Please summarize the words As illustrated in 30 when solving $R$ with least square methods after updating the induced current 7 contains localeffect term $leftbarbarbeta cdot barI_j+barE_j^text inc ri

The article discusses electromagnetic inverse scattering problems (ISPs) and the challenges of accurately estimating contrast in the presence of strong scatterers. It is well-known that the nonlinearity of electromagnetic ISPs is mainly due to multiple scattering effects, which can be represented by a global-effect term $\left(\bar{\bar{A}}\left(\bar{I}_j\right)\right)$ in the least square methods used to solve for the unknown contrast $R$ after updating the induced current.

For weak or not-too-strong scatterers, the contribution from the global-effect term is small compared with the incident field, and the estimation of contrast is accurate enough to lead the inversion to the global solution. However, for strong scatterers, the global-effect term becomes a significant portion of the problem and can influence the entire domain through the global Green's operator $\bar{\bar{A}}$. Since there is no information about the induced current $\bar{I}_j$ at the beginning of the inversion, the estimation of contrast becomes less accurate and farther from the global minimum.

To address this issue, the article proposes a strategy of gradually reducing the value of the regularized parameter $\beta$ in the nested inversion, which effectively suppresses the global effect caused by the $\bar{\bar{A}}\left(\bar{I}_j\right)$ term. By choosing large values of $\beta$ at the beginning of the inversion, the modified local-effect term dominates over the global-effect term, leading to accurate estimation of contrast. As the inversion progresses, the value of $\beta$ is gradually reduced, allowing the global effect to be gradually introduced and leading to a more accurate estimation of contrast.

This approach, known as the Conjugate-Gradient Iterative Eshelby (CIE-I) method, effectively alleviates the global multiple scattering contributions in estimating contrast in CSI-type methods. The article refers the reader to reference [30] for more details on the approach