zbMATH — the first resource for mathematics

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.

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
PDF BibTeX Cite
Full Text: DOI