# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2022011
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## Reconstruction of singularities in two-dimensional quasi-linear biharmonic operator

 Research Unit of Mathematical Sciences, P.O. BOX 3000, FIN-90014 University of Oulu, Finland

*Corresponding author: Jaakko Kultima

Received  September 2021 Revised  January 2022 Early access March 2022

The inverse backscattering Born approximation for two-dimensional quasi-linear biharmonic operator is studied. We prove the precise formulae for the first nonlinear term of the Born sequence. We prove also that all other terms in this sequence are $H^t-$functions for any $t<1$. These formulae and estimates allow us to conclude that all main singularities of a certain combination of unknown coefficients, in particular, $L^p-$singularities for $2\le p<\infty$, can be uniquely reconstructed using the inverse backscattering Born approximation. In addition, it is shown that the jumps ($L^{\infty}-$singularities) over smooth curves are uniquely determined by the backscattering data and can be recovered from the Born approximation. We present a numerical method for the reconstruction of these singularities.

Citation: Jaakko Kultima, Valery Serov. Reconstruction of singularities in two-dimensional quasi-linear biharmonic operator. Inverse Problems and Imaging, doi: 10.3934/ipi.2022011
##### References:
 [1] 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. [2] G. Fotopoulos, M. Harju and V. Serov, Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D, Inverse Probl. Imaging, 7 (2013), 183-197.  doi: 10.3934/ipi.2013.7.183. [3] 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. [4] F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-12245-3. [5] P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal., 5 (1996), 403-415. [6] M. Harju, J. Kultima, V. Serov and T. Tyni, Two-dimensional inverse scattering for quasi-linear biharmonic operator, Inverse Probl. Imaging, 15 (2021), 1015-1033.  doi: 10.3934/ipi.2021026. [7] V. Serov, Inverse fixed angle scattering and backscattering problems in two dimensions, Inverse Problems, 24 (2008), 065002, 14 pp. doi: 10.1088/0266-5611/24/6/065002. [8] V. Serov, An inverse Born approximation for the general nonlinear Schrödinger operator on the line, J. Phys. A, 42 (2009), 332002, 7 pp. doi: 10.1088/1751-8113/42/33/332002. [9] V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D, J. Math. Phys., 53 (2012), 123522, 16 pp. doi: 10.1063/1.4769825. [10] V. Serov and J. Sandhu, Inverse backscattering problem for the generalized nonlinear Schrödinger operator in two dimensions, J. Phys. A, 43 (2010), 325206, 15 pp. doi: 10.1088/1751-8113/43/32/325206. [11] T. Tyni, Numerical results for Saito's uniqueness theorem in inverse scattering theory, Inverse Problems, 36 (2020), 065002, 14 pp. doi: 10.1088/1361-6420/ab7d2d. [12] T. Tyni and M. Harju, Inverse backscattering problem for perturbations of biharmonic operator, Inverse Problems, 33 (2017), 105002, 20 pp. doi: 10.1088/1361-6420/aa873e. [13] T. Tyni and V. Serov, Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Probl. Imaging, 12 (2018), 205-227.  doi: 10.3934/ipi.2018008. [14] T. Tyni and V. Serov, Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line, Inverse Probl. Imaging, 13 (2019), 159-175.  doi: 10.3934/ipi.2019009. [15] E. Zeidler, Applied Functional Analysis, Springer-Verlag, New York, 1995.

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##### References:
 [1] 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. [2] G. Fotopoulos, M. Harju and V. Serov, Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D, Inverse Probl. Imaging, 7 (2013), 183-197.  doi: 10.3934/ipi.2013.7.183. [3] 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. [4] F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-12245-3. [5] P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal., 5 (1996), 403-415. [6] M. Harju, J. Kultima, V. Serov and T. Tyni, Two-dimensional inverse scattering for quasi-linear biharmonic operator, Inverse Probl. Imaging, 15 (2021), 1015-1033.  doi: 10.3934/ipi.2021026. [7] V. Serov, Inverse fixed angle scattering and backscattering problems in two dimensions, Inverse Problems, 24 (2008), 065002, 14 pp. doi: 10.1088/0266-5611/24/6/065002. [8] V. Serov, An inverse Born approximation for the general nonlinear Schrödinger operator on the line, J. Phys. A, 42 (2009), 332002, 7 pp. doi: 10.1088/1751-8113/42/33/332002. [9] V. Serov, M. Harju and G. Fotopoulos, Direct and inverse scattering for nonlinear Schrödinger equation in 2D, J. Math. Phys., 53 (2012), 123522, 16 pp. doi: 10.1063/1.4769825. [10] V. Serov and J. Sandhu, Inverse backscattering problem for the generalized nonlinear Schrödinger operator in two dimensions, J. Phys. A, 43 (2010), 325206, 15 pp. doi: 10.1088/1751-8113/43/32/325206. [11] T. Tyni, Numerical results for Saito's uniqueness theorem in inverse scattering theory, Inverse Problems, 36 (2020), 065002, 14 pp. doi: 10.1088/1361-6420/ab7d2d. [12] T. Tyni and M. Harju, Inverse backscattering problem for perturbations of biharmonic operator, Inverse Problems, 33 (2017), 105002, 20 pp. doi: 10.1088/1361-6420/aa873e. [13] T. Tyni and V. Serov, Scattering problems for perturbations of the multidimensional biharmonic operator, Inverse Probl. Imaging, 12 (2018), 205-227.  doi: 10.3934/ipi.2018008. [14] T. Tyni and V. Serov, Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line, Inverse Probl. Imaging, 13 (2019), 159-175.  doi: 10.3934/ipi.2019009. [15] E. Zeidler, Applied Functional Analysis, Springer-Verlag, New York, 1995.
Combination $\beta$ consisting three disjoint parts is depicted on the left hand side. In the middle we have the TSVD reconstruction of the function $\beta_B^b$ and on the right the difference $\beta - \beta_B^b$
The second example includes different types of non-linearities compared with the first example. In particular, here the function $V$ is independent on $|u|$ and therefore this example highlights that the linear potentials are included in our considerations. In the middle we have the TSVD reconstruction of function $\beta_B^b$ and on the right the difference $\beta - \beta_B^b$
In the third example, only the first order potential $\overrightarrow{W}$ is presented. The unknown combination $\beta = -\frac{1}{2}\nabla\cdot \overrightarrow{W}$ (on the left-hand side) in this case has jump-discontinuities over smooth boundaries and as Corollary 1 suggests, such jump-discontinuities of $\beta$ can be recovered, by using the backscattering Born approximation. In the middle we have the TSVD-reconstruction of function $\beta^b_B$ and on the right-hand side the difference $\beta - \beta_B^b$
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