Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems.

*(English)*Zbl 1080.65045The 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 FrĂ©chet 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.

Reviewer: Vladimir Gorbunov (Ul’yanovsk)

##### 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 |