| Title:
|
Reconstruction algorithms for an inverse medium problem (English) |
| Author:
|
Liu, Ji-Chuan |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
63 |
| Issue:
|
2 |
| Year:
|
2018 |
| Pages:
|
195-216 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
inverse medium problem |
| Keyword:
|
Levenberg-Marquardt algorithm |
| Keyword:
|
trust-region-reflective algorithm |
| Keyword:
|
ill-posed problem |
| MSC:
|
65N20 |
| MSC:
|
65N21 |
| MSC:
|
92C55 |
| idZBL:
|
Zbl 06890305 |
| idMR:
|
MR3795246 |
| DOI:
|
10.21136/AM.2018.0114-17 |
| . |
| Date available:
|
2018-05-09T08:55:13Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147189 |
| . |
| Reference:
|
[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. Zbl 1156.35101, MR 2424952, 10.1137/070686408 |
| Reference:
|
[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. Zbl 1302.65284, MR 3223378, 10.1090/S0002-9939-2014-12090-9 |
| Reference:
|
[3] Ammari, H., Bossy, E., Garnier, J., Seppecher, L.: Acousto-electromagnetic tomography.SIAM J. Appl. Math. 72 (2012), 1592-1617. Zbl 1268.78015, MR 3022278, 10.1137/120863654 |
| Reference:
|
[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. Zbl 1235.31006, MR 2873260, 10.1137/110828241 |
| Reference:
|
[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. Zbl 1187.92058, MR 2511278, 10.1017/S0956792509007888 |
| Reference:
|
[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. Zbl 06256850, MR 3169761, 10.1080/03605302.2013.803483 |
| Reference:
|
[7] Bal, G., Schotland, J. C.: Inverse scattering and acousto-optic imaging.Phys. Rev. Lett. 104 (2010), Article ID 043902. 10.1103/physrevlett.104.043902 |
| Reference:
|
[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. Zbl 1273.35308, MR 3084700, 10.1002/cpa.21453 |
| Reference:
|
[9] Bao, G., Triki, T.: Error estimates for the recursive linearization of inverse medium problems.J. Comput. Math. 28 (2010), 725-744. Zbl 1240.35574, MR 2765913, 10.4208/jcm.1003-m0004 |
| Reference:
|
[10] Choulli, M., Triki, F.: New stability estimates for the inverse medium problem with internal data.SIAM J. Math. Anal. 47 (2015), 1778-1799. Zbl 1335.35294, MR 3345935, 10.1137/140988577 |
| Reference:
|
[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. Zbl 0842.90106, MR 1305566, 10.1007/BF01582221 |
| Reference:
|
[12] Coleman, T., Li, Y.: An interior, trust region approach for nonlinear minimization subject to bounds.SIAM J. Optim. 6 (1996), 418-445. Zbl 0855.65063, MR 1387333, 10.1137/0806023 |
| Reference:
|
[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). Zbl 0522.35001, MR 0700400 |
| Reference:
|
[14] Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory.Applied Mathematical Sciences 93, Springer, Berlin (1992). Zbl 0760.35053, MR 1183732, 10.1007/978-3-662-02835-3 |
| Reference:
|
[15] Hanke, M., Rundell, W.: On rational approximation methods for inverse source problems.Inverse Probl. Imaging 5 (2011), 185-202. Zbl 1215.35166, MR 2773431, 10.3934/ipi.2011.5.185 |
| Reference:
|
[16] Isakov, V.: Inverse Problems for Partial Differential Equations.Applied Mathematical Sciences 127, Springer, New York (2006). Zbl 1092.35001, MR 2193218, 10.1007/0-387-32183-7 |
| Reference:
|
[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. Zbl 1241.78025, MR 2876854, 10.1088/0266-5611/28/2/025003 |
| Reference:
|
[18] Kress, R.: Linear Integral Equations.Applied Mathematical Sciences 82, Springer, New York (1999). Zbl 0920.45001, MR 1723850, 10.1007/978-1-4612-0559-3 |
| Reference:
|
[19] Levenberg, K.: A method for the solution of certain non-linear problems in least squares.Q. Appl. Math. 2 (1944), 164-168. Zbl 0063.03501, MR 0010666, 10.1090/qam/10666 |
| Reference:
|
[20] Marquardt, D. W.: An algorithm for least-squares estimation of nonlinear parameters.J. Soc. Ind. Appl. Math. 11 (1963), 431-441. Zbl 0112.10505, MR 0153071, 10.1137/0111030 |
| Reference:
|
[21] McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations.Cambridge University Press, Cambridge (2000). Zbl 0948.35001, MR 1742312 |
| Reference:
|
[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. Zbl 0372.65022, MR 0483445, 10.1007/bfb0067700 |
| Reference:
|
[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. Zbl 1345.92090, MR 3024668, 10.1007/978-3-642-22990-9_1 |
| Reference:
|
[24] Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem.Ann. Math. (2) 125 (1987), 153-169. Zbl 0625.35078, MR 0873380, 10.2307/1971291 |
| Reference:
|
[25] Triki, F.: Uniqueness and stability for the inverse medium problem with internal data.Inverse Probl. 26 (2010), Article ID 095014, 11 pages. Zbl 1200.35333, MR 2679551, 10.1088/0266-5611/26/9/095014 |
| Reference:
|
[26] Widlak, T., Scherzer, O.: Stability in the linearized problem of quantitative elastography.Inverse Probl. 31 (2015), Article ID 035005, 27 pages. Zbl 1309.92050, MR 3319371, 10.1088/0266-5611/31/3/035005 |
| . |
