A regularized dynamical system method for nonlinear ill-posed Hammerstein type operator equations. (English) Zbl 1253.65082
The authors consider nonlinear ill-posed Hammerstein type operator equations \(KF(x)=y\), where \(F:D(F)\subset X \to Z\) is nonlinear and \(K: Z \to Y\) is a bounded linear operator, and \(X\), \(Y\), \(Z\) are Hilbert spaces. Instead of \(y \in Y\) an approximation \(y^{\delta}\) is available, \(\| y^{\delta}-y\| \leq \delta\). The aim of the paper is to analyze the following solution method for the original equation: \(z_{\alpha}^{\delta}=(K^*K+\alpha I)^{-1} K^* y^{\delta}\), \(x^{\prime}(t)=-(F^{\prime}(x_0)+\varepsilon(t)I)^{-1} (F(x)-z_{\alpha}^{\delta})\), \(x(0)=x_0\); \(\alpha=\alpha(\delta)>0\), \(0\leq \varepsilon(t) \leq K\).
Reviewer: Mikhail Yu. Kokurin (Yoshkar-Ola)
MSC:
65J15 | Numerical solutions to equations with nonlinear operators |
65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
47J06 | Nonlinear ill-posed problems |
47J25 | Iterative procedures involving nonlinear operators |
47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |