A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems

Pornsarp Pornsawad,Christine Böckmann,Nantawan Sapsakul

doi:10.3390/math7050419

Abstract

In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥ F ( x δ ( T ) ) − y δ ∥ = τ δ + for some δ + > δ , and an appropriate source condition. We yield the optimal rate of convergence.

Highlights

Let X and Y be infinite-dimensional real Hilbert space with inner products h·, ·i and norms k · k.Let us consider a nonlinear operator equation: F ( x ) = y, (1)where F : D ( F ) ⊂ X → Y is a nonlinear operator between the Hilbert space X and Y
Due to the minimal assumptions for the convergence analysis of the modified iterative Runge–Kutta-type methods (RKTM), we studied in detail the additional term in the continuous version written as: ẋ δ (t) = F 0 ( x δ (t))∗ [yδ − F ( x δ (t))] − ( x δ (t) − x ), 0 < t ≤ T, x δ (0) = x, (7)
At the end of this section, we show that the proposed method provides a stable approximation x δ ( T ∗ ) of F ( x δ ) = yδ if a unique solution T ∗ is chosen by the discrepancy principle (16)

Summary

Introduction

Let X and Y be infinite-dimensional real Hilbert space with inner products h·, ·i and norms k · k. Detailed studies of inverse ill-posed problems may be found, e.g., in [3] and [4,5,6,7] It is well-known that the asymptotic regularization is a continuous version of the Landweber iteration. Using a priori and a posteriori stopping rules, the convergence rate resultes of the RKTM are obtained under a Hölder-type sourcewise condition if the Fréchet derivative is properly scaled. Due to the minimal assumptions for the convergence analysis of the modified iterative RKTM, we studied in detail the additional term in the continuous version written as:.

Preliminaries

Convergence Results

Convergence Rates

Conclusions