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Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems. (English) Zbl 1080.65045
The paper is devoted to the problem of solving nonlinear equations $$F(x)=y$$, where $$F: X\rightarrow Y$$, $$X$$ and $$Y$$ are Hilbert spaces, $$F$$ is a Fréchet differentiable operator and its range is not closed. Instead of the exact data $$y$$ a noisy ones $$y_\delta$$ are given with an estimation $$\| y_\delta - y\| \leq\delta.$$ The initial ill-posed problem is replaced by the minimization of the functional $$J(x)=\| F(x)-y_\delta\| ^2/2$$. The numerical scheme is based on the following variant of the trust region method. On the iterations $$x_{k+1}=x_k+\xi_k$$ the functional $$J(x)$$ is approximated by the quadratic one $$\psi_k(\xi)=(\nabla J(x_k),\xi)+(F'(x_k)^*F'(x_k)\xi,\xi)$$. The step $$\xi_k$$ is defined as a minimum of $$\psi_k(\xi)$$ at the constraint $$\| \xi\| \leq\Delta_k.$$ An algorithm of recalculating the parameter $$\Delta_k$$ that yields convergence of the scheme for exact data is suggested. For noisy data a regularizing stopping rule is also suggested. The proposed regularization method is demonstrated on the example of inverse gravimetry.

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 65K10 Numerical optimization and variational techniques 65R30 Numerical methods for ill-posed problems for integral equations 47J06 Nonlinear ill-posed problems 45G10 Other nonlinear integral equations
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