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A logarithmic estimate for the inverse source scattering problem with attenuation in a two-layered medium

  • Mozhgan N. Entekhabi EMAIL logo and Ajith Gunaratne

Abstract

The paper aims a logarithmic stability estimate for the inverse source problem of the one-dimensional Helmholtz equation with attenuation factor in a two layer medium. We establish a stability by using multiple frequencies at the two end points of the domain which contains the compact support of the source functions.

Award Identifier / Grant number: HRD-1824267

Funding statement: This research is supported in part by NSF Award HRD-1824267.

References

[1] H. Ammari, G. Bao and J. L. Fleming, An inverse source problem for Maxwell’s equations in magnetoencephalography, SIAM J. Appl. Math. 62 (2002), no. 4, 1369–1382. 10.1137/S0036139900373927Search in Google Scholar

[2] D. Aralumallige Subbarayappa and V. Isakov, Increasing stability of the continuation for the Maxwell system, Inverse Problems 26 (2010), no. 7, Article ID 074005. Search in Google Scholar

[3] C. Balanis, Antenna Theory - Analysis and Design, Wiley, Hoboken, 2005. Search in Google Scholar

[4] G. Bao, J. Lin and F. Triki, A multi-frequency inverse source problem, J. Differential Equations 249 (2010), no. 12, 3443–3465. 10.1016/j.jde.2010.08.013Search in Google Scholar

[5] G. Bao, J. Lin and F. Triki, An inverse source problem with multiple frequency data, C. R. Math. Acad. Sci. Paris 349 (2011), no. 15–16, 855–859. 10.1016/j.crma.2011.07.009Search in Google Scholar

[6] G. Bao, S. Lu, W. Rundell and B. Xu, A recursive algorithm for multifrequency acoustic inverse source problems, SIAM J. Numer. Anal. 53 (2015), no. 3, 1608–1628. 10.1137/140993648Search in Google Scholar

[7] J. Chen, D. Fan and C. Zhang, Estimates for damped fractional wave equations and applications, Electron. J. Differential Equations2015 (2015), Paper No. 162. Search in Google Scholar

[8] J. Cheng, V. Isakov and S. Lu, Increasing stability in the inverse source problem with many frequencies, J. Differential Equations 260 (2016), no. 5, 4786–4804. 10.1016/j.jde.2015.11.030Search in Google Scholar

[9] R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II: Partial Differential Equations, John Wiley & Sons, New York, 1962. Search in Google Scholar

[10] M. Eller and N. P. Valdivia, Acoustic source identification using multiple frequency information, Inverse Problems 25 (2009), no. 11, Article ID 115005. 10.1088/0266-5611/25/11/115005Search in Google Scholar

[11] M. N. Entekhabi, Increasing stability in the two dimensional inverse source scattering problem with attenuation and many frequencies, Inverse Problems 34 (2018), no. 11, Article ID 115001. 10.1088/1361-6420/aad677Search in Google Scholar

[12] M. N. Entekhabi and V. Isakov, On increasing stability in the two dimensional inverse source scattering problem with many frequencies, Inverse Problems 34 (2018), no. 5, Article ID 055005. 10.1088/1361-6420/aab465Search in Google Scholar

[13] M. N. Entekhabi and V. Isakov, Increasing stability in acoustic and elastic inverse source problems, SIAM. J. Math. Anal., to appear. 10.1137/19M1279885Search in Google Scholar

[14] V. Isakov, Inverse Problems for Partial Differential Equations, 3rd ed., Appl. Math. Sci. 127, Springer, Cham, 2017. 10.1007/978-3-319-51658-5Search in Google Scholar

[15] V. Isakov and S. Kindermann, Subspaces of stability in the Cauchy problem for the Helmholtz equation, Methods Appl. Anal. 18 (2011), no. 1, 1–29. 10.4310/MAA.2011.v18.n1.a1Search in Google Scholar

[16] V. Isakov and S. Lu, Increasing stability in the inverse source problem with attenuation and many frequencies, SIAM J. Appl. Math. 78 (2018), no. 1, 1–18. 10.1137/17M1112704Search in Google Scholar

[17] V. Isakov and S. Lu, Inverse source problems without (pseudo) convexity assumptions, Inverse Probl. Imaging 12 (2018), no. 4, 955–970. 10.3934/ipi.2018040Search in Google Scholar

[18] F. John, Continuous dependence on data for solutions of partial differential equations with a presribed bound, Comm. Pure Appl. Math. 13 (1960), 551–585. 10.1002/cpa.3160130402Search in Google Scholar

[19] F. John, Partial Differential Equations, 4th ed., Appl. Math. Sci. 1, Springer, New York, 1982. 10.1007/978-1-4684-9333-7Search in Google Scholar

[20] S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan 47 (1995), no. 4, 617–653. 10.2969/jmsj/04740617Search in Google Scholar

[21] P. Li and G. Yuan, Increasing stability for the inverse source scattering problem with multi-frequencies, Inverse Probl. Imaging 11 (2017), no. 4, 745–759. 10.3934/ipi.2017035Search in Google Scholar

[22] Y. Zhao and P. Li, Stability on the one-dimensional inverse source scattering problem in a two-layered medium, Appl. Anal. 98 (2019), no. 4, 682–692. 10.1080/00036811.2017.1399365Search in Google Scholar

Received: 2019-03-25
Revised: 2019-12-06
Accepted: 2020-02-02
Published Online: 2020-03-28
Published in Print: 2020-08-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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