An inverse random source problem for the Helmholtz equation
Authors:
Gang Bao, ShuiNee Chow, Peijun Li and Haomin Zhou
Journal:
Math. Comp. 83 (2014), 215233
MSC (2010):
Primary 65N21, 78A46
Published electronically:
June 10, 2013
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Abstract: This paper is concerned with an inverse random source problem for the onedimensional stochastic Helmholtz equation, which is to reconstruct the statistical properties of the random source function from boundary measurements of the radiating random electric field. Although the emphasis of the paper is on the inverse problem, we adapt a computationally more efficient approach to study the solution of the direct problem in the context of the scattering model. Specifically, the direct model problem is equivalently formulated into a twopoint spatially stochastic boundary value problem, for which the existence and uniqueness of the pathwise solution is proved. In particular, an explicit formula is deduced for the solution from an integral representation by solving the twopoint boundary value problem. Based on this formula, a novel and efficient strategy, which is entirely done by using the fast Fourier transform, is proposed to reconstruct the mean and the variance of the random source function from measurements at one boundary point, where the measurements are assumed to be available for many realizations of the source term. Numerical examples are presented to demonstrate the validity and effectiveness of the proposed method.
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 M. Badieirostami, A. Adibi, H. Zhou, and S. Chow, Model for efficient simulation of spatially incoherent light using the Wiener chaos expansion method, Opt. Lett., 32 (2007), 31883190.
 [2]
 G. Bao, S.N. Chow, P. Li, and H. Zhou, Numerical solution of an inverse medium scattering problem with a stochastic source, Inverse Problems, 26 (2010), 074014.
 [3]
 Gang Bao, Junshan Lin, and Faouzi Triki, A multifrequency inverse source problem, J. Differential Equations 249 (2010), no. 12, 34433465. MR 2737437 (2012c:35464), http://dx.doi.org/10.1016/j.jde.2010.08.013
 [4]
 Gang Bao, Junshan Lin, and Faouzi Triki, Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data, Mathematical and statistical methods for imaging, Contemp. Math., vol. 548, Amer. Math. Soc., Providence, RI, 2011, pp. 4560. MR 2868487 (2012j:65375), http://dx.doi.org/10.1090/conm/548/10835
 [5]
 Gang Bao, Junshan Lin, and Faouzi Triki, An inverse source problem with multiple frequency data, C. R. Math. Acad. Sci. Paris 349 (2011), no. 1516, 855859 (English, with English and French summaries). MR 2835891 (2012h:35372), http://dx.doi.org/10.1016/j.crma.2011.07.009
 [6]
 Guillaume Bal, Central limits and homogenization in random media, Multiscale Model. Simul. 7 (2008), no. 2, 677702. MR 2443008 (2010b:35508), http://dx.doi.org/10.1137/070709311
 [7]
 Guillaume Bal and Kui Ren, Physicsbased models for measurement correlations: application to an inverse SturmLiouville problem, Inverse Problems 25 (2009), no. 5, 055006, 13. MR 2501024 (2010d:81278), http://dx.doi.org/10.1088/02665611/25/5/055006
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 Matthias Eller and Nicolas P. Valdivia, Acoustic source identification using multiple frequency information, Inverse Problems 25 (2009), no. 11, 115005, 20. MR 2546000, http://dx.doi.org/10.1088/02665611/25/11/115005
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Additional Information
Gang Bao
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, China — and — Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email:
bao@math.msu.edu
ShuiNee Chow
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email:
chow@math.gatech.edu
Peijun Li
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email:
lipeijun@math.purdue.edu
Haomin Zhou
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email:
hmzhou@math.gatech.edu
DOI:
http://dx.doi.org/10.1090/S002557182013027305
PII:
S 00255718(2013)027305
Keywords:
Inverse source problem,
Helmholtz equation,
stochastic differential equation
Received by editor(s):
June 24, 2010
Received by editor(s) in revised form:
October 22, 2011
Published electronically:
June 10, 2013
Additional Notes:
The first author’s research was supported in part by the NSF grants DMS0908325, CCF0830161, EAR0724527, DMS0968360, DMS1211292, the ONR grant N000141210319, a Key Project of the Major Research Plan of NSFC (No. 91130004), and a special research grant from Zhejiang University.
The third author’s research was supported in part by NSF grants DMS0914595 and DMS1042958
The fourth author’s research was supported in part by NSF Faculty Early Career Development (CAREER) Award DMS0645266 and DMS1042998
Article copyright:
© Copyright 2013
American Mathematical Society
