On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems
For researchers in inverse problems, this provides theoretical justification for using the discrepancy principle with Newton-type methods, extending known results to weaker smoothness conditions.
The paper proves convergence and optimal convergence rates for Newton-type methods solving nonlinear ill-posed inverse problems when the iteration is stopped by the discrepancy principle, even under only Lipschitz continuity of the Fréchet derivative if the initial error is smooth.
We consider the computation of stable approximations to the exact solution $x^†$ of nonlinear ill-posed inverse problems $F(x)=y$ with nonlinear operators $F:X\to Y$ between two Hilbert spaces $X$ and $Y$ by the Newton type methods $$ x_{k+1}^δ=x_0-g_{α_k} (F'(x_k^δ)^*F'(x_k^δ)) F'(x_k^δ)^* (F(x_k^δ)-y^δ-F'(x_k^δ)(x_k^δ-x_0)) $$ in the case that only available data is a noise $y^δ$ of $y$ satisfying $\|y^δ-y\|\le δ$ with a given small noise level $δ>0$. We terminate the iteration by the discrepancy principle in which the stopping index $k_δ$ is determined as the first integer such that $$ \|F(x_{k_δ}^δ)-y^δ\|\le τδ<\|F(x_k^δ)-y^δ\|, \qquad 0\le k<k_δ$$ with a given number $τ>1$. Under certain conditions on $\{α_k\}$, $\{g_α\}$ and $F$, we prove that $x_{k_δ}^δ$ converges to $x^†$ as $δ\to 0$ and establish various order optimal convergence rate results. It is remarkable that we even can show the order optimality under merely the Lipschitz condition on the Fréchet derivative $F'$ of $F$ if $x_0-x^†$ is smooth enough.