Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
inverse medium problem; Levenberg-Marquardt algorithm; trust-region-reflective algorithm; ill-posed problem
Summary:
In this paper, we consider a two-dimensional inverse medium problem from noisy observation data. We propose effective reconstruction algorithms to detect the number, the location and the size of the piecewise constant medium within a body, and then we try to recover the unknown shape of inhomogeneous media. This problem is nonlinear and ill-posed, thus we should consider stable and elegant approaches in order to improve the corresponding approximation. We give several examples to show the viability of our proposed algorithms.
References:
[1] Ammari, H., Bonnetier, E., Capdeboscq, Y., Tanter, M., Fink, M.: Electrical impedance tomography by elastic deformation. SIAM J. Appl. Math. 68 (2008), 1557-1573. DOI 10.1137/070686408 | MR 2424952 | Zbl 1156.35101
[2] Ammari, H., Bossy, E., Garnier, J., Nguyen, L. H., Seppecher, L.: A reconstruction algorithm for ultrasound-modulated diffuse optical tomography. Proc. Am. Math. Soc. 142 (2014), 3221-3236. DOI 10.1090/S0002-9939-2014-12090-9 | MR 3223378 | Zbl 1302.65284
[3] Ammari, H., Bossy, E., Garnier, J., Seppecher, L.: Acousto-electromagnetic tomography. SIAM J. Appl. Math. 72 (2012), 1592-1617. DOI 10.1137/120863654 | MR 3022278 | Zbl 1268.78015
[4] Ammari, H., Capdeboscq, Y., Gournay, F. de, Rozanova-Pierrat, A., Triki, F.: Microwave imaging by elastic deformation. SIAM J. Appl. Math. 71 (2011), 2112-2130. DOI 10.1137/110828241 | MR 2873260 | Zbl 1235.31006
[5] Ammari, H., Capdeboscq, Y., Kang, H., Kozhemyak, A.: Mathematical models and reconstruction methods in magneto-acoustic imaging. Eur. J. Appl. Math. 20 (2009), 303-317. DOI 10.1017/S0956792509007888 | MR 2511278 | Zbl 1187.92058
[6] Ammari, H., Garnier, J., Nguyen, L. H., Seppecher, L.: Reconstruction of a piecewise smooth absorption coefficient by an acousto-optic process. Commun. Partial Differ. Equations 38 (2013), 1737-1762. DOI 10.1080/03605302.2013.803483 | MR 3169761 | Zbl 06256850
[7] Bal, G., Schotland, J. C.: Inverse scattering and acousto-optic imaging. Phys. Rev. Lett. 104 (2010), Article ID 043902. DOI 10.1103/physrevlett.104.043902
[8] Bal, G., Uhlmann, G.: Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions. Commun. Pure Appl. Math. 66 (2013), 1629-1652. DOI 10.1002/cpa.21453 | MR 3084700 | Zbl 1273.35308
[9] Bao, G., Triki, T.: Error estimates for the recursive linearization of inverse medium problems. J. Comput. Math. 28 (2010), 725-744. DOI 10.4208/jcm.1003-m0004 | MR 2765913 | Zbl 1240.35574
[10] Choulli, M., Triki, F.: New stability estimates for the inverse medium problem with internal data. SIAM J. Math. Anal. 47 (2015), 1778-1799. DOI 10.1137/140988577 | MR 3345935 | Zbl 1335.35294
[11] Coleman, T. F., Li, Y.: On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds. Math. Program. 67 (1994), 189-224. DOI 10.1007/BF01582221 | MR 1305566 | Zbl 0842.90106
[12] Coleman, T., Li, Y.: An interior, trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6 (1996), 418-445. DOI 10.1137/0806023 | MR 1387333 | Zbl 0855.65063
[13] Colton, D., Kress, R.: Integral Equation Methods in Scattering Theory. Pure and Applied Mathematics. A Wiley-Interscience Publication. John Wiley & Sons, New York (1983). MR 0700400 | Zbl 0522.35001
[14] Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences 93, Springer, Berlin (1992). DOI 10.1007/978-3-662-02835-3 | MR 1183732 | Zbl 0760.35053
[15] Hanke, M., Rundell, W.: On rational approximation methods for inverse source problems. Inverse Probl. Imaging 5 (2011), 185-202. DOI 10.3934/ipi.2011.5.185 | MR 2773431 | Zbl 1215.35166
[16] Isakov, V.: Inverse Problems for Partial Differential Equations. Applied Mathematical Sciences 127, Springer, New York (2006). DOI 10.1007/0-387-32183-7 | MR 2193218 | Zbl 1092.35001
[17] Ito, K., Jin, B., Zou, J.: A direct sampling method to an inverse medium scattering problem. Inverse Probl. 28 (2012), Article ID 025003, 11 pages. DOI 10.1088/0266-5611/28/2/025003 | MR 2876854 | Zbl 1241.78025
[18] Kress, R.: Linear Integral Equations. Applied Mathematical Sciences 82, Springer, New York (1999). DOI 10.1007/978-1-4612-0559-3 | MR 1723850 | Zbl 0920.45001
[19] Levenberg, K.: A method for the solution of certain non-linear problems in least squares. Q. Appl. Math. 2 (1944), 164-168. DOI 10.1090/qam/10666 | MR 0010666 | Zbl 0063.03501
[20] Marquardt, D. W.: An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11 (1963), 431-441. DOI 10.1137/0111030 | MR 0153071 | Zbl 0112.10505
[21] McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000). MR 1742312 | Zbl 0948.35001
[22] Moré, J. J.: The Levenberg-Marquardt algorithm: implementation and theory. Numerical Analysis G. A. Watson et al. Lecture Notes in Mathematics 630, Springer, Berlin (1978), 105-116. DOI 10.1007/bfb0067700 | MR 0483445 | Zbl 0372.65022
[23] Schotland, J. C.: Direct reconstruction methods in optical tomography. Mathematical Modeling in Biomedical Imaging II H. Ammari et al. Lecture Notes in Mathematics 2035, Springer, Berlin (2012), 1-29. DOI 10.1007/978-3-642-22990-9_1 | MR 3024668 | Zbl 1345.92090
[24] Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. (2) 125 (1987), 153-169. DOI 10.2307/1971291 | MR 0873380 | Zbl 0625.35078
[25] Triki, F.: Uniqueness and stability for the inverse medium problem with internal data. Inverse Probl. 26 (2010), Article ID 095014, 11 pages. DOI 10.1088/0266-5611/26/9/095014 | MR 2679551 | Zbl 1200.35333
[26] Widlak, T., Scherzer, O.: Stability in the linearized problem of quantitative elastography. Inverse Probl. 31 (2015), Article ID 035005, 27 pages. DOI 10.1088/0266-5611/31/3/035005 | MR 3319371 | Zbl 1309.92050
Partner of
EuDML logo