Iterative Runge-Kutta-Type Methods with Convex Penalty for Inverse Problems in Hilbert Spaces (CSIAM Transactions on Applied Mathematics, 2023)
An s-stage Runge-Kutta-type iterative method with the convex penalty forsolving nonlinear ill-posed problems is proposed and analyzed in this paper. Theapproach is developed by using a family of Runge-Kutta-type methods to solve theasymptotical regularization method, which can be seen as an ODE with the initialvalue. The convergence and regularity of the proposed method are obtained undercertain conditions. The reconstruction results of the proposed method for some special cases are studied through numerical experiments on both parameter identificationin inverse potential problem and diffuse optical tomography. The numerical resultsindicate that the developed methods yield stable approximations to true solutions, especially the implicit schemes have obvious advantages on allowing a wider range ofstep length, reducing the iterative numbers, and saving computation time.