Improved Primal–Dual Interior-Point Method Using the Lawson-Norm for Inverse Problems

Wenjing Shang,Yidong Xu,Wei Xue,Yingsong Li

doi:10.1109/access.2020.2976727

Abstract

Electrical impedance tomography (EIT), geophysics and undersea target reconstruction are typical non-linear ill-posed inverse problems, and in many cases, the anomalous bodies feature with a clear boundary. Thus, it is suitable to obtain sharp boundaries and blocky features with the Total Variation (TV) functional regularization. However, the TV function regularization leads to a non-differentiable objective function at zero in the inverse formulation and reduces the algorithm robustness. In this paper, we propose an improved primal-dual interior-point method (PD-IPM) based on the Lawson norm to get sharp spatial profiles of the anomalous bodies. Furthermore, the impact of the smooth parameter is investigated to get the inverse results. Numerical experiment using simulated data is setup to support our claim.

Highlights

Electrical impedance tomography, geophysics, and undersea target reconstruction are recovering the conductivity distribution from quasi-static current and voltage measurements [1]–[3]
Canonical smooth optimization algorithms, such as the steepest descent or Newton method, can be applied with an approximation to solve an inverse problem based on the l1-norm regularization techniques [7]. (2) The l1-norm regularization can be approached by using the iteratively re-weighted least squares (IRLS) algorithm [25]
We find that the differences between the Ekblom-norm based traditional primal-dual interior-point method (PD-IPM) and the proposed Lawson-norm PD-IPM are the expressions of the vector s, the matrix E and S

Summary

INTRODUCTION

Electrical impedance tomography, geophysics, and undersea target reconstruction are recovering the conductivity distribution from quasi-static current and voltage measurements [1]–[3]. L2-norm regularization technique is unsuitable for recovering anomalous bodies which have clear boundaries between the entity and the background To overcome such drawbacks, the l1-norm (e.g. the total variation (TV) regularization [13]) acting as the regularization term has been applied in many reconstructions as it reduces the redundancy information and yields a sparse parameter representation which leads to better edge sharpness and fewer image artifacts [10], [14]–[16]. The l1-norm or TV regularization shows more advantages in recovering blocky anomalous body in EIT, geophysical research and underwater applications. Canonical smooth optimization algorithms, such as the steepest descent or Newton method, can be applied with an approximation to solve an inverse problem based on the l1-norm regularization techniques [7]. The more detailed description of quasi-static electric field survey, electrode configuration and applications can be found in [10]

FORWARD MODEL

CONCLUSION