R-K type Landweber method for nonlinear ill-posed problems
The paper concerns a regularization method for solving a nonlinear ill-posed problem \(F(x)=y\) in a Hilbert space. The Runge-Kutta (R-K) method of order 2 is applied to the continuous analog of the Landweber method: \(\dot{x}^\delta(t)=F'(x^\delta(t))^*[y^\delta-F(x^\delta(t))]\), \(0<t\leq T\), \(x^\delta(0)=x_0\). Here \(y^\delta\) are available noisy data with \(\| y-y^\delta\| \leq\delta\). This method, called by the authors the Runge-Kutta type Landweber method, gives a sequence of regularized solutions \(\{x^\delta_k\}\), where \(k\) plays the role of regularization parameter. Applying a generalized discrepancy principle as the stopping rule (\(k\leq k_*(\delta)\)) the authors prove the convergence when \(\delta\rightarrow 0\). Moreover, a convergence rate is obtained under a source condition and usual additional assumptions on \(F\). Finally, numerical results of the considered method applied to the convolution equation are compared with the corresponding results for the Landweber method.
scientific article