DSM for solving ill-conditioned linear algebraic systems

A standard way to solve linear algebraic systems $Au=f,\,\,(*)$ with ill-conditioned matrices $A$ is to use variational regularization. This leads to solving the equation $(A^*A+aI)u=A^*f_\d$, where $a$ is a regularization parameter, and $f_\d$ are noisy data, $||f-f_\d||\leq \d$. Numerically it requires to calculate products of matrices $A^*A$ and inversion of the matrix $A^*A+aI$ which is also ill-conditioned if $a>0$ is small. We propose a new method for solving (*) stably, given noisy data $f_\d$. This method, the DSM (Dynamical Systems Method) is developed in this paper for selfadjoint $A$. It consists in solving a Cauchy problem for systems of ordinary differential equations.