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An inverse random source problem for the Helmholtz equation

Authors: Gang Bao, Shui-Nee Chow, Peijun Li and Haomin Zhou
Journal: Math. Comp. 83 (2014), 215-233
MSC (2010): Primary 65N21, 78A46
Published electronically: June 10, 2013
MathSciNet review: 3120587
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Abstract: This paper is concerned with an inverse random source problem for the one-dimensional 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 two-point 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 two-point 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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Additional Information

Gang Bao
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, China — and — Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Shui-Nee Chow
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Peijun Li
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Haomin Zhou
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

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 DMS-0908325, CCF-0830161, EAR-0724527, DMS-0968360, DMS-1211292, the ONR grant N00014-12-1-0319, 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 DMS-0914595 and DMS-1042958
The fourth author’s research was supported in part by NSF Faculty Early Career Development (CAREER) Award DMS-0645266 and DMS-1042998
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