Solving Severely Ill⁃Posed Linear Systems with Time Discretization Based Iterative Regularization Methods

Recently,inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications.After the discretization,many of inverse problems are reduced to linear systems.Due to the typical ill-posedness of inverse problems,the reduced linear systems are often illposed,especially when their scales are large.This brings great computational difficulty.Particularly,a small perturbation in the right side of an ill-posed linear system may cause a dramatical change in the solution.Therefore,regularization methods should be adopted for stable solutions.In this paper,a new class of accelerated iterative regularization methods is applied to solve this kind of large-scale ill-posed linear systems.An iterative scheme becomes a regularization method only when the iteration is early terminated.And a Morozov’s discrepancy principle is applied for the stop criterion.Compared with the conventional Landweber iteration,the new methods have acceleration effect,and can be compared to the well-known acceleratedν-method and Nesterov method.From the numerical results,it is observed that using appropriate discretization schemes,the proposed methods even have better behavior when comparing withν-method and Nesterov method.

Downward continuation of airborne geomagnetic data based on two iterative regularization methods in the frequency domain

Downward continuation is a key step in processing airborne geomagnetic data. However,downward continuation is a typically ill-posed problem because its computation is unstable; thus, regularization methods are needed to realize effective continuation. According to the Poisson integral plane approximate relationship between observation and continuation data, the computation formulae combined with the fast Fourier transform（FFT）algorithm are transformed to a frequency domain for accelerating the computational speed. The iterative Tikhonov regularization method and the iterative Landweber regularization method are used in this paper to overcome instability and improve the precision of the results. The availability of these two iterative regularization methods in the frequency domain is validated by simulated geomagnetic data, and the continuation results show good precision.

An Adaptive Strategy for the Restoration of Textured Images using Fractional Order Regularization

Total variation regularization has good performance in noise removal and edge preservation but lacks in texture restoration.Here we present a texture-preserving strategy to restore images contaminated by blur and noise.According to a texture detection strategy,we apply spatially adaptive fractional order diffusion.A fast algorithm based on the half-quadratic technique is used to minimize the resulting objective function.Numerical results show the effectiveness of our strategy.

Estimation of a source term in a quasi steady twodimensional heat transfer problem： application to an electron beam welding

In previous work, we have analyzed the feasibility of the estimation for a source term S（x, y, z） in a transversal section, The present study is concerned with a twodimensional inverse phase change problem. The goal is the estimation of the dissipated heat flux in the liquid zone （reconstruction of a source term in the energy equation） from experimentally measured temperatures in the solid zone. This work has an application in the electron beam welding of steels of thickness about 8cm. The direct thermometallurgical problem is treated in a quasi steady two-dimensional longitudinal section （x, y）. The beam displacement is normally in the y direction. But in the quasi steady simulation, the beam is steady in the study section. The sample is divided in the axial direction z in few sections. At each section, a source term is defined with a part of the beam and creates a vaporized zone and a fused zone. The goal of this work is the rebuilding of the complete source term with the estimations at each section. In this paper, we analyze the feasibility of the estimation. For this work, we use only the simulated measurements without noise.