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
References:

show all references

References:
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 $v_{\rm comp}(\cdot, k = 1.5)$
 [1] Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367 [2] Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006 [3] Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108 [4] Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021002 [5] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [6] Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004 [7] Shahede Omidi, Jafar Fathali. Inverse single facility location problem on a tree with balancing on the distance of server to clients. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021017 [8] Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005 [9] Weihong Guo, Yifei Lou, Jing Qin, Ming Yan. IPI special issue on "mathematical/statistical approaches in data science" in the Inverse Problem and Imaging. Inverse Problems & Imaging, 2021, 15 (1) : I-I. doi: 10.3934/ipi.2021007 [10] Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046 [11] Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250 [12] Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020389 [13] George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003 [14] Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1 [15] Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021006 [16] Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292 [17] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [18] Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107 [19] Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386 [20] Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

2019 Impact Factor: 1.373