  Multi-source localization based on received signal strength (RSS) has drawn
great interest in wireless sensor networks. However, the shadow fading term
caused by obstacles cannot be separated from the received signal, which leads
to severe error in location estimate. In this paper, we approximate the
log-normal sum distribution through Fenton-Wilkinson method to formulate a
non-convex maximum likelihood (ML) estimator with unknown shadow fading factor.
In order to overcome the difficulty in solving the non-convex problem, we
propose a novel algorithm to estimate the locations of sources. Specifically,
the region is divided into $N$ grids firstly, and the multi-source localization
is converted into a sparse recovery problem so that we can obtain the sparse
solution. Then we utilize the K-means clustering method to obtain the rough
locations of the off-grid sources as the initial feasible point of the ML
estimator. Finally, an iterative refinement of the estimated locations is
proposed by dynamic updating of the localization dictionary. The proposed
algorithm can efficiently approach a superior local optimal solution of the ML
estimator. It is shown from the simulation results that the proposed method has
a promising localization performance and improves the robustness for
multi-source localization in unknown shadow fading environments. Moreover, the
proposed method provides a better computational complexity from $O(K^3N^3)$ to
$O(N^3)$.
