A novel mixed group preserving scheme for the inverse Cauchy problem of elliptic equations in annular domains

In this paper, the inverse Cauchy problems for elliptic equations, including the Laplace equation, the Poisson equation, and the Helmholtz equation, defined in annular domains are investigated. When the outer boundary of an annulus is imposed by overspecified boundary data, we seek unknown data in the inner boundary through a combination of the spring-damping regularization method (SDRM) and the mixed group-preserving scheme (MGPS). Several numerical examples are examined to show that the MGPS plus the SDRM can overcome the ill-posed behavior of this highly ill-conditioned inverse Cauchy problem. The presently proposed novel algorithm has good efficiency and stability against the disturbance from large random noise even up to 50%, and the computational cost of MGPS is very time saving.     

Regularized solutions with a singular point for the inverse biharmonic boundary value problem by the method of fundamental solutions

Two numerical methods for the Cauchy problem of the biharmonic equation are proposed. The solution of the problem does not continuously depend on given Cauchy data since the problem is ill-posed. A small noise contained in the Cauchy data sensitively affects on the accuracy of the solution. Our problem is directly discretized by the method of fundamental solutions (MFS) to derive an ill-conditioned matrix equation. As another method, our problem is decomposed into two Cauchy problems of the Laplace and the Poisson equations, which are discretized by the MFS and the method of particular solutions (MPS), respectively. The Tikhonov regularization and the truncated singular value decomposition are applied to the matrix equation to stabilize a numerical solution of the problem for the given Cauchy data with high noises. The L-curve and the generalized cross-validation determine a suitable regularization parameter for obtaining an accurate solution. Based on numerical experiments, it is concluded that the numerical method proposed in this paper is effective for the problem that has an irregular domain and the Cauchy data with high noises. Furthermore, our latter method can successfully solve the problem whose solution has a singular point outside the computational domain.     

An inverse Stokes problem using interior pressure data

An inverse boundary value problem associated to the Stokes equations in a domain of two dimensions is considered. This problem requires the determination of the unspecified surface fluid velocity, or one of its components, over a part of its boundary by introducing extra interior pressure measurements. The problem is discretised numerically using the boundary element method (BEM) and the resulting ill-conditioned system of linear algebraic equations is solved using the Tikhonov regularisation method, with the choice of the regularisation parameter based on the L-curve criterion. The numerical technique is validated for some test examples with known analytical solutions. The accuracy of the numerical solutions is checked by comparison with their corresponding exact values and an investigation into stability of the numerical solution is undertaken by the addition of random noise into the interior pressure measurements. It is shown that the BEM provides a stable numerical solution of the Stokes problem which converges to the exact solution as the magnitude of error in the interior data decreases.     

Wave-front recovery from two orthogonal sheared interferograms

We present a new technique for using the information of two orthogonal lateral-shear interferograms to estimate an aspheric wave front. The wave-front estimation from sheared inteferometric data may be considered an ill-posed problem in the sense of Hadamard. We apply Thikonov regularization theory to estimate the wave front that has produced the lateral sheared interferograms as the minimizer of a positive definite-quadratic cost functional. The introduction of the regularization term permits one to find a well-defined and stable solution to the inverse shearing problem over the wave-front aperture as well as to reduce wave-front noise as desired.     

An Optimal Multi-Vector Iterative Algorithm in a Krylov Subspace for Solving the Ill-Posed Linear Inverse Problems

An optimal m-vector descent iterative algorithm in a Krylov subspace is developed, of which the m weighting parameters are optimized from a properly defined objective function to accelerate the convergence rate in solving an ill-posed linear problem. The optimal multi-vector iterative algorithm (OMVIA) is convergent fast and accurate, which is verified by numerical tests of several linear inverse problems, including the backward heat conduction problem, the heat source identification problem, the inverse Cauchy problem, and the external force recovery problem. Because the OMVIA has a good filtering effect, the numerical results recovered are quite smooth with small error, even under a large noise up to 10%.     

数值求解Symm积分方程的正则化MINRES方法

Symm积分方程在位势理论中具有重要应用,它是Hadamard意义下的不适定问题。离散该方程将产生对称线性不适定系统。基于GCV准则,并应用截断奇异值分解,本文提出数值求解Symm积分方程的正则化MINRES方法。与Tikhonov正则化方法相比,在数据出现噪声的情况下,新方法能有效地求得Symm积分方程的数值解。     

Image reconstruction in three-dimensional magnetostatic permeability tomography

Magnetostatic permeability tomography is an imaging technique that attempts to reconstruct the permeability distribution of an object using magnetostatic measurement data. The data for image reconstruction are external magnetic field measurements on the surface of the object due to an applied magnetostatic field. Theoretically, the normal and tangential components of the magnetic field in the surface uniquely define the internal isotropic permeability distributions. However, the inverse permeability problem is an ill-posed nonlinear problem. Regularization is needed for a stable solution. In this paper, we present a numerical method to solve the reconstruction problem in three dimensions using a regularized Gauss-Newton scheme. We have solved the forward problem using an edge finite-element method and we have employed an efficient technique to calculate the Jacobian matrix. The permeability of the object is assumed to be linear and isotropic. We present the reconstruction results for the permeability using synthetically generated data with additive noise.     

Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation

In this paper, the boundary knot method is extended to the solution of inhomogeneous equations, and it is applied to the Cauchy problem associated with the inhomogeneous Helmholtz equation. Here, we assume that the boundary condition is specified only on a part of the boundary, and the boundary conditions on the remaining part of the boundary are to be determined with the assistance of additional data. Since the resulting matrix equation is highly ill-conditioned, a regularized solution is obtained by employing the truncated singular value decomposition to solve the matrix equation arising from the boundary knot method, with the regularization parameter determined by the L-curve method. Numerical results are presented for several examples with smooth and piecewise smooth boundaries. The numerical verification shows that the proposed numerical scheme is accurate, stable with respect to data noise, and convergent with respect to decreasing the amount of noise in the data.     

自适应模糊滑模控制裹包机PMSM交流伺服系统

针对接缝式裹包机的交流伺服系统控制精度差的问题,提出了采用自适应模糊滑模控制器,控制具有高效率的永磁铁同步电机(PMSM)伺服驱动系统.仿真和实验结果表明,系统对于参数变化和外面负载扰动的鲁棒性大大提高,而且控制力大为降低.     

基于滑模控制投影混沌同步在隔振系统中的应用研究

线谱是潜艇辐射噪声谱中最显著的特征,即使在潜艇以低速航行,产生最低的噪声辐射的情况下也可以探测到。利用混沌的宽频特性来降低潜艇辐射噪声线谱,提高其隐身能力已被广泛研究,但由于混沌的内在不确定特性,当系统处于混沌状态时,整体隔振性能不能得到保证。提出应用滑模变结构控制实现隔振系统和混沌系统投影同步的新方法,在保持原有隔振系统隔振性能的前提下有效的消除了线谱。该方法不用计算Lyapunov指数,且对参数失配具有很强的鲁棒性。通过对两自由度隔振系统和Duffing方程的数值仿真,验证了该方法可成功消除线谱且明显改善隔振系统的隔振性能。     

Interacting convection modes in a saturated porous medium of nearly square planform: a special case

The problem of differentiating non-smooth functions of specimen displacements, which are measured during the material removal, is discussed. This problem arises when employing the layer removal method, namely a method of rings and strips, for residual stress depth profiling. It is shown that this problem is ill-posed and special solution methods are required in order to obtain a stable solution. The stability of the solution affects to a high extent the resulting accuracy of the residual stress evaluation in the investigated material. The presented study discusses a numerical approach to solving such ill-posed problems. The proposed approach, which is based on the Tikhonov regularization and a regularized finite difference method, provides a stable approximate solution, including its pointwise error estimation. The advantage of this approach is that it does not require any knowledge about the unknown exact solution; the pointwise error estimation of the measured data is the only prior information that must be available. In addition, this approach provides a convergence of the approximate solution to the unknown exact one when the perturbation of the initial data approaches zero.     

Necessary and sufficient invariance conditions in mismatched uncertain variable structure systems

Abstract

This paper investigates new invariance conditions in mismatched uncertain variable structure systems associated with a new sliding mode control, at the same time, retaining the benefits achieved in conventional variable structure systems design, namely, fast response, good robustness, and stability. Based on new equivalent state idea and two sets of switching surfaces, necessary and sufficient invariance conditions are derived such that matched and mismatched uncertainties completely vanish from the sliding mode dynamics. In terms of linear matrix inequalities, we give explicit formulas of linear switching surfaces to guarantee that the system in the new sliding mode is quadratically stable. Additionally, we give a control law to perform the new sliding mode. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.     

Optimally Generalized Regularization Methods for Solving Linear Inverse Problems

In order to solve ill-posed linear inverse problems, we modify the Tikhonov regularization method by proposing three different preconditioners, such that the resultant linear systems are equivalent to the original one, without dropping out the regularized term on the right-hand side. As a consequence, the new regularization methods can retain both the regularization effect and the accuracy of solution. The preconditioned coefficient matrix is arranged to be equilibrated or diagonally dominated to derive the optimal scales in the introduced preconditioning matrix. Then we apply the iterative scheme to find the solution of ill-posed linear inverse problem. Two theorems are proved that the iterative sequences are monotonically convergent to the true solution. The presently proposed optimally generalized regularization methods are able to overcome the ill-posedness of linear inverse problems, and provide rather accurate numerical solution.     

