Increasing stability in the linearized inverse Schrödinger potential problem with power type nonlinearities. (English) Zbl 1489.35308
Summary: We consider increasing stability in the inverse Schrödinger potential problem with power type nonlinearities at a large wavenumber. Two linearization approaches, with respect to small boundary data and small potential function, are proposed and their performance on the inverse Schrödinger potential problem is investigated. It can be observed that higher order linearization for small boundary data can provide an increasing stability for an arbitrary power type nonlinearity term if the wavenumber is chosen large. Meanwhile, linearization with respect to the potential function leads to increasing stability for a quadratic nonlinearity term, which highlights the advantage of nonlinearity in solving the inverse Schrödinger potential problem. Noticing that both linearization approaches can be numerically approximated, we provide several reconstruction algorithms for the quadratic and general power type nonlinearity terms, where one of these algorithms is designed based on boundary measurements of multiple wavenumbers. Several numerical examples shed light on the efficiency of our proposed algorithms.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J61 | Semilinear elliptic equations |
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
Keywords:
increasing stability; inverse Schrödinger potential problem; power type nonlinearities; reconstruction algorithms; linearization approachesReferences:
| [1] | Alessandrini, G., Stable determination of conductivity by boundary measurements, Appl. Anal., 27, 153-172 (1988) · Zbl 0616.35082 · doi:10.1080/00036818808839730 |
| [2] | Borges, C.; Greengard, L., Inverse obstacle scattering in two dimensions with multiple frequency data and multiple angles of incidence, SIAM J. Imaging Sci., 8, 280-298 (2015) · Zbl 1317.31006 · doi:10.1137/140982787 |
| [3] | Bao, G.; Li, P.; Lin, J.; Triki, F., Inverse scattering problems with multi-frequencies, Inverse Problems, 31 (2015) · Zbl 1332.78019 · doi:10.1088/0266-5611/31/9/093001 |
| [4] | Bao, G.; Lu, S.; Rundell, W.; Xu, B., A recursive algorithm for multifrequency acoustic inverse source problems, SIAM J. Numer. Anal., 53, 1608-1628 (2015) · Zbl 1321.65166 · doi:10.1137/140993648 |
| [5] | Bao, G.; Lin, J.; Triki, F., A multi-frequency inverse source problem, J. Differ. Equ., 249, 3443-3465 (2010) · Zbl 1205.35336 · doi:10.1016/j.jde.2010.08.013 |
| [6] | Bao, G.; Li, P.; Zhao, Y., Stability for the inverse source problems in elastic and electromagnetic waves, J. Math. Appl., 134, 122-178 (2020) · Zbl 1433.35376 · doi:10.1016/j.matpur.2019.06.006 |
| [7] | Bao, G.; Triki, F., Error estimates for the recursive linearization of inverse medium problems, J. Comput. Math., 28, 725-744 (2010) · Zbl 1240.35574 · doi:10.4208/jcm.1003-m0004 |
| [8] | Bao, G.; Triki, F., Stability for the multifrequency inverse medium problem, J. Differ. Equ., 269, 7106-7128 (2020) · Zbl 1443.35180 · doi:10.1016/j.jde.2020.05.021 |
| [9] | Calderón, A. P., On an inverse boundary value problem, Seminear on Numerical Analysis and its Application to Continuum Physics, 65-73 (1980), Rio de Jeneiro: Soc. Brasileira de Matèmatica, Rio de Jeneiro |
| [10] | Cârstea, C. I.; Feizmohammadi, A.; Kian, Y.; Krupchyk, K.; Uhlmann, G., The Calderón inverse problem for isotropic quasilinear conductivities, Adv. Math., 391 (2021) · Zbl 1479.35943 · doi:10.1016/j.aim.2021.107956 |
| [11] | Cheng, J.; Isakov, V.; Lu, S., Increasing stability in the inverse source problem with many frequencies, J. Differ. Equ., 260, 4786-4804 (2016) · Zbl 1401.35339 · doi:10.1016/j.jde.2015.11.030 |
