Multilevel Linear Sampling Method for Inverse Scattering Problems

Abstract

A novel multilevel algorithm is presented for implementing the widely used linear sampling method in inverse obstacle scattering problems. The new method is shown to possess asymptotically optimal computational complexity. For an $n\times n$ sampling mesh in $\mathbb{R}^2$ or an $n\times n\times n$ sampling mesh in $\mathbb{R}^3$, the proposed algorithm requires one to solve only $\mathcal{O}(n^{N-1})$ far-field equations for a $\mathbb{R}^N$ problem (N=2,3), and this is in sharp contrast to the original version of the method which needs to solve $n^N$ far-field equations. Numerical experiments are presented to illustrate the promising feature of the algorithm in significantly reducing the computational cost of the linear sampling method.