# The Levenberg-Marquardt Method for Acousto-Electric Tomography on Different Conductivity Contrast
## 2 The Reconstruction Algorithm Based on Levenberg-Marquardt Method
### The Reconstruction Algorithm
The instability caused by the conductivity contrast can be improved with a proper Tikhonov regularization parameter. A simple dynamic strategy is used in this paper for choosing the proper value of Tikhonov regularization parameter \(\alpha_{k}\), which requires the ground truth of the conductivity distribution. Although it works quite well in all the numerical results presented in the paper, it is not practical, as the ground truth is generally unknown in most cases. However, one can use the Morozov's discrepancy principle for choosing this proper value, which should work better in theory. Conceptualization, C.L. and K.Z. ; methodology, C.L. ; software, C.L. and K.A. ; formal analysis, C.L. and K.A. investigation, C.L. and K.A. ; writing--original draft preparation, C.L. and K.A. ; writing--review and editing, K.Z. ; All authors have read and agreed to the published version of the manuscript. This research was funded by National Nature Science Foundation of China, grant number 61801388 and 61971351. The authors declare no conflicts of interest. ## References

* _[PERSON] et al. (2019)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON]; et al. A review of electrical impedance tomography in lung applications: Theory and algorithms for absolute images. _Annu. Rev. Control._**2019**, _48_, 442-471. [CrossRef] [PubMed]
* _[PERSON] et al. (2008)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Electrical impedance tomography by elastic deformation. _SIAM J. Appl. Math._**2008**, _68_, 1557-1573. [CrossRef]
* _[PERSON] (2018)_ [PERSON] [PERSON]; [PERSON] [PERSON] Review on biomedical techniques for imaging electrical impedance. _IRBM_**2018**, _39_, 243-250. [CrossRef]
* _[PERSON] (2018)_ [PERSON]; [PERSON] A new optimization approach to sparse reconstruction of log-conductivity in acousto-electric tomography. _SIAM J. Imaging Sci._**2018**, _11_, 1759-1784. [CrossRef]
* _[PERSON] et al. (2009)_ [PERSON] [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] Imaging by modification: Numerical reconstruction of local conductivities from corresponding power density measurements. _SIAM J. Imaging Sci._**2009**, \(2\), 1003-1030. [CrossRef]
* _[PERSON] et al. (2018)_ [PERSON]; [PERSON]; [PERSON]; [PERSON], [PERSON] Image reconstruction in electrical impedance tomography based on structure-aware sparse Bayesian learning. _IEEE Trans. Med. Imaging_**2018**, _37_, 2090-2102. [CrossRef]
* _[PERSON] et al. (2019)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Accelerated structure-aware sparse Bayesian learning for three-dimensional electrical impedance tomography. _IEEE Trans. Ind. Inform._**2019**, _15_, 5033-5041. [CrossRef]
* _[PERSON] et al. (2019)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] A B-Scan imaging method of conductivity variation detection for magneto-acousto-electrical tomography. _IEEE Access_**2019**, \(7\), 26881-26891. [CrossRef]
* _[PERSON] et al. (2020)_ [PERSON]; [PERSON]; [PERSON] Dual modality EIT-UTT for water dominate three-phase material imaging. _IEEE Access_**2020**, \(8\), 14523-14530. [CrossRef]
* _[PERSON] et al. (2015)_ [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography. _J. Math. Pures Appl._**2015**, _103_, 1390-1409. [CrossRef]
* _[PERSON] et al. (2017)_ [PERSON]; [PERSON] [PERSON] Smoothened complete electrode model. _SIAM J. Appl. Math._**2017**, _77_, 2250-2271. [CrossRef]
* _[PERSON] et al. (2018)_ [PERSON]; [PERSON]; [PERSON]; [PERSON] Enhanced imaging of piezoresistive nanocomposites through the incorporation of nonlocal conductivity changes in electrical impedance tomography. _J. Intell. Mater. Syst. Struct._**2018**, _29_, 1850-1861. [CrossRef]* _[PERSON] et al. (2015)_ [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] An application of impediography to the high sensitivity and high resolution identification of structural damage. _Smart Mater. Struct._**2015**, _24_, 065044.