Discrepancy principle for the dynamical systems method (Q1883515)

For the linear ill--posed equation \(Au=f\) with a bounded operator \(A \in L(H)\) and approximately given \(f\) in a Hilbert space \(H\), the author proposes and studies the solution method \({\dot u}=-u+(A^* A+ \epsilon(t))^{-1} A^* f_{\delta}\), \(u(0)=u_0 \in H\). Here \(u=u(t), t \geq 0\), \(\| f_{\delta}-f \| \leq \delta\), \(\| f_{\delta}\| >\delta\), \(\epsilon(t)>0, \lim_{t \to \infty} \epsilon(t)=0\); a minimal-norm solution \(y\) to the original equation is approximated by the element \(u(t_{\delta})\), where the stopping point \(t=t_{\delta}\) is determined by the discrepancy principle: \(\| A(A^* A+\epsilon(t))^{-1} A^* f_{\delta} -f_{\delta}\|=\delta\). The author establishes that \(\lim_{\delta \to 0} \| u(t_{\delta})-y \|=0\), \(\lim_{\delta \to 0} t_{\delta}=\infty\). The result is generalized to the case where \(A\) is a nonlinear, monotone, continuous operator on \(H\).