This paper concerns the A smooth regularization method for linear ill posed equations in the presence of perturbed operators and noisy data. The semi and full a posteriori Morozov discrepancy principles for...This paper concerns the A smooth regularization method for linear ill posed equations in the presence of perturbed operators and noisy data. The semi and full a posteriori Morozov discrepancy principles for choosing the regularization parameter are proposed, which lead to satisfactory results.展开更多
This article presents a fast convergent method of iterated regularization based on the idea of Landweber iterated regularization, and a method for a-posteriori choice by the Morozov discrepancy principle and the optim...This article presents a fast convergent method of iterated regularization based on the idea of Landweber iterated regularization, and a method for a-posteriori choice by the Morozov discrepancy principle and the optimum asymptotic convergence order of the regularized solution is obtained. Numerical test shows that the method of iterated regularization can quicken the convergence speed and reduce the calculation burden efficiently.展开更多
In this paper, we consider an unknown source problem for the modified Helmholtz equation. The Tikhonov regularization method in Hilbert scales is extended to deal with ill-posedness of the problem. An a priori strateg...In this paper, we consider an unknown source problem for the modified Helmholtz equation. The Tikhonov regularization method in Hilbert scales is extended to deal with ill-posedness of the problem. An a priori strategy and an a posteriori choice rule have been present to obtain the regularization parameter and corresponding error estimates have been obtained. The smoothness parameter and the a priori bound of exact solution are not needed for the a posteriori choice rule. Numerical results are presented to show the stability and effectiveness of the method.展开更多
Abstract This paper presents a restarted conjugate gradient iterative algorithm for solving ill-posed problems. The damped Morozov's discrepancy principle is used as a stopping rule. Numerical experiments are give...Abstract This paper presents a restarted conjugate gradient iterative algorithm for solving ill-posed problems. The damped Morozov's discrepancy principle is used as a stopping rule. Numerical experiments are given to illustrate the efficiency of the method.展开更多
Abstract In this paper, we introduce a modified Landweber iteration to solve the sideways parabolic equation, which is an inverse heat conduction problem （IHCP） in the quarter plane and is severely ill-posed. We sha...Abstract In this paper, we introduce a modified Landweber iteration to solve the sideways parabolic equation, which is an inverse heat conduction problem （IHCP） in the quarter plane and is severely ill-posed. We shall show that our method is of optimal order under both a priori and a posteriori stopping rule. Furthermore, if we use the discrepancy principle we can avoid the selection of the a priori bound. Numerical examples show that the computation effect is satisfactory.展开更多
We study multi-parameter regularization(multiple penalties) for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. We revisit a balancing principle for choosing regu...We study multi-parameter regularization(multiple penalties) for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. We revisit a balancing principle for choosing regularization parameters from the viewpoint of augmented Tikhonov regularization, and derive a new parameter choice strategy called the balanced discrepancy principle. A priori and a posteriori error estimates are provided to theoretically justify the principles, and numerical algorithms for efficiently implementing the principles are also provided. Numerical results on deblurring are presented to illustrate the feasibility of the balanced discrepancy principle.展开更多
The symmetric kernel-driven operator equations play an important role in mathematical physics, engineering, atmospheric image processing and remote sensing sciences. Such problems are usually ill-posed in the sense th...The symmetric kernel-driven operator equations play an important role in mathematical physics, engineering, atmospheric image processing and remote sensing sciences. Such problems are usually ill-posed in the sense that even if a unique solution exists, the solution need not depend continuously on the input data. One common technique to overcome the difficulty is applying the Tikhonov regularization to the symmetric kernel operator equations, which is more generally called the Lavrentiev regularization. It has been shown that the iterative implementation of the Tikhonov regularization can improve the rate of convergence. Therefore in this paper, we study the iterative Lavrentiev regularization method in a similar way when applying it to symmetric kernel problems which appears frequently in applications, say digital image restoration problems. We first prove the convergence property, and then under the widely used Morozov discrepancy principle(MDP), we prove the regularity of the method. Numerical performance for digital image restoration is included to confirm the theory. It seems that the iterated Lavrentiev regularization with the MDP strategy is appropriate for solving symmetric kernel problems.展开更多
文摘This paper concerns the A smooth regularization method for linear ill posed equations in the presence of perturbed operators and noisy data. The semi and full a posteriori Morozov discrepancy principles for choosing the regularization parameter are proposed, which lead to satisfactory results.
文摘This article presents a fast convergent method of iterated regularization based on the idea of Landweber iterated regularization, and a method for a-posteriori choice by the Morozov discrepancy principle and the optimum asymptotic convergence order of the regularized solution is obtained. Numerical test shows that the method of iterated regularization can quicken the convergence speed and reduce the calculation burden efficiently.
文摘In this paper, we consider an unknown source problem for the modified Helmholtz equation. The Tikhonov regularization method in Hilbert scales is extended to deal with ill-posedness of the problem. An a priori strategy and an a posteriori choice rule have been present to obtain the regularization parameter and corresponding error estimates have been obtained. The smoothness parameter and the a priori bound of exact solution are not needed for the a posteriori choice rule. Numerical results are presented to show the stability and effectiveness of the method.
基金Supported by CAS Hundred Talents Program and Digital Earth (KZCX2-312)also partially supported by National Natural Science Foundation of China (No.19731010).
文摘Abstract This paper presents a restarted conjugate gradient iterative algorithm for solving ill-posed problems. The damped Morozov's discrepancy principle is used as a stopping rule. Numerical experiments are given to illustrate the efficiency of the method.
基金Supported by the National Natural Science Foundation of China (No. 10971019)Scientific Research Fund of Guangxi Education Department Grant No. 201012MS067
文摘Abstract In this paper, we introduce a modified Landweber iteration to solve the sideways parabolic equation, which is an inverse heat conduction problem （IHCP） in the quarter plane and is severely ill-posed. We shall show that our method is of optimal order under both a priori and a posteriori stopping rule. Furthermore, if we use the discrepancy principle we can avoid the selection of the a priori bound. Numerical examples show that the computation effect is satisfactory.
基金supported by the Army Research Office under DAAD19-02-1-0394,US-ARO grant 49308MA and US-AFSOR grant FA9550-06-1-0241
文摘We study multi-parameter regularization(multiple penalties) for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. We revisit a balancing principle for choosing regularization parameters from the viewpoint of augmented Tikhonov regularization, and derive a new parameter choice strategy called the balanced discrepancy principle. A priori and a posteriori error estimates are provided to theoretically justify the principles, and numerical algorithms for efficiently implementing the principles are also provided. Numerical results on deblurring are presented to illustrate the feasibility of the balanced discrepancy principle.
基金the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education M inistry.
文摘The symmetric kernel-driven operator equations play an important role in mathematical physics, engineering, atmospheric image processing and remote sensing sciences. Such problems are usually ill-posed in the sense that even if a unique solution exists, the solution need not depend continuously on the input data. One common technique to overcome the difficulty is applying the Tikhonov regularization to the symmetric kernel operator equations, which is more generally called the Lavrentiev regularization. It has been shown that the iterative implementation of the Tikhonov regularization can improve the rate of convergence. Therefore in this paper, we study the iterative Lavrentiev regularization method in a similar way when applying it to symmetric kernel problems which appears frequently in applications, say digital image restoration problems. We first prove the convergence property, and then under the widely used Morozov discrepancy principle(MDP), we prove the regularity of the method. Numerical performance for digital image restoration is included to confirm the theory. It seems that the iterated Lavrentiev regularization with the MDP strategy is appropriate for solving symmetric kernel problems.
