Iterative Runge-Kutta-Type Methods with Convex Penalty for Inverse Problems in Hilbert Spaces

An s-stage Runge-Kutta-type iterative method with the convex penalty for solving nonlinear ill-posed problems is proposed and analyzed in this paper. The approach is developed by using a family of Runge-Kutta-type methods to solve the asymptotical regularization method, which can be seen as an ODE with the initial value. The convergence and regularity of the proposed method are obtained under certain conditions. The reconstruction results of the proposed method for some special cases are studied through numerical experiments on both parameter identification in inverse potential problem and diffuse optical tomography. The numerical results indicate that the developed methods yield stable approximations to true solutions, especially the implicit schemes have obvious advantages on allowing a wider range of step length, reducing the iterative numbers, and saving computation time.