| Regularized Fixed-Point Iterations for Nonlinear Inverse Problems (2005) | |||||||||||||
Abstract | |||||||||||||
| In this paper we introduce a derivative-free, iterative method for solving nonlinear ill-posed problems $Fx=y$, where instead of $y$ noisy data $y_delta$ with $| y-y_delta |leqdelta$ are given and $F:, D(F)subseteq Xrightarrow Y$ is a nonlinear operator between Hilbert spaces $X$ and $Y$. This method is defined by splitting the operator $F$ into a linear part $A$ and a nonlinear part $G$, such that $F=A+G$. Then iterations are organized as $A u_{k+1}=y_delta-Gu_k$. In the context of ill-posed problems we consider the situation when $A$ does not have a bounded inverse, thus each iteration needs to be regularized. Under some conditions on the operators $A$ and $G$ we study the behavior of the iteration error. We obtain its stability with respect to the iteration number $k$ as well as the optimal convergence rate with respect to the noise level $delta$, provided that the solution satisfies a generalized source condition. As an example, we consider an inverse problem of initial temperature reconstruction for a nonlinear heat equation, where the nonlinearity appears due to radiation effects. The obtained iteration error in the numerical results has the theoretically expected behavior. The theoretical assumptions are illustrated by a computational experiment. | |||||||||||||
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