An optimization approach for the Cauchy problem in linear elasticity

An energy error functional is introduced in the context of the ill-posed problem of boundary data recovery in linear elasticity, which is well known as the Cauchy problem. The problem is converted into one of optimization; the computation of the gradients of the energy functional is given for both the continuous and the discrete problems. Links with existing methods for data completion are described and numerical experiments highlight the efficiency of the proposed method as well as its robustness in the case of singular data.     

Rapid distributed model predictive control design using singular value decomposition for linear systems

The issue of model predictive control design of distribution systems using a popular singular value decomposition (SVD) technique is addressed. Namely, projection to a set of conjugate structure is dealt with in this paper. The structure of the resulting predictive model is decomposed into small sets of subsystems. The optimal inputs can be separately designed at each subsystem in parallel without any interaction problems. The optimal inputs can be directly obtained and the communication among the subsystems can be significantly reduced. In addition, the design of distribution model predictive control (DMPC) with constraints using the SVD framework is also presented. The unconstraint inputs are checked in parallel in the conjugate space. Without solving the QP problem of each subsystem, the suboptimal solution can be quickly obtained by selecting the bigger singular values and discarding the small singular values in the singular value space. The convergence condition of the proposed algorithm is also proved. Two case studies are used to illustrate the distribution control systems using the suggested approach. Comparisons between the centralized model predictive control method and the proposed DMPC method are carried out to show the advantages of the newly proposed method.     

An iterative method for the Cauchy problem in linear elasticity with fading regularization effect

In this paper, an iterative method for solving the Cauchy problem in linear elasticity is introduced. This problem consists in recovering missing data (displacements and forces) on some parts of a domain boundary from the knowledge of overspecified data (displacements and forces) on the remaining parts. The algorithm reads as a least square fitting of the given data, with a regularization term whose effect fades as the iterations go on. So the algorithm converges to the solution of the Cauchy problem. Numerical simulations using the finite element method highlight the algorithm’s efficiency, accuracy, robustness to noisy data as well as its ability to deblur noisy data.     

Numerical solution of singular boundary value problems using advanced Adomian decomposition method

Engineering with Computers - In this work, we present a powerful method for the numerical solution of non-linear singular boundary value problems, namely the advanced Adomian decomposition method...     

Point cloud matching using singular value decomposition

We propose a simple, fast three-dimensional (3D) matching method that determines the best rotation matrix between non-corresponding point clouds (PCs) with no iterations. An estimated rotation matrix can be derived by the two following steps: (1) the singular value decomposition is applied to a measured data matrix, and a database matrix is constructed from the PC datasets; (2) the inner product of each left singular vector is used to produce the estimated rotation. Through experimentation, we demonstrate that the proposed method executes 3D PC matching with <4 % of the computational time of the iterative closest point algorithm with nearly identical accuracy.     

A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity

We propose two algorithms involving the relaxation of either the given Dirichlet data (boundary displacements) or the prescribed Neumann data (boundary tractions) on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. [16] applied to Cauchy problems in linear elasticity. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed method.     

Explicit solution to the singular discrete-time stationary linear filtering problem

A closed form solution to the stationary discrete-time linear filtering problem is obtained explicitly in terms of the system state-space matrices in the limiting singular case where the measurement noise tends to zero. Simple expressions, in closed form, are obtained for the Kalman gain matrix both for uniform and nonuniform rank systems and the explicit eigenstructure of the Kalman filter closed loop matrix is derived. The minimum error covariance matrices of the a priori and a posteriori filtered estimates are obtained using this special eigenstructure, and a remarkably different behavior of the solution in the minimum- and nonminimum-phase cases is found.     

A simple solution to the singular linear minimum-variance estimation problem

A simple expression is obtained for the transfer function matrix of the minimum-variance estimator for the states of a linear stationary continuous-time right invertible system whose output is measured perfectly. Using thes-domain approach, it is shown that the optimal estimator first finds the driving noise input that achieves, when applied on the minimum-phase image model of the system, an output spectrum which is identical to the measurement spectrum. This input is then applied on a state-space representation of the minimum-phase image to produce the optimal estimate.     

Reducing variables in Model Predictive Control of linear system with disturbances using singular value decomposition

Arising from the need to reduce online computations of Model Predictive Controller, this paper proposes an approach for a linear system with bounded additive disturbance using fewer variables than the standard. The new variables are chosen so that they transfer the maximal energy to the control inputs. Several other features are introduced. These include an auxiliary state to ensure recursive feasibility, an initialization procedure that recovers a substantial portion of the original domain of attraction arising from the use of fewer variables. A comparison of the domains of attraction associated with the new variables is also discussed. Run-time computational advantage of more than an order of magnitude compared with standard approach is demonstrated using several numerical examples although a more expensive initialization is needed.     

Generalizing discriminant analysis using the generalized singular value decomposition

Discriminant analysis has been used for decades to extract features that preserve class separability. It is commonly defined as an optimization problem involving covariance matrices that represent the scatter within and between clusters. The requirement that one of these matrices be nonsingular limits its application to data sets with certain relative dimensions. We examine a number of optimization criteria, and extend their applicability by using the generalized singular value decomposition to circumvent the nonsingularity requirement. The result is a generalization of discriminant analysis that can be applied even when the sample size is smaller than the dimension of the sample data. We use classification results from the reduced representation to compare the effectiveness of this approach with some alternatives, and conclude with a discussion of their relative merits.     

