Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green’s formulation, the problem can be transformed into a moment problem. Then we propose a modified Tikhonov regularization algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method.     

Regularizing properties of a truncated newton-cg algorithm for nonlinear inverse problems

This paper develops truncated Newton methods as an appropriate tool for nonlinear inverse problems which are ill-posed in the sense of Hadamard. In each Newton step an approximate solution for the linearized problem is computed with the conjugate gradient method as an inner iteration. The conjugate gradient iteration is terminated when the residual has been reduced to a prescribed percentage. Under certain assumptions on the nonlinear operator it is shown that the algorithm converges and is stable if the discrepancy principle is used to terminate the outer iteration. These assumptions are fulfilled, e.g., for the inverse problem of identifying the diffusion coefficient in a parabolic differential equation from distributed data.     

Fast regularized linear sampling for inverse scattering problems

A new numerical procedure is proposed for the reconstruction of the shape and volume of unknown objects from measurements of their radiation in the far field. This procedure is a variant and the linear sampling method has a very acceptable computational load and is fully automated. It is based on exploiting an iteratively computed truncated singular‐value decomposition and heuristics to extract the desired signal from the background noise. Its performance on a battery of examples of different types is shown to be promising. Copyright © 2010 John Wiley & Sons, Ltd.     

Preconditioners for ill‐posed Toeplitz matrices with differentiable generating functions

Both theoretical analysis and numerical experiments in the literature have shown that the Tyrtyshnikov circulant superoptimal preconditioner for Toeplitz systems can speed up the convergence of iterative methods without amplifying the noise of the data. Here we study a family of Tyrtyshnikov‐based preconditioners for discrete ill‐posed Toeplitz systems with differentiable generating functions. In particular, we show that the distribution of the eigenvalues of these preconditioners has good regularization features, since the smallest eigenvalues stay well separated from zero. Some numerical results confirm the regularization effectiveness of this family of preconditioners. Copyright © 2009 John Wiley & Sons, Ltd.     

Casimir effect for a scalar field via Krein quantization

In this work, we present a rather simple method to study the Casimir effect on a spherical shell for a massless scalar field with Dirichlet boundary condition by applying the indefinite metric field (Krein) quantization technique. In this technique, the field operators are constructed from both negative and positive norm states. Having understood that negative norm states are un-physical, they are only used as a mathematical tool for renormalizing the theory and then one can get rid of them by imposing some proper physical conditions.     

位势平面问题的新的规则化边界积分方程

广泛实践集中在直接变量边界积分方程的规则化研究,其本质是利用简单解消除边界积分的奇异性．然而,至今关于平面位势问题的第一类边界积分方程的规则化研究尚未涉足．致力于间接变量边界积分方程的规则化方法研究,基于一种新的思想和观点,确立平面位势问题的间接变量规则边界积分方程,它不包含CPV强奇异积分和HFP超奇异积分．数值算例表明现在的方法可取得很好的精度和效率,特别是边界量的计算．     

Positive solutions for mixed problems of singular fractional differential equations

We investigate the existence of positive solutions to the singular fractional boundary value problem: $^c\hspace{-1.0pt}D^{\alpha }u +f(t,u,u^{\prime },^c\hspace{-2.0pt}D^{\mu }u)=0$, u′(0) = 0, u(1) = 0, where 1 < α < 2, 0 < μ < 1, f is a L^q‐Carathéodory function, $q > \frac{1}{\alpha -1}$, and f(t, x, y, z) may be singular at the value 0 of its space variables x, y, z. Here $^c \hspace{-1.0pt}D$ stands for the Caputo fractional derivative. The results are based on combining regularization and sequential techniques with a fixed point theorem on cones.     

非正则线性增广拉格朗日乘子法的收敛速率研究CSCD

Recently,an indefinite linearized augmented Lagrangian method(IL-ALM)was proposed for the convex programming problems with linear constraints.The IL-ALM differs from the linearized augmented Lagrangian method in that the augmented Lagrangian is linearized by adding an indefinite quadratic proximal term.But,it preserves the algorithmic feature of the linearized ALM and usually has the advantage to improve the performance.The IL-ALM is proved to be convergent from contraction perspective,but its convergence rate is still missing.This is mainly because that the indefinite setting destroys the structures when we directly employ the contraction frameworks.In this paper,we derive the convergence rate for this algorithm by using a different analysis.We prove that a worst-case O(1/t)convergence rate is still hold for this algorithm,where t is the number of iterations.Additionally we show that the customized proximal point algorithm can employ larger step sizes by proving its equivalence to the linearized ALM.     

On the numerical solution of ill‐conditioned linear systems by regularization and iteration

We propose to reduce the (spectral) condition number of a given linear system by adding a suitable diagonal matrix to the system matrix, in particular by shifting its spectrum. Iterative procedures are then adopted to recover the solution of the original system. The case of real symmetric positive definite matrices is considered in particular, and several numerical examples are given. This approach has some close relations with Riley's method and with Tikhonov regularization. Moreover, we identify approximately the aforementioned procedure with a true action of preconditioning.     

自适应非凸稀疏正则化下自适应光学系统加性噪声的去除

自适应光学系统可以实时测量并校正波前信息,但是系统中大量的噪声严重影响了系统的探测精度.自适应光学系统中一般为加性噪声,本文提出一种全新的变分处理模型去除加性噪声,该模型采用自适应非凸正则项.非凸正则项在保持图像细节上较凸正则项具有更好的效果,能更好地保持点源目标的完整性.另外,根据不同区域的噪声水平自适应地构建正则化参数,使不同区域的像素点受到不同程度的噪声抑制,可以更好地保持目标的边缘细节.在算法实现上,为了解决非凸正则项收敛性较差的缺陷,采用分裂Bregman算法及增广拉格朗日对偶算法进行计算.实验及数值仿真结果都表明,该方法能够较好地去除系统中的加性噪声,且光斑信号保存得较为完整,处理后的质心探测精度及信噪比较高.