| [12] | Evéquoz, G.; Weth, T., Real solutions to the nonlinear Helmholtz equation with local nonlinearity, Arch. Ration. Mech. Anal., 211, 359-388 (2014) · Zbl 1285.35003 · doi:10.1007/s00205-013-0664-2 |
| [13] | Feizmohammadi, A.; Oksanen, L., An inverse problem for a semi-linear elliptic equation in Riemannian geometries, J. Differ. Equ., 269, 4683-4719 (2020) · Zbl 1448.58016 · doi:10.1016/j.jde.2020.03.037 |
| [14] | Fibich, G.; Tsynkov, S., Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comput. Phys., 210, 183-224 (2005) · Zbl 1076.78008 · doi:10.1016/j.jcp.2005.04.015 |
| [15] | Isakov, V., Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map, Discrete Continuous Dyn. Syst. S, 4, 631-640 (2011) · Zbl 1210.35289 · doi:10.3934/dcdss.2011.4.631 |
| [16] | Isakov, V.; Lu, S., Increasing stability in the inverse source problem with attenuation and many frequencies, SIAM J. Appl. Math., 78, 1-18 (2018) · Zbl 1391.35419 · doi:10.1137/17m1112704 |
| [17] | Isakov, V.; Lai, R-Y; Wang, J-N, Increasing stability for the conductivity and attenuation coefficients, SIAM J. Math. Anal., 48, 569-594 (2016) · Zbl 1338.35496 · doi:10.1137/15m1019052 |
| [18] | Isakov, V.; Lu, S.; Xu, B., Linearized inverse Schrödinger potential problem at a large wavenumber, SIAM J. Appl. Math., 80, 338-358 (2020) · Zbl 1430.35267 · doi:10.1137/18m1226932 |
| [19] | Isakov, V.; Wang, J. N., Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map, Inverse Problems Imaging, 8, 1139-1150 (2014) · Zbl 1328.35312 · doi:10.3934/ipi.2014.8.1139 |
| [20] | Karamehmedović, M.; Kirkeby, A.; Knudsen, K., Stable source reconstruction from a finite number of measurements in the multi-frequency inverse source problem, Inverse Problems, 34 (2018) · Zbl 1404.65238 · doi:10.1088/1361-6420/aaba83 |
| [21] | Kian, Y.; Krupchyk, K.; Uhlmann, G., Partial data inverse problems for quasilinear conductivity equations (2020) |
| [22] | Kurylev, Y.; Lassas, M.; Uhlmann, G., Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Invent. Math., 212, 781-857 (2018) · Zbl 1396.35074 · doi:10.1007/s00222-017-0780-y |
| [23] | Krupchyk, K.; Uhlmann, G., A remark on partial data inverse problems for semilinear elliptic equations, Proc. Am. Math. Soc., 148, 681-685 (2020) · Zbl 1431.35246 · doi:10.1090/proc/14844 |
| [24] | Lassas, M.; Liimatainen, T.; Lin, Y-H; Salo, M., Inverse problems for elliptic equations with power type nonlinearities, J. Math. Pure Appl., 145, 44-82 (2021) · Zbl 1460.35395 · doi:10.1016/j.matpur.2020.11.006 |
| [25] | Lassas, M.; Liimatainen, T.; Lin, Y-H; Salo, M., Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations, Rev. Mat. Iberoam., 37, 1553-1580 (2021) · Zbl 1473.35653 · doi:10.4171/rmi/1242 |
| [26] | Liimatainen, T.; Lin, Y-H; Salo, M.; Tyni, T., Inverse problems for elliptic equations with fractional power type nonlinearities, J. Differ. Equ., 306, 189-219 (2022) · Zbl 1482.35276 · doi:10.1016/j.jde.2021.10.015 |
| [27] | Nagayasu, S.; Uhlmann, G.; Wang, J-N, Increasing stability in an inverse problem for the acoustic equation, Inverse Problems, 29 (2013) · Zbl 1302.65243 · doi:10.1088/0266-5611/29/2/025012 |
| [28] | Wu, H.; Zou, J., Finite element method and its analysis for a nonlinear Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 56, 1338-1359 (2018) · Zbl 1393.65047 · doi:10.1137/17m111314x |
| [29] | Xu, Z.; Bao, G., A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects, J. Opt. Soc. Am. A, 27, 2347-2353 (2010) · doi:10.1364/josaa.27.002347 |
| [30] | Yuan, L.; Lu, Y. Y., Robust iterative method for nonlinear Helmholtz equation, J. Comput. Phys., 343, 1-9 (2017) · Zbl 1380.65394 · doi:10.1016/j.jcp.2017.04.046 |
| [31] | Zhang, B.; Zhang, H., Recovering scattering obstacles by multi-frequency phaseless far-field data, J. Comput. Phys., 345, 58-73 (2017) · Zbl 1378.35210 · doi:10.1016/j.jcp.2017.05.022 |
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