On the computational complexity of the conjugate-gradient method for solving inverse scattering problems

This paper presents an efficient implementation of the inversion algorithm based on the conjugate-gradient method (CGM) for solving inverse scattering problems. The original CGM provides good image reconstruction and robustness to noise-corrupted data. This method requires the solution of the forward scattering problem and the Fréchet derivative operator of the cost function at each iteration step. However, these procedures can make the computational cost prohibitive, even for moderately sized problems. To avoid the computational burden, a two-step conjugate gradient fast Fourier transform (CG-FFT) procedure is proposed. Such an approach reduces the computational cost and storage requirements of the CGM implementation. The efficient CGM is found to share a computational complexity similar to the distorted-Born iterative method (DBIM). Thus the convergence speed and accuracy of the CGM is compared with the DBIM. Numerical tests using both synthetic and experimental data show effectiveness for solving 2D inverse scattering problems.