面载荷识别的TSVD正则化方法

线性系统的响应可以表示为单位脉冲响应函数与动态载荷的卷积分,经离散后得到载荷识别反问题。引入TSVD正则化方法处理反问题的不适定性,对不同噪声的实测响应进行了面载荷的反演求解,并对识别精度做出了分析。算例表明,TSVD正则化方法对响应误差的适应性强,识别精度高。     

附加广义函数的几个导数

作者证明了广义函数（ｘ±ｉ０）^λｌｎ^k（ｘ±ｉ０）的表示定理,给出了附加广义函数的导数：（ｌｎ^kｘ_±）＇,（ｘ^λ_±ｌｎ^kｘ_±）＇,（ｘ^-n_±ｌｎ^k＇ｘ_±）＇,（ｄ／ｄｘ）｛（ｘ±ｉ０）^λｌｎ^k（ｘ±ｉ０     

Processing the capillary viscometry data of fluids with yield stress

The capillary viscometer is used to measure the shear stress-shear rate relationship of a wide range of purely viscous fluids. It is however not considered as an appropriate instrument for obtaining the yield stress and the post-yield behaviour of fluids that have a yield stress. This is partly because conventional methods of processing the capillary viscometry data of purely viscous fluids cannot be applied to similar data of fluids with yield stress. The unavoidable experimental noise in the capillary data, particularly at low shear rates, also makes it difficult to obtain a reliable estimate of the yield stress from capillary data. In this investigation the problem of converting the capillary viscometry data of yield stress fluids into a shear stress-shear rate curve and a yield stress is formulated as a Volterra integral equation of the first kind. This is an ill-posed problem i.e. noise in the data will be amplified by inappropriate methods of data processing. A method, based on Tikhonov regularisation that takes into account the ill-posed nature of the problem, is then developed to solve this problem for fluids with yield stress. The performance of this method is assessed by applying it to a set of “synthetic” capillary viscometry data with added random noise and to a set of experimental data for a concentrated suspension of TiO₂ taken from the literature. In both cases Tikhonov regularisation was able to extract the complete shear properties of these fluids from capillary viscometry data alone. Received: 22 November 1999/Accepted: 17 December 1999     

材料物性参数识别的梯度正则化方法

本文对梯度正则化方法（Ｇｒａｄｉｅｎｔ－Ｒｅｇｕｌａｒｉｚａｔｉｏｎ Ｍｅｔｈｏｄ）作了进一步的研究,给出一种建立了梯度正则化迭代算法和选择正参数的简明实用方法。文中椭圆算子方程参数识别算例不仅说明了ＧＲ法具有广泛的适应性和一定的抗噪声能力,而且收敛速度较快,具有较大的收敛范围。     

Globally Convergent BFGS Method for Nonsmooth Convex Optimization1

We propose an implementable BFGS method for solving a nonsmooth convex optimization problem by converting the original objective function into a once continuously differentiable function by way of the Moreau–Yosida regularization. The proposed method makes use of approximate function and gradient values of the Moreau-Yosida regularization instead of the corresponding exact values. We prove the global convergence of the proposed method under the assumption of strong convexity of the objective function.     

An L-ribbon for large underdetermined linear discrete ill-posed problems

The L-curve is a popular aid for determining a suitable value of the regularization parameter when solving ill-conditioned linear systems of equations with a right-hand side vector, which is contaminated by errors of unknown size. However, for large problems, the computation of the L-curve can be quite expensive, because the determination of a point on the L-curve requires that both the norm of the regularized approximate solution and the norm of the corresponding residual vector be available. Recently, an approximation of the L-curve, referred to as the L-ribbon, was introduced to address this difficulty. The present paper discusses how to organize the computation of the L-ribbon when the matrix of the linear system of equations has many more columns than rows. Numerical examples include an application to computerized tomography.     

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．     

一类热传导问题源项识别的快速稳定算法

利用正算子的性质和Lax-Milgram定理,将一维热传导方程源项识别问题转化为适定的、第二类Volterra方程的求解,给出一个新的快速稳定算法,并进行理论分析.采用两种实现后验准则的新途径,在输入数据的误差水平已知和未知的情况下,快速地决定正则参数.数值实验证实了算法的有效性.     

The Local Conditional Stability Estimate for an Inverse Contact Problem in the Elasticity

In this paper, we consider an inverse problem in elasticity of determining stress on a contact domain from measurements outside the contact domain. The local conditional stability estimate for determining the stress vector is obtained. This conditional stability can give the convergence rate for Tikhonov's regularized solutions. This revised version was published online in July 2006 with corrections to the Cover Date.     

Regularization of a two-dimensional strongly damped wave equation with statistical discrete data

In this paper, we consider an inverse problem for a strongly damped wave equation in two dimensional with statistical discrete data. Firstly, we give a representation for the solution and then present a discretization form of the Fourier coefficients. Secondly, we show that the solution does not depend continuously on the data by stating a concrete example, which makes the solution be not stable and thus the present problem is ill-posed in the sense of Hadamard. Next, we use the trigonometric least squares method associated with the Fourier truncation method to regularize the instable solution of the problem. Finally, the convergence rate of the error between the regularized solution and the sought solution is estimated and also investigated numerically.