Novel Methods for Solving Severely Ill-posed Linear Equations System

We treat an ill-posed system of linear equations by transforming it into a linear system of stiff ordinary differential equations (SODEs), adding a differential term on the left-hand side. In order to overcome the difficulty of numerical instability when integrating the SODEs, Liu [20] has combined nonstandard finite difference method and group-preserving scheme, namely the nonstandard group-preserving scheme (NGPS), to obtain an unconditional stable numerical method for SODEs. This paper applies the NGPS to the SODEs resulting from the ill-posed linear equations, and proves that the new algorithms are unconditional stable. To strengthen accuracy, an L-curve is used to select a suitable regularization parameter. Moreover, we also combine the NGPS with a newly developed fictitious time integration method (FTIM) from Liu and Atluri [29] to solve the ill-posed linear equations. Several numerical examples are examined and compared with exact solutions, revealing that the new algorithms have better computational efficiency and accuracy even for the highly ill-conditioned linear equations with a large disturbance on the given data.