基于GaBP算法的快速潮流计算方法

研究在潮流迭代求解过程中雅可比矩阵方程组的迭代求解方法及其收敛性。首先利用PQ分解法进行潮流迭代求解,并针对求解过程中雅可比矩阵对称且对角占优的特性,对雅可比矩阵方程组采用高斯置信传播算法(GaBP)进行求解,再结合Steffensen加速迭代法以提高GaBP算法的收敛性。对IEEE118、IEEE300节点标准系统和两个波兰互联大规模电力系统进行仿真计算后结果表明：随着系统规模的增长,使用Steffensen加速迭代法进行加速的GaBP算法相对于基于不完全LU的预处理广义极小残余方法(GMRES)具有更好的收敛性,为大规模电力系统潮流计算的快速求解提供了一种新思路。     

Convergence and numerical analysis of a family of two-step steffensen's methods

We provide sufficient conditions for the semilocal convergence of a family of two-step Steffensen's iterative methods on a Banach space. The main advantage of this family is that it does not need to evaluate neither any Fréchet derivative nor any bilinear operator, but having a high speed of convergence. Some numerical experiments are also presented.     

Fehlerabschätzung für ein Runge-Kutta-Verfahren mit mehrfachen Knoten

Kastlunger-Wanner [2] recently studied Runge-Kutta methods with multiple nodes. Their paper using Taylor-expansion generalized results ofButcher [1]. In this paper we consider a Runge-Kutta method of order four with two double nodes. Inserting appropriate integration formulas we deduce a clearly arranged representation for the truncation error. Moreover one can easily derive the conditions for the coefficients. The obtained error bound has a rather simple form — compared with the error bound for the usual (classical) Runge-Kutta method — and in some cases carries a better result.     

A Survey of Model Reduction by Balanced Truncation and Some New Results

Balanced truncation is one of the most common model reduction schemes. In this note, we present a survey of balancing related model reduction methods and their corresponding error norms, and also introduce some new results. Five balancing methods are studied: (1) Lyapunov balancing, (2) stochastic balancing, (3) bounded real balancing, (4) positive real balancing and (5) frequency weighted balancing. For positive real balancing, we introduce a multiplicative-type error bound. Moreover, for a certain subclass of positive real systems, a modified positive-real balancing scheme with an absolute error bound is proposed. We also develop a new frequency-weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians. Two numerical examples are illustrated to verify the efficiency of the proposed methods.     

Close tracking of equilibrium paths

A method to control a generic system of nonlinear ordinary differential equations between equilibrium states is analyzed. The objective is to ensure that the system's state space trajectory closely tracks an equilibrium path. The control law is obtained via time parameterization of the corresponding equilibrium control path. Conditions which guarantee that the system's state space trajectory closely tracks the equilibrium path are proved using two approaches. One approach uses the mean value theorem, and the other uses the slowly time‐varying systems theory. Importantly, both methods provide relationships between the control rate norm and the tracking error norm. These allow computation of upper bounds on the control rate norm which guarantee a desired upper bound on the tracking error norm. They also enable computation of upper bounds on the tracking error norm for a given upper bound on the control rate norm. Examples illustrate the theoretical results.     

Non-linear measurement feedback H8-control of time-periodic systems with application to tracking control of robot manipulators

In this paper, we propose a practical and effective approach to compute the worst-case norm of finite-dimensional convolution systems. System inputs are modelled to have bounded magnitude and rate limit. The computation of the worst-case norm is formulated as a fixed-terminal-time optimal control problem. Applying Pontryagin's maximum principle with the generalized Karush–Kuhn–Tucker theorem, we obtain necessary conditions which are subsequently exploited to characterize the worst-case input. Furthermore, we develop a novel algorithm called successive pang interval search (SPIS) to construct the worst-case input for general finite-dimensional convolution systems. The algorithm is guaranteed to converge and give an accurate solution within a prescribed error bound. To verify the accuracy of the algorithm, we derive bounds on computational errors including the truncation error and the discretization error. Then, the bounds on the errors yielded by our algorithm are compared with those of a comparative discrete-time method. This suggests that SPIS is deemed to be more accurate, analytically. Numerical results based on second-order linear systems show that both approaches give the worst-case norms with comparable errors, but SPIS requires much less computation time than the discrete-time method.     

