Two-dimensional inverse scattering for quasi-linear biharmonic operator

Markus Harju; Jaakko Kultima; Valery Serov; Teemu Tyni

doi:10.3934/ipi.2021026

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October 2021, 15(5): 1015-1033. doi: 10.3934/ipi.2021026

Two-dimensional inverse scattering for quasi-linear biharmonic operator

Markus Harju ^1, , Jaakko Kultima ^2,, , Valery Serov ^2, and Teemu Tyni ^3,

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

^* Corresponding author: Jaakko Kultima

Received October 2020 Revised February 2021 Published October 2021 Early access March 2021

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.

Keywords: Nonlinear biharmonic operator, inverse problem, scattering problem, Saito's formula, numerical method.

Mathematics Subject Classification: Primary: 35P25, 65M32; Secondary: 35R30, 35J91.

Citation: Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems and Imaging, 2021, 15 (5) : 1015-1033. doi: 10.3934/ipi.2021026

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.

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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.

Figure 1. The potentials $ \beta $ for Examples 1 (top left), 2 (top right), 3 (bottom left) and 4 (bottom right). In Example 1 we have only potential $ V $ which is a characteristic function of an ellipse. In Example 2 $ V $ is the characteristic function of an L-shaped domain and $ \overrightarrow{W} = (0,\varphi_2)\sin(|u|) $ has one component, where $ \varphi_2 $ is a smooth bump function in a circular domain. In Example 3 both components of $ \overrightarrow{W} = (\varphi_1\frac{|u|^2}{1+|u|^2},\varphi_2|u|^2) $ are multiplied by smooth bump functions $ \varphi_1 $ and $ \varphi_2 $ supported in ellipses located at the top and bottom right in the figure, respectively. The coefficient $ \varphi_3 $ of potential $ V $ is also a smooth bump function supported in an ellipse, located in the middle-left side of the figure. In example 4 all coefficients are smooth bump functions (see also Figure 6), but their supports are intersecting

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Figure 2. The scattered fields for Examples 1 (top left), 2 (top right), 3 (bottom left) and 4 (bottom right) with $ k = 25 $. The locations of the supports of the potentials are presented in black. Here the incident field is travelling from the left to the right

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Figure 3. Example 1. Left: The unknown target $ \beta $. Right: The numerical reconstruction $ \beta_ \mathrm{num} $

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Figure 4. Example 2. Left: The unknown target $ \beta $. Right: The numerical reconstruction $ \beta_ \mathrm{num} $. This example shows recovery of corners and recovery of a shape with piece-wise smooth boundary

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Figure 5. Example 3. Left: The unknown target $ \beta $. Right: The numerical reconstruction $ \beta_ \mathrm{num} $. We see that weak potentials are quite difficult to detect while stronger potentials are clearly visible in comparison, as is expected

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Figure 6. Example 4. Top left $ V(x,1) $, top middle $ W_1(x,1) $ and top right $ W_2(x,1) $. Bottom left: The unknown target $ \beta $. Bottom right: The numerical reconstruction $ \beta_ \mathrm{num} $. In this example the supports of potentials $ V $, $ W_1 $ and $ W_2 $ overlap. We can not distinguish these functions from each other

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Two-dimensional inverse scattering for quasi-linear biharmonic operator

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