Non-destructive identification of residual stresses in steel under thermal loadings

We discuss computational aspects of the inverse and ill-posed problem of identifying residual stresses in steel structures under thermal loading. This corresponds to an inverse source problem in linear thermo-elasticity. The studies aim in investigating whether thermal loadings for the excitation of structures are sufficient in order to detect reliably inherent residual stresses. These stresses may result from the construction process or later thermal or mechanical treatment of the structure-like welding. By answering the raised question positively, our method provides an important basis for successful thermal straightenings. The quality of the solution of the inverse problem depends on a series of parameters, like material parameters, noise in the measurements, and the experimental setup. We numerically study the effects of these parameters and quantify the uncertainties in the results of the inverse problems by means of Sobol indices.     

Boundary element method for the Cauchy problem in linear elasticity

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput Maths Math Phys 32 (1991) 45] for obtaining approximate solutions to ill-posed boundary value problems in linear elasticity is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularizing criterion is given and in addition, the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. Analytical formulae for the integration constants resulting from the direct application of the BEM for an isotropic linear elastic medium are also presented.     

结构损伤识别的一种反馈岭估计方法

利用含有测量噪声的数据进行结构损伤识别时,经常出现病态最小二乘问题,可能导致计算结果完全失真。为了显著提高计算精度,在岭估计的基础上,进一步提出了一种反馈岭估计方法,以获得精确并具稳定的损伤识别结果。所提的反馈岭估计方法主要分为三个步骤:对结构损伤评估中的线性方程组进行第一次岭估计计算,得到损伤参数的粗略解;根据损伤参数的粗略解,设计一个新的对角矩阵,用于随后的反馈岭估计(即第二次岭估计)计算中;对损伤评估线性方程组进行第二次岭估计计算(即反馈岭估计),最终获得损伤参数的高精度解,据此来对结构中的损伤位置和严重程度进行判定。以一个梁结构作为数值算例,讨论了所提方法在10%噪声水平下的有效性,并把计算结果与普通岭估计和奇异值截断法进行了比较,结果表明:所提反馈岭估计方法大幅度提高了计算精度,即使在10%的噪声水平下,该方法也能获得精度很高的计算结果。     

Adaptive error estimation technique of the Trefftz method for solving the over-specified boundary value problem

In this paper, numerical solutions are investigated based on the Trefftz method for an over-specified boundary value problem contaminated with artificial noise. The main difficulty of the inverse problem is that divergent results occur when the boundary condition on over-specified boundary is contaminated by artificial random errors. The mechanism of the unreasonable result stems from its ill-posed influence matrix. The accompanied ill-posed problem is remedied by using the Tikhonov regularization technique and the linear regularization method, respectively. This remedy will regularize the influence matrix. The optimal parameter λ of the Tikhonov technique and the linear regularization method can be determined by adopting the adaptive error estimation technique. From this study, convergent numerical solutions of the Trefftz method adopting the optimal parameter can be obtained. To show the accuracy of the numerical solutions, we take the examples as numerical examination. The numerical examination verifies the validity of the adaptive error estimation technique. The comparison of the Tikhonov regularization technique and the linear regularization method was also discussed in the examples.     

基于奇异摄动理论的电液伺服系统Backstepping滑模自适应控制

针对电液伺服位置跟踪系统中存在的非线性特性、系统参数和外部负载的非匹配不确定性,提出了基于奇异摄动理论的电液伺服系统的Backstepping滑模自适应控制。利用奇异摄动中双时间刻度理论将原系统分解为快慢变子系统,分别设计快变和慢变子系统的控制律,再合成得到复合控制器。应用Backstepping的逆向递推方法有效地解决了高阶非线性系统的控制问题,用滑模方法抑制系统的外部扰动,对系统的不确定性参数进行自适应估计。数字仿真的结果验证了所设计控制器的正确性和有效性。     

非均匀损伤介质中利用波传播理论对损伤参数的反演

介绍了波在非均匀损伤介质中的传播的反问题,根据波的响应来反演介质的损伤。首先应用Newton迭代法把非线性问题化为线性问题,得到一组不适定的线性方程式组,采用Tikhonov正则化求解,对于正则参数选取采用L线曲线准则,求出了原问题的稳定的、有效的解,获得了与真实损伤度较为吻合的反演参数。工程实例的模拟分析计算证明了上述算法的有效性。     

ATMD-结构带有补偿器的模糊滑模控制

针对基于指数趋近律的滑模控制算法设计ATMD系统中的主动控制力偏大、控制系统抖振过大等不足,采用一种带有补偿器的滑模控制算法来计算ATMD系统的主动控制力。考虑到外部能量有限问题,引入了相应的饱和控制器。同时,为了减小控制系统的抖振,设计了一种基于模糊自适应调整的滑模控制律。以一个单自由度结构模型为例进行了相应的数值分析,与相应的由未含补偿器的指数趋近律方法设计的ATMD控制系统以及TMD和AMD控制系统的控制效果进行对比来验证所提方法的有效性。