American Institute of Mathematical Sciences

October  2019, 13(5): 1067-1094. doi: 10.3934/ipi.2019048

A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media

 Department of Mathematics and Statistics, University of North Carolina, Charlotte, Charlotte, NC, 28223, USA

* Corresponding author: Loc H. Nguyen

Received  January 2019 Revised  April 2019 Published  July 2019

Fund Project: The work of Nguyen and Klibanov was supported by US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044. In addition, the effort of Nguyen and Li was supported by research funds FRG 111172 provided by The University of North Carolina at Charlotte

A new numerical method to solve an inverse source problem for the Helmholtz equation in inhomogenous media is proposed. This method reduces the original inverse problem to a boundary value problem for a coupled system of elliptic PDEs, in which the unknown source function is not involved. The Dirichlet boundary condition is given on the entire boundary of the domain of interest and the Neumann boundary condition is given on a part of this boundary. To solve this problem, the quasi-reversibility method is applied. Uniqueness and existence of the minimizer are proven. A new Carleman estimate is established. Next, the convergence of those minimizers to the exact solution is proven using that Carleman estimate. Results of numerical tests are presented.

Citation: Loc H. Nguyen, Qitong Li, Michael V. Klibanov. A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media. Inverse Problems & Imaging, 2019, 13 (5) : 1067-1094. doi: 10.3934/ipi.2019048
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The comparison of the true function $v(\cdot , k = 1.5) = \sum_{m = 1}^{\infty }v_{m}(\mathbf{x})\Psi _{m}(k)$ and the test function $\sum_{m = 1}^{10}v_{m}(\cdot )\Psi _{m}(k)$ in Test 5, see Section 4. In this test, we consider the case $n = 2$ and $\Omega = (-2, 2)^{2}$. On $\Omega ,$ we arrange a uniform grid of $121\times 121$ points in $\Omega$. Those points are numbered from $1$ to $121^{2}$. In (a) and (b), we respectively show the real and imaginary parts of the two functions at 300 points numbered from 7170 to 7470. It is evident that reconstructing the first 10 terms of the Fourier coefficients of $v(\mathbf{x }, k)$ is sufficient to solve our inverse source problems
Test 1. The true and reconstructed source functions and the true and reconstructed functions $v(\mathbf{x}, k) = u(\mathbf{x}, k)/g(k)$ when $k = 1.5.$ The reconstructed positive value of the source function is 2.76 (relative error 10.5%). The reconstructed negative value of the source function is -2.17 (relative error 8.5%). (A) The function $f_{\rm true}$; (B) The real part of the function $v_{\rm true}(\cdot, k = 1.5)$; (C) The imaginary part of the function $v_{\rm true}(\cdot, k = 1.5)$; (D) The function $f_{\rm comp}$; (E) The real part of the function $v_{\rm comp}(\cdot, k = 1.5)$; (F) The imaginary part of the function $v_{\rm comp}(\cdot, k = 1.5)$
Test 2. The true and reconstructed source functions and the true and reconstructed functions $v(\mathbf{x}, k) = u(\mathbf{x}, k)/g(k)$ when $k = 1.5.$ The reconstructed positive value of the source function is 1.11 (relative error 11.1%). The reconstructed negative value of the source function is -1.11 (relative error 11.1%). A) The function $f_{\rm true}$; (B) The real part of the function $v_{\rm true}(\cdot, k = 1.5)$; (C) The imaginary part of the function $v_{\rm true}(\cdot, k = 1.5)$; (D) The function $f_{\rm comp}$; (E) The real part of the function $v_{\rm comp}(\cdot, k = 1.5)$; (F) The imaginary part of the function $v_{\rm comp}(\cdot, k = 1.5)$
Test 3. The true and reconstructed source functions and the true and reconstructed functions $v(\mathbf{x}, k) = u(\mathbf{x}, k)/g(k)$ when $k = 1.5.$ The reconstructed positive value of the source function is 1.09 (relative error 9.0%). The reconstructed negative value of the source function is -0.89 (relative error 11.0%). A) The function $f_{\rm true}$; (B) The real part of the function $v_{\rm true}(\cdot, k = 1.5)$; (C) The imaginary part of the function $v_{\rm true}(\cdot, k = 1.5)$; (D) The function $f_{\rm comp}$; (E) The real part of the function $v_{\rm comp}(\cdot, k = 1.5)$; (F) The imaginary part of the function $v_{\rm comp}(\cdot, k = 1.5)$
Test 4. The true and reconstructed source functions and the true and reconstructed functions $v(\mathbf{x}, k) = u(\mathbf{x}, k)/g(k)$ when $k = 1.5.$ The reconstructed positive value of the source function is 1.12 (relative error 12.0%). The reconstructed negative value of the source function is -1.94 (relative error 3.0%). A) The function $f_{\rm true}$; (B) The real part of the function $v_{\rm true}(\cdot, k = 1.5)$; (C) The imaginary part of the function $v_{\rm true}(\cdot, k = 1.5)$; (D) The function $f_{\rm comp}$; (E) The real part of the function $v_{\rm comp}(\cdot, k = 1.5)$; (F) The imaginary part of the function $v_{\rm comp}(\cdot, k = 1.5)$
Test 5. The true and reconstructed source functions and the true and reconstructed functions $v(\mathbf{x}, k) = u(\mathbf{x}, k)/g(k)$ when $k = 1.5.$ The true and reconstructed maximal positive value of the source function are 8.10 and 7.36 (relative error 9.1%) respectively. The true and reconstructed minimal negative value of the source function are -6.55 and -5.48 (relative error 16.0%) respectively. A) The function $f_{\rm true}$; (B) The real part of the function $v_{\rm true}(\cdot, k = 1.5)$; (C) The imaginary part of the function $v_{\rm true}(\cdot, k = 1.5)$; (D) The function $f_{\rm comp}$; (E) The real part of the function $v_{\rm comp}(\cdot, k = 1.5)$; (F) The imaginary part of the function $v_{\rm comp}(\cdot, k = 1.5)$
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