Simplified Levenberg-Marquardt method in Hilbert spaces (Q6157123)

Let \( F : D(F) \subset X \to Y \) be a nonlinear operator between Hilbert spaces \( X \) and \( Y \). For the stable approximate solution of an ill-posed equation \( F(x) = y \), in this paper a frozen Levenberg-Marquardt method of the form \[ x_{n+1}^\delta = x_{n}^\delta - (\alpha_n I + F^\prime(x_0)^* F^\prime(x_0))^{-1} F^\prime(x_0)^* (F(x_n^\delta) - y^\delta) \] is considered. Here, \( \{ \alpha_n\} \) denotes a decreasing sequence of positive parameters converging geometrically to 0, and \( y^\delta \in Y \) are perturbed data satisfying \( \Vert y^\delta - y \Vert \le \delta \). In addition, \( F^\prime(x_0) \) denotes the Fréchet derivative of \( F \) at some initial guess \(x_0 = x_0^\delta \), where \(x_0\) is sufficiently close to an exact solution \( x^\dagger \in X \) of \( F(x) = y \). Let \( n_\delta \) be chosen such that \( \Vert F(x_{n_\delta}^\delta) - y^\delta \Vert \le \tau \delta < \Vert F(x_n^\delta) - y^\delta \Vert \) holds for \( 0 \le n < n_\delta \), where \( \tau > 1 \) is fixed. It is shown that, under a generalized tangential cone condition and other technical assumptions, this discrepancy principle serves as regularization scheme, i.e., we have \( \Vert x_{n_\delta}^\delta - x^\dagger \Vert \to 0 \) as \( \delta \to 0 \). In the concluding section, results of some numerical experiments are presented.