Volume 17, Issue 9

A combination of Newton's method and a regularization method has been considered for obtaining a stable approximate solution for ill-posed Hammerstein type operator equation. By choosing the regularization parameter according to an adaptive scheme considered by Pereverzev and Schock (SIAM. J. Numer. Anal. 43: 2060–2076, 2005) an order optimal error estimate has been obtained. Moreover the method that we consider gives quadratic convergence compared to the linear convergence obtained by George and Nair (J. Complexity 24: 228–240, 2008).

The goal of this paper is to investigate the Tikhonov–Phillips method for semi-discrete linear ill-posed problems in order to determine tight error bounds and to obtain a good parameter choice. We consider the equation A ƒ = g , where the operator A is known, noisy discrete data with can be observed and a solution ƒ* is sought. Assuming that ƒ* is an element of a Sobolev space, we use the well-known theory of optimal recovery in Hilbert spaces for the reconstruction process. We then provide L 2 -error estimates in terms of the data density and derive an a priori selection for the regularization parameter, which guarantees an optimal compromise between approximation and stability. Finally, we illustrate the parameter selection with a simple example.

In this paper we investigate the numerical complexity to solve nonlinear ill-posed problems when the operator equations F ( x ) = y δ are solved by the iteratively regularized Gauss–Newton method (IRGNM) with inner CG-iteration. Additionally we consider a preconditioned version of the IRGNM and compare the complexity of the standard IRGNM and its preconditioned version. In the case of exponentially ill-posed problems we show the superiority of the preconditioned IRGNM, that is we prove that the preconditioning techniques presented in this paper yield a significant reduction of the total complexity.

We study a Cauchy problem for a parabolic equation in multiple dimensions, which is naturally a generalization of some one-dimensional and two-dimensional inverse heat conduction problems. This is a severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data. After simply analyzing the ill-posedness of the Cauchy problem in the frequency space, from a new viewpoint we propose two regularization methods: Tikhonov method and Fourier truncation method. We give and prove the convergence estimate between the exact solution and its regularized approximation. We also discuss the relationship of these two and other regularization methods. At last, we employ some numerical examples to illustrate the behavior of the proposed methods.

In this paper, a simple and convenient new regularization method which is called modified quasi-boundary value method for solving nonlinear backward heat equation is given. Some new quite sharp error estimates between the approximate solution are provided and generalize the results in our paper [Trong, Quan, Khanh, and Tuan, Zeitschrift Analysis und ihre Anwendungen 26: 231–245, 2007, Trong and Tuan, Electron. J. Diff. Eqns. 4: 1–10, 2006, Trong and Tuan, Electron. J. Diff. Eqns. 84: 1–12, 2008]. The approximation solution is calculated by the contraction principle. A numerical experiment is given.