Global convergence for ill-posed equations with monotone operators: the   dynamical systems method
  Consider an operator equation $F(u)=0$ in a real Hilbert space.   Let us call this equation ill-posed if the operator   $F'(u)$ is not boundedly invertible, and well-posed otherwise.   If $F$ is monotone $C^2_{loc}(H)$ operator, then we construct a Cauchy problem, which has the following properties:   1) it has a global solution for an arbitrary initial data,   2) this solution tends to a limit as time tends to infinity,   3) the limit is the minimum norm solution to the equation $F(u)=0$.   Example of applications to linear ill-posed operator equation is given. 
A.G.Ramm