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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
Software:
GQTPAR
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