Selecting a restoration technique to minimize OCR error

This paper introduces a learning problem related to the task of converting printed documents to ASCII text files. The goal of the learning procedure is to produce a function that maps documents to restoration techniques in such a way that on average the restored documents have minimum optical character recognition error. We derive a general form for the optimal function and use it to motivate the development of a nonparametric method based on nearest neighbors. We also develop a direct method of solution based on empirical error minimization for which we prove a finite sample bound on estimation error that is independent of distribution. We show that this empirical error minimization problem is an extension of the empirical optimization problem for traditional M-class classification with general loss function and prove computational hardness for this problem. We then derive a simple iterative algorithm called generalized multiclass ratchet (GMR) and prove that it produces an optimal function asymptotically (with probability 1). To obtain the GMR algorithm we introduce a new data map that extends Kesler's construction for the multiclass problem and then apply an algorithm called Ratchet to this mapped data, where Ratchet is a modification of the Pocket algorithm . Finally, we apply these methods to a collection of documents and report on the experimental results.     

Semiparametric mixed-effect least squares support vector machine for analyzing pharmacokinetic and pharmacodynamic data

In this paper we propose a semiparametric mixed-effect least squares support vector machine (LS-SVM) regression model for the analysis of pharmacokinetic (PK) and pharmacodynamic (PD) data. We also develop the generalized cross-validation (GCV) method for choosing the hyperparameters which affect the performance of the proposed LS-SVM. The performance of the proposed LS-SVM is compared with those of NONMEM and the regular semiparametric LS-SVM via four measures, which are mean squared error (MSE), mean absolute error (MAE), mean relative absolute error (MRAE) and mean relative prediction error (MRPE). Through paired-t test statistic we find that the absolute values of four measures of the proposed LS-SVM are significantly smaller than those of NONMEM for PK and PD data. We also investigate the coefficient of determinations R²'s of predicted and observed values. The R²'s of NONMEM are 0.66 and 0.59 for PK and PD data, respectively, while the R²'s of the proposed LS-SVM are 0.94 and 0.96. Through cross validation technique we also find that the proposed LS-SVM shows better generalization performance than the regular semiparametric LS-SVM for PK and PD data. These facts indicate that the proposed LS-SVM is an appealing tool for analyzing PK and PD data.     

An incomplete factorization iterative technique for elliptic equations

This paper describes efficient iterative techniques for solving the large sparse symmetric linear systems that arise from application of finite difference approximations to self-adjoint elliptic equations. We use an incomplete factorization technique with the method of D'Yakonov type, generalized conjugate gradient and Chebyshev semi-iterative methods. We compare these methods with numerical examples. Bounds for the 4-norm of the error vector of the Chebyshev semi-iterative method in terms of the spectral radius of the iteration matrix are derived.     

Balanced truncation reduced models of quadratic-bilinear systems in time interval

Accumulative error along with time impedes the application of many computational methods on high-precision simulation; therefore, time-interval methods emerge. In this paper, a new model reduction method of quadratic-bilinear (QB) systems based on time-interval Gramians (TIGs) is presented. The conditions for generalized Lyapunov equations whose solutions are exactly TIGs to be solvable are derived. Lyapunov stability and error bound are discussed to demonstrate the succession and advance of time-interval balanced truncation (TIBT). The accuracy and robustness are improved, as is illustrated in numerical results.     

Multiscale methods for the valuation of American options with stochastic volatility

This paper deals with the efficient valuation of American options. We adopt Heston's approach for a model of stochastic volatility, leading to a generalized Black–Scholes equation called Heston's equation. Together with appropriate boundary conditions, this can be formulated as a parabolic boundary value problem with a free boundary, the optimal exercise price of the option. For its efficient numerical solution, we employ, among other multiscale methods, a monotone multigrid method based on linear finite elements in space and display corresponding numerical experiments.     

