August  2017, 11(4): 745-759. doi: 10.3934/ipi.2017035

Increasing stability for the inverse source scattering problem with multi-frequencies

1. 

Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA

2. 

KLAS, School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China

* Corresponding author: Ganghua Yuan

Received  July 2016 Revised  March 2017 Published  June 2017

Fund Project: The research of PL was supported in part by the NSF grant DMS-1151308. The research of GY was supported in part by NSFC grants 10801030,11671072,11571064, the Ying Dong Fok Education Foundation under grant 141001, and the Fundamental Research Funds for the Central Universities under grant 2412015BJ011.

Consider the scattering of the two-or three-dimensional Helmholtz equation where the source of the electric current density is assumed to be compactly supported in a ball. This paper concerns the stability analysis of the inverse source scattering problem which is to reconstruct the source function. Our results show that increasing stability can be obtained for the inverse problem by using only the Dirichlet boundary data with multi-frequencies.

Citation: Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems and Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035
References:
[1]

S. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93.  doi: 10.1088/0266-5611/15/2/022.

[2]

C. A. Balanis, Antenna Theory-Analysis and Design, Wiley, Hoboken, NJ, 2005.

[3]

G. Bao, P. Li, J. Lin and F. Triki, Inverse scattering problems with multi-frequencies Inverse Problems, 31 (2015), 093001, 21pp. doi: 10.1088/0266-5611/31/9/093001.

[4]

G. BaoJ. Lin and F. Triki, A multi-frequency inverse source problem, J. Differential Equations, 249 (2010), 3443-3465.  doi: 10.1016/j.jde.2010.08.013.

[5]

G. BaoJ. Lin and F. Triki, Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data, Contemp. Math., 548 (2011), 45-60.  doi: 10.1090/conm/548/10835.

[6]

G. BaoS. LuW. Rundell and B. Xu, A recursive algorithm for multifrequency acoustic inverse source problems, SIAM J. Numer. Anal., 53 (2015), 1608-1628.  doi: 10.1137/140993648.

[7]

J. ChengV. Isakov and S. Lu, Increasing stability in the inverse source problem with many frequencies, J. Differential Equations, 260 (2016), 4786-4804.  doi: 10.1016/j.jde.2015.11.030.

[8]

A. Devaney and G. Sherman, Nonuniqueness in inverse source and scattering problems, IEEE Trans. Antennas Propag., 30 (1982), 1034-1042.  doi: 10.1109/TAP.1982.1142902.

[9]

M. Eller and N. P. Valdivia, Acoustic source identification using multiple frequency information Inverse Problems, 25 (2009), 115005, 20pp. doi: 10.1088/0266-5611/25/11/115005.

[10]

K.-H. HauerL. Kühn and R. Potthast, On uniqueness and non-uniqueness for current reconstruction from magnetic fields, Inverse Problems, 21 (2005), 955-967.  doi: 10.1088/0266-5611/21/3/010.

[11]

V. Isakov, Inverse Source Problems, AMS, Providence, RI, 1990. doi: 10.1090/surv/034.

[12]

V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 2006.

[13]

V. Isakov, Increasing stability in the continuation for the Helmholtz equation with variable coefficient, Contemp. Math., 426 (2007), 255-267.  doi: 10.1090/conm/426/08192.

[14]

V. Isakov, Increasing stability for the Schödinger potential from the Dirichlet-to-Neumann map, DCDS-S, 4 (2011), 631-640.  doi: 10.3934/dcdss.2011.4.631.

[15]

P. Li and G. Yuan, Stability on the inverse random source scattering problem for the one-dimensional Helmholtz equation, J. Math. Anal. Appl., 450 (2017), 872-887.  doi: 10.1016/j.jmaa.2017.01.074.

[16]

P. Stefanov and G. Uhlmann, Themoacoustic tomography arising in brain imaging Inverse Problems, 27 (2011), 075011, 26pp. doi: 10.1088/0266-5611/27/4/045004.

[17]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944.

show all references

References:
[1]

S. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93.  doi: 10.1088/0266-5611/15/2/022.

[2]

C. A. Balanis, Antenna Theory-Analysis and Design, Wiley, Hoboken, NJ, 2005.

[3]

G. Bao, P. Li, J. Lin and F. Triki, Inverse scattering problems with multi-frequencies Inverse Problems, 31 (2015), 093001, 21pp. doi: 10.1088/0266-5611/31/9/093001.

[4]

G. BaoJ. Lin and F. Triki, A multi-frequency inverse source problem, J. Differential Equations, 249 (2010), 3443-3465.  doi: 10.1016/j.jde.2010.08.013.

[5]

G. BaoJ. Lin and F. Triki, Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data, Contemp. Math., 548 (2011), 45-60.  doi: 10.1090/conm/548/10835.

[6]

G. BaoS. LuW. Rundell and B. Xu, A recursive algorithm for multifrequency acoustic inverse source problems, SIAM J. Numer. Anal., 53 (2015), 1608-1628.  doi: 10.1137/140993648.

[7]

J. ChengV. Isakov and S. Lu, Increasing stability in the inverse source problem with many frequencies, J. Differential Equations, 260 (2016), 4786-4804.  doi: 10.1016/j.jde.2015.11.030.

[8]

A. Devaney and G. Sherman, Nonuniqueness in inverse source and scattering problems, IEEE Trans. Antennas Propag., 30 (1982), 1034-1042.  doi: 10.1109/TAP.1982.1142902.

[9]

M. Eller and N. P. Valdivia, Acoustic source identification using multiple frequency information Inverse Problems, 25 (2009), 115005, 20pp. doi: 10.1088/0266-5611/25/11/115005.

[10]

K.-H. HauerL. Kühn and R. Potthast, On uniqueness and non-uniqueness for current reconstruction from magnetic fields, Inverse Problems, 21 (2005), 955-967.  doi: 10.1088/0266-5611/21/3/010.

[11]

V. Isakov, Inverse Source Problems, AMS, Providence, RI, 1990. doi: 10.1090/surv/034.

[12]

V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 2006.

[13]

V. Isakov, Increasing stability in the continuation for the Helmholtz equation with variable coefficient, Contemp. Math., 426 (2007), 255-267.  doi: 10.1090/conm/426/08192.

[14]

V. Isakov, Increasing stability for the Schödinger potential from the Dirichlet-to-Neumann map, DCDS-S, 4 (2011), 631-640.  doi: 10.3934/dcdss.2011.4.631.

[15]

P. Li and G. Yuan, Stability on the inverse random source scattering problem for the one-dimensional Helmholtz equation, J. Math. Anal. Appl., 450 (2017), 872-887.  doi: 10.1016/j.jmaa.2017.01.074.

[16]

P. Stefanov and G. Uhlmann, Themoacoustic tomography arising in brain imaging Inverse Problems, 27 (2011), 075011, 26pp. doi: 10.1088/0266-5611/27/4/045004.

[17]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944.

Figure 1.  Problem geometry of the inverse source scattering problem
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