Fast correlation registration method using singular value decomposition

A new, fast template-matching method using the Singular Value Decomposition (SVD) is presented. This approach involves a two-stage algorithm, which can be used to increase the speed of the matching process. In the first stage, the reference image is orthogonally separated by the SVD and then low-cost pseudo-correlation values are calculated. This reduces the number of computations to 2*N*L instead of N²L², where L × L is the size of the reference image and N × N is the original image size. At the second stage, a small group of values near the maximum pseudo-correlation is selected. the true correlation for the small number of pixels in this group is them computed precisely in the second stage. Experimental and analytic results are presented to show how the computation complexity is greatly improved.     

Boundary element sensitivity evaluation for elasticity problems using complex variable method

The complex variable method is used to evaluate sensitivities in two-dimensional elasticity problems, using the BEM as numerical method. The method shows negligible dependency on the perturbation magnitude, and can be easily incorporated to existing codes which handle complex algebra. Numerical results for elasticity problems show that the numerical accuracy of the sensitivities obtained with complex variable differentiation is competitive with other methods, and can be easily implemented in any other approximation methods as well.     

Video summarization and retrieval using singular value decomposition

In this paper, we propose novel video summarization and retrieval systems based on unique properties from singular value decomposition (SVD). Through mathematical analysis, we derive the SVD properties that capture both the temporal and spatial characteristics of the input video in the singular vector space. Using these SVD properties, we are able to summarize a video by outputting a motion video summary with the user-specified length. The motion video summary aims to eliminate visual redundancies while assigning equal show time to equal amounts of visual content for the original video program. On the other hand, the same SVD properties can also be used to categorize and retrieve video shots based on their temporal and spatial characteristics. As an extended application of the derived SVD properties, we propose a system that is able to retrieve video shots according to their degrees of visual changes, color distribution uniformities, and visual similarities.     

大规模最小二乘奇异值分解的并行处理方法

大规模最小二乘问题求解中,直接进行奇异值分解会产生巨大的内存需求以及漫长的计算时间。为解决该问题,提出了一种基于迭代的并行处理方法。该方法利用奇异值分解降维的特性,通过迭代不断减小矩阵规模,直到可以直接使用奇异值分解求解。在迭代过程中,将矩阵分解为许多足够小的子矩阵,并行处理其奇异值分解过程,从而提升运行速度。实验结果表明,该方法即使是串行处理,也使得大规模最小二乘奇异值分解的时间成本及空间成本大大降低;而并行处理在双机条件下加速比接近200％。     

Control problem structure and the numerical solution of linear singular systems

Recently, a general numerical procedure has been developed for solvable systems of singular differential equationsE(t)x′(t)+F(t)x(t)=f(t). This paper shows how to exploit the structure present in many control problems to reduce the computational effort substantially. An example is worked which shows that additional reductions are possible in some cases. This research was supported in part by the Air Force Office of Scientific Research under Grant No. 84-0240.     

Boundary point method for linear elasticity using constant and quadratic moving elements

Based on the boundary integral equations and stimulated by the work of Young et al. [J Comput Phys 2005;209:290–321], the boundary point method (BPM) is a newly developed boundary-type meshless method enjoying the favorable features of both the method of fundamental solution (MFS) and the boundary element method (BEM). The present paper extends the BPM to the numerical analysis of linear elasticity. In addition to the constant moving elements, the quadratic moving elements are introduced to improve the accuracy of the stresses near the boundaries in the post processing and to enhance the analysis for thin-wall structures. Numerical tests of the BPM are carried out by benchmark examples in the two- and three-dimensional elasticity. Good agreement is observed between the numerical and the exact solutions.     

Form-finding of tensegrity structures using double singular value decomposition

A numerical form-finding procedure of tensegrity structures is developed. The only required information is the topology and the types of members. The singular value decompositions of the force density and equilibrium matrices are performed iteratively to find the feasible sets of nodal coordinates and force densities which satisfy the minimum required deficiencies of these two matrices, respectively. An approach of defining a unique configuration of tensegrity structure by specifying an independent set of nodal coordinates is provided. An explanation is given for the preservation in self-equilibrium status of the tensegrity structures under affine transformation. Two- and three-dimensional examples are illustrated to demonstrate the efficiency and robustness of the proposed method in searching stable self-equilibrium configurations of tensegrity structures.     

Handwritten digit classification using higher order singular value decomposition

In this paper we present two algorithms for handwritten digit classification based on the higher order singular value decomposition (HOSVD). The first algorithm uses HOSVD for construction of the class models and achieves classification results with error rate lower than 6%. The second algorithm uses the HOSVD for tensor approximation simultaneously in two modes. Classification results for the second algorithm are almost down at 5% even though the approximation reduces the original training data with more than 98% before the construction of the class models. The actual classification in the test phase for both algorithms is conducted by solving a series least squares problems. Considering computational amount for the test presented the second algorithm is twice as efficient as the first one.     

Approximate factorization of multivariate polynomials using singular value decomposition

We describe the design, implementation and experimental evaluation of new algorithms for computing the approximate factorization of multivariate polynomials with complex coefficients that contain numerical noise. Our algorithms are based on a generalization of the differential forms introduced by W. Ruppert and S. Gao to many variables, and use singular value decomposition or structured total least squares approximation and Gauss–Newton optimization to numerically compute the approximate multivariate factors. We demonstrate on a large set of benchmark polynomials that our algorithms efficiently yield approximate factorizations within the coefficient noise even when the relative error in the input is substantial (10⁻³).     

De-identifying facial images using singular value decomposition and projections

In this paper, two methods are presented that manipulate images to hinder automatic face identification. They partly degrade image quality, so that humans can identify the persons in a scene, while face identification algorithms fail to do so. The approaches used involve: a) singular value decomposition (SVD) and b) image projections on hyperspheres. Simulation experiments verify that these methods reduce the percentage of correct face identification rate by over 90 %. Additionally, the final image is not degraded beyond recognition by humans, in contrast with the majority of other de-identification methods.