Optimizing resources in model selection for support vector machine

Tuning support vector machine (SVM) hyperparameters is an important step in achieving a high-performance learning machine. It is usually done by minimizing an estimate of generalization error based on the bounds of the leave-one-out (LOO) such as radius-margin bound and on the performance measures such as generalized approximate cross-validation (GACV), empirical error, etc. These usual automatic methods used to tune the hyperparameters require an inversion of the Gram-Schmidt matrix or a resolution of an extra-quadratic programming problem. In the case of a large data set these methods require the addition of huge amounts of memory and a long CPU time to the already significant resources used in SVM training. In this paper, we propose a fast method based on an approximation of the gradient of the empirical error, along with incremental learning, which reduces the resources required both in terms of processing time and of storage space. We tested our method on several benchmarks, which produced promising results confirming our approach. Furthermore, it is worth noting that the gain time increases when the data set is large.     

Estimating time and score uncertainty in generating successful learning paths under time constraints

This paper addresses the problem of course (path) generation when a learner's available time is not enough to follow the complete course. We propose a method to recommend successful paths regarding a learner's available time and his/her knowledge background. Our recommender is an instance of long term goal recommender systems (LTRS). This method, after locating a target learner in a course graph, applies a depth‐first search algorithm to find all paths for the learner given a time limitation. In addition, our method estimates learning time and score for all paths. It also indicates the probability of error for the estimated time and score for each path. Finally, our method recommends a path that satisfies the learner's time restriction while maximizing expected learning score. In order to evaluate our proposals for time and score estimation, we used the mean absolute error and average MAE. We have evaluated time and score estimation methods, including one proposed in the literature, on two E‐learning datasets.     

An Isoperimetric Theorem for Sequences Generated by Feedback and Feedback-Codes for Unequal Error Protection

We derive an isoperimetric theorem for sequences generated by feedback and consider block codes for a binary broadcast channel with two receivers and noiseless feedback. By applying the isoperimetric theorem, we get an upper bound on the achievable rates for special cases of these codes with unequal error protection. We get a lower bound with a generalized Varshamov–Gilbert construction.     

Newton's method revisited: How accurate do we have to be?

We analyze the convergence of quasi-Newton methods in exact and finite precision arithmetic using three different techniques. We derive an upper bound for the stagnation level and we show that any sufficiently exact quasi-Newton method will converge quadratically until stagnation. In the absence of sufficient accuracy, we are likely to retain rapid linear convergence. We confirm our analysis by computing square roots and solving bond constraint equations in the context of molecular dynamics. In particular, we apply both a symmetric variant and Forsgren's variant of the simplified Newton method. This work has implications for the implementation of quasi-Newton methods regardless of the scale of the calculation or the machine.     

Region based interpolation error expansion algorithm for reversible image watermarking

In this paper, we propose a new reversible image watermarking method based on interpolation-error expansion called region based interpolation error expansion (RBIEE). We improved Thodi's prediction error expansion (PEE) technique by using a novel interpolation algorithm which exploits interpixel correlation better. Furthermore, interpolation error histogram is divided into two regions. The parameters of each region are determined separately and iteratively to reach a given embedding capacity more precisely. However, adaptive embedding strategy is utilized to get better capacity-distortion performance. Advantage of the proposed method over the other state-of-the-art methods in terms of capacity and visual quality is demonstrated experimentally.     

On a weakly singular Volterra integral equation

The need for providing reliable numerical methods for the solution of weakly singular Volterra integral equations ofI ^st Kind stems from the fact that they are connected to important problems in the theory and applications of stochastic processes. In the first section are briefly sketched the above problems and some peculiarities of such equations. Section 2 described the method for obtaining an approximate solution whose properties are described in section 3: such properties guarantee that our approximate solution always oscillates around the rigorous one. Section 4 discusses the applicability to our case of some classical bounds on the errors. The remaining sections are all devoted to the construction of upper bounds on the oscillating error in order to reach a high degree of reliability for our solution. All the bounds are independent on the numerical method which is employed for obtaining the numerical solution. In section 5 is derived a Volterra II Kind integral equation by subtracting to the original kernel the weak singularity, while in section 6 is given an upper bound to the error in the case of Wiener and Ornstein-Ühlenbeck kernels with constant barriers. Such a bound is generalized to other kinds of barriers in section 7 while in section 8 is suggested an approximation of the Kernel for the O. Ü. case with constant barriers and by means of it is given an explicit bound for the error in terms of Abel's transform of the known term in the original integral equation. A rough estimation of the error is also given under the assumption that \(y(t) - \int\limits_0^t {K(t,\tau )\tilde x(\tau )d_\