Comparative analysis of inverse coefficient problems for parabolic equations. Part III: Conjugate gradient method and coarse-fine grid algorithm

This article presents a computational analysis of the Conjugate Gradient Method (CGM), and a comparative analysis of the method (CGM) and coarse-fine grid algorithm (CFGA) for parabolic inverse coefficient problems (ICPs) based on boundary measured data. The adjoint problem approach is applied to obtain formal gradients of each ICPs as the L-2-scalar product of the derivatives u(x)(x, t; k) and phi(x)(x, t; k) of the corresponding direct and adjoint problems. Then the CGM is applied to the least-squares formulation of the inverse coefficient problems. Detailed numerical study of the method for each ICP is presented for various types of type input data concentrated at the boundary. Comparative computational analysis of the CGM and CFGA shows the limits of applications and effectiveness of each these methods.