Several Regularization Methods For Linear Ill-posed Problems

Ill-posed problems arise from many scientific areas,such as geophysical,biomedicine,celestial mechanics and other fields.The main difficulty for solving such problems is the instability of the approximate solution,i.e.,the small perturbation of initial data will lead to large deviation between the exact solution and the approximate one.It is very important to design effective algorithms combined with some regular-ization techniques.In this dissertation,we investigate some regularization techniques to solve linear ill-posed problems(i.e.,Fredholm integral equation of the first kind)with the aim of improving the existing algorithms to establish new ones.The main results of our work are listed as follows.In the first part of the dissertation(Chapter 2),we give a modified regularization method.The singular value decomposition theory shows that the ill-posedness of the ill-conditioned system lies in that the singular values of the coefficient matrix decay rapidly to zero.We consider to construct some proper regularization filter factor to weaken or filter the influence caused by the smaller singular values.Motivated by this idea,by introducing a regularization parameter,we construct a new regularization matrix,thus get a modified Tikhonov regularization method,and the parameter selec-tion technique is discussed.We also study two regularization methods,one combines the proposed method with Amoldi process,and the other with augmented Krylov sub-space method,respectively.Numerical experiments show that,compared with some classical regularization methods,the proposed method can effectively reduce the rel-ative error of the regular solution.In the second part of the dissertation(Chapters 3-5),we study the image denois-ing and deblurring problem which is an important topic in the field of image process-ing.Based on the theory of optimization of the energy functional variational method,an energy functional is defined for the given image problem,then the original image problem is converted to an optimization problem of searching for the minimum of the energy functional.In Chapter 3,to overcome the limitation of the total variation(TV),we consider a hybrid model which combines the first-order and second-order total variation,i.e.,?||u||TV+(1-?)||u||HTV.To avoid solving a high-order problem,a two-step algo-rithm is designed to solve this model.In the first step,we use the variable splitting technique two times to reduce this hybrid model.In the second step,we apply the augmented Lagrangian method and split Bregman method to solve the reduced mod-el.The convergence of the algorithm is proved and the effectiveness is confirmed by some numerical experiments.In Chapter 4,we study the lp-lq problems minu{?/p||?u||pp+1/q||Au-f||qq},O<p,q?2,which is a challenging problem in image restoration and reconstruction.In 2014,R_H.Chan et al.proposed a Half-Quadratic algorithm for this model.In this algorithm,the authors applied conjugate gradient(CG)to an equation with symmetric and positive definite coefficient matrix.It will greatly increase the calculation of this algorithm.To improve the performance of this algorithm and take advantage of the spectral structure of the total variation and the blurring operator,we propose to use the alternating direction method to solve this equation.The convergence analysis of our algorithm is also discussed.Experiments prove that the proposed algorithm effectively improves the signal-noise ratio of the image with less CPU time.In Chapter 5,we consider the image restoration problem for impulse noise re-moval.It is well known that images,especially natural images,can be regard-ed as piecewise smooth functions,and wavelet frames can usually provide good s-parse approximations to piecewise smooth functions.By virtue of the edge pre-serving property of TV,we propose a Framelet-based hybrid regularization model minu{||Hu-g||1+?||u||TV+?||Wu||1},then the alternating direction method is used to solve this model.Experiments show that this model can significantly lessen staircase artifacts while well preserve the valuable edge information of the image.