polyharmonic operator, Trans Amer. Math. Soc., 366 (2014), 95-112.
doi: 10.1090/S0002-9947-2013-05713-3. |
[12] |
N. N. Lebedev, Special Functions and Their Applications, Dover Publications, 1972. |
[13] |
B. Pausander, Scattering for the beam equation in low dimensions, Indiana Univ. Math. J., 59 (2010), 791–822. arXiv: 0903.3777v2 [math.AP].
doi: 10.1512/iumj.2010.59.3966. |
[14] |
V. S. Serov, An inverse Born approximation for the general nonlinear Schrödinger operator on the line, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 332002.
doi: 10.1088/1751-8113/42/33/332002. |
[15] |
V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D, Journal of Mathematical Physics, 53 (2012), 123522.
doi: 10.1063/1.4769825. |
[16] |
T. Tyni, Numerical results for Saito's uniqueness theorem in inverse scattering theory, Inverse Problems, 35 (2020), 065002.
doi: 10.1088/1361-6420/ab7d2d. |
[17] |
T. Tyni and V. Serov,
Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Problems and Imaging, 12 (2018), 205-227.
doi: 10.3934/ipi.2018008. |
[18] |
T. Tyni and M. Harju, Inverse backscattering problem for perturbations of biharmonic operator, Inverse Problems, 33 (2017), 105002.
doi: 10.1088/1361-6420/aa873e. |
[19] |
T. Tyni and V. Serov,
Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line, Inverse Problems and Imaging, 13 (2019), 159-175.
doi: 10.3934/ipi.2019009. |
[20] |
E. Zeidler, Applied Functional Analysis, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-0821-1. |
show all references
References:
[1] |
S. Agmon,
Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (1975), 151-218.
|
[2] |
T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-Beam CT, Springer, Berlin Heidelberg, 2008. |
[3] |
F. Cakoni and D. Colton, A Qualitative Approach in Inverse Scattering Theory, Springer, New York, 2014.
doi: 10.1007/978-1-4614-8827-9. |
[4] |
G. Fotopoulos and M. Harju,
Inverse scattering with fixed observation angle data in 2D, Inv. Prob. Sci. Eng., 25 (2017), 1492-1507.
doi: 10.1080/17415977.2016.1267170. |
[5] |
G. Fotopoulos, M. Harju and V. Serov,
Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D, Inverse Problems ans Imaging, 7 (2013), 183-197.
doi: 10.3934/ipi.2013.7.183. |
[6] |
G. Fotopoulos and V. Serov,
Inverse fixed energy scattering problem for the two-dimensional nonlinear Schrödinger operator, Inv. Prob. Sci. Eng., 24 (2016), 692-710.
doi: 10.1080/17415977.2015.1055263. |
[7] |
F. Gazzola, H.-Ch. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin Heidelberg, 2010.
doi: 10.1007/978-3-642-12245-3. |
[8] |
L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, New Jersey, 2004. |
[9] |
M. Harju, On the Direct and Inverse Scattering Problems for a Nonlinear Three-Dimensional Schrödinger Equation, PhD-thesis, University of Oulu, 2010. |
[10] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Determining a first order perturbation of the biharmonic operator by partial boundary measurements, Journal of Functional Analysis, 262 (2012), 1781-1801.
doi: 10.1016/j.jfa.2011.11.021. |
[11] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Inverse boundary value problems for the perturbed polyharmonic operator, Trans Amer. Math. Soc., 366 (2014), 95-112.
doi: 10.1090/S0002-9947-2013-05713-3. |
[12] |
N. N. Lebedev, Special Functions and Their Applications, Dover Publications, 1972. |
[13] |
B. Pausander, Scattering for the beam equation in low dimensions, Indiana Univ. Math. J., 59 (2010), 791–822. arXiv: 0903.3777v2 [math.AP].
doi: 10.1512/iumj.2010.59.3966. |
[14] |
V. S. Serov, An inverse Born approximation for the general nonlinear Schrödinger operator on the line, Journal of Physics A: Mathematical and Theoretical, 42 (2009), 332002.
doi: 10.1088/1751-8113/42/33/332002. |
[15] |
V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D, Journal of Mathematical Physics, 53 (2012), 123522.
doi: 10.1063/1.4769825. |
[16] |
T. Tyni, Numerical results for Saito's uniqueness theorem in inverse scattering theory, Inverse Problems, 35 (2020), 065002.
doi: 10.1088/1361-6420/ab7d2d. |
[17] |
T. Tyni and V. Serov,
Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Problems and Imaging, 12 (2018), 205-227.
doi: 10.3934/ipi.2018008. |
[18] |
T. Tyni and M. Harju, Inverse backscattering problem for perturbations of biharmonic operator, Inverse Problems, 33 (2017), 105002.
doi: 10.1088/1361-6420/aa873e. |
[19] |
T. Tyni and V. Serov,
Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line, Inverse Problems and Imaging, 13 (2019), 159-175.
doi: 10.3934/ipi.2019009. |
[20] |
E. Zeidler, Applied Functional Analysis, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-0821-1. |






[1] |
Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems and Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009 |
[2] |
Siamak RabieniaHaratbar. Inverse scattering and stability for the biharmonic operator. Inverse Problems and Imaging, 2021, 15 (2) : 271-283. doi: 10.3934/ipi.2020064 |
[3] |
Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems and Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291 |
[4] |
Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems and Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027 |
[5] |
Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems and Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681 |
[6] |
Weishi Yin, Jiawei Ge, Pinchao Meng, Fuheng Qu. A neural network method for the inverse scattering problem of impenetrable cavities. Electronic Research Archive, 2020, 28 (2) : 1123-1142. doi: 10.3934/era.2020062 |
[7] |
Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems and Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029 |
[8] |
Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741 |
[9] |
Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems and Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042 |
[10] |
Yunwen Yin, Weishi Yin, Pinchao Meng, Hongyu Liu. The interior inverse scattering problem for a two-layered cavity using the Bayesian method. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2021069 |
[11] |
Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems and Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551 |
[12] |
Teemu Tyni, Valery Serov. Scattering problems for perturbations of the multidimensional biharmonic operator. Inverse Problems and Imaging, 2018, 12 (1) : 205-227. doi: 10.3934/ipi.2018008 |
[13] |
Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems and Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029 |
[14] |
Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455 |
[15] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems and Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
[16] |
Loc H. Nguyen, Qitong Li, Michael V. Klibanov. A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media. Inverse Problems and Imaging, 2019, 13 (5) : 1067-1094. doi: 10.3934/ipi.2019048 |
[17] |
Michael V. Klibanov, Dinh-Liem Nguyen, Loc H. Nguyen, Hui Liu. A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data. Inverse Problems and Imaging, 2018, 12 (2) : 493-523. doi: 10.3934/ipi.2018021 |
[18] |
Michael V. Klibanov, Loc H. Nguyen, Anders Sullivan, Lam Nguyen. A globally convergent numerical method for a 1-d inverse medium problem with experimental data. Inverse Problems and Imaging, 2016, 10 (4) : 1057-1085. doi: 10.3934/ipi.2016032 |
[19] |
Michael V. Klibanov, Thuy T. Le, Loc H. Nguyen, Anders Sullivan, Lam Nguyen. Convexification-based globally convergent numerical method for a 1D coefficient inverse problem with experimental data. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2021068 |
[20] |
Benoît Pausader, Walter A. Strauss. Analyticity of the nonlinear scattering operator. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 617-626. doi: 10.3934/dcds.2009.25.617 |
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]