A coupled complex boundary method for the Cauchy problem

Considered in this paper is a Cauchy problem governed by an elliptic partial differential equation. In the Cauchy problem, one wants to recover the unknown Neumann and Dirichlet data on a part of the boundary from the measured Neumann and Dirichlet data, usually contaminated with noise, on the remaining part of the boundary. The Cauchy problem is an inverse problem with severe ill-posedness. In this paper, a coupled complex boundary method (CCBM), originally proposed in [Cheng XL, Gong RF, Han W, et al. A novel coupled complex boundary method for solving inverse source problems. Inverse Prob. 2014;30:055002], is applied to solve the Cauchy problem stably. With the CCBM, all the data, including the known and unknown ones on the boundary are used in a complex Robin boundary on the whole boundary. As a result, the Cauchy problem is transferred into a complex Robin boundary problem of finding the unknown data such that the imaginary part of the solution equals zero in the domain. Then the Tikhonov regularization is applied to the resulting new formulation. Some theoretical analysis is performed on the CCBM-based Tikhonov regularization framework. Moreover, through the adjoint technique, a simple solver is proposed to compute the regularized solution. The finite-element method is used for the discretization. Numerical results are given to show the feasibility and effectiveness of the proposed method.     

The geometric element transformation method for mixed mesh smoothing

The geometric element transformation method (GETMe) is a geometry-based smoothing method for mixed and non-mixed meshes. It is based on a simple geometric transformation applicable to elements bounded by polygons with an arbitrary number of nodes. The transformation, if applied iteratively, leads to a regularization of the polygons. Global mesh smoothing is accomplished by averaging the new node positions obtained by local element transformations. Thereby, the choice of transformation parameters as well as averaging weights can be based on the element quality which leads to high quality results. In this paper, a concept of an enhanced transformation approach is presented and a proof for the regularizing effect of the transformation based on eigenpolygons is given. Numerical examples confirm that the GETMe approach leads to superior mesh quality if compared to other geometry-based methods. In terms of quality it can even compete with optimization-based techniques, despite being conceptually significantly simpler.     

Experience-consistent modeling: Regression and classification problems

In this study, we are concerned with system modeling which involves limited data and reconciles the developed model with some previously acquired domain knowledge being captured in the format of already constructed models. Each of these previously available models was formed on a basis of extensive data sets which are not available for the current identification pursuits. To emphasize the nature of modeling being guided by the reconciliation mechanisms, we refer to this mode of identification as experience-consistent modeling. The paper presents the conceptual and algorithmic framework by focusing on regression models. By forming a certain extended form of the performance index, it is shown that the domain knowledge captured by regression models can play a similar role as a regularization component used quite commonly in system identification. Experimental results involve both synthetic low-dimensional data and selected data coming from Machine Learning repository. The data used in the experiments tackle regression models as well as classification problems (two-class classifiers).     

基于遗传算法和广义交叉原理求解正则参数

本文研究了正则化方法中正则参数的求解问题,提出了一种新的正则参数求解策略,即利用遗传算法基于广义交叉检验准则求解正则参数,数值模拟验证了该方法的可行性和有效性。     

Some First-Order Algorithms for Total Variation Based Image Restoration

This paper deals with first-order numerical schemes for image restoration. These schemes rely on a duality-based algorithm proposed in 1979 by Bermùdez and Moreno. This is an old and forgotten algorithm that is revealed wider than recent schemes (such as the Chambolle projection algorithm) and able to improve contemporary schemes. Total variation regularization and smoothed total variation regularization are investigated. Algorithms are presented for such regularizations in image restoration. We prove the convergence of all the proposed schemes. We illustrate our study with numerous numerical examples. We make some comparisons with a class of efficient algorithms (proved to be optimal among first-order numerical schemes) recently introduced by Y. Nesterov.     

A meshless method to the numerical solution of an inverse reaction–diffusion–convection problem

In this paper, the determination of the source term in a reaction–diffusion convection problem is investigated. First with suitable transformations, the problem is reduced, then a new meshless method based on the use of the heat polynomials as basis functions is proposed to solve the inverse problem. Due to the ill-posed inverse problem, the Tikhonov regularization method with a generalized cross-validation criterion is employed to obtain a numerical stable solution. Finally, some numerical examples are presented to show the accuracy and effectiveness of the algorithm.     

双相介质二维波动方程孔隙率反演的扰动方法

本文考虑在双相介质的二维波动方程中，应用扰动方法对其中的参数孔隙率进行反演，并应用正则化方法解反演中出现的不稳定问题，方法简洁，迭代过程稳定，收敛效率高，文中给出的算例说明此方法是可行的。     

一种改进的正则化自适应匹配追踪算法

针对压缩感知中未知稀疏度信号的重构问题,提出了一种改进的正则化自适应匹配追踪算法。它通过自适应变步长迭代对信号稀疏度进行估计,并将其作为初始支撑集长度,然后在分阶段迭代中正则化筛选原子,最终实现信号的精确重构。仿真结果表明,该算法重构信号的性能和效率均优于子空间追踪算法、正交匹配追踪算法和稀疏度自适应匹配追踪算法。     

RKRLS及在混炼胶质量建模与预报中的应用研究

对递推最小二乘(RLS)进行了非线性的核(kernel)变换,并采用正则化技术改写了目标范函,提出了一种正则核变换递推最小二乘(regularized kernel,RLS)算法.获得了RKRLS模型的系数和误差表达式,分析了算法的推广能力并证明了KeYnel RLS算法为其特例,进而导出了RKRLS算法在限定、增长和缩减记忆三种不同模式下的递推公式均无需进行求逆计算.RKRLS算法具有三个特性:小样本、可控的推广能力和速度快,因而非常适合于工业应用场合.通过对橡胶混炼质量的门尼指标进行建模和预测分析表明,本算法具有较好的跟踪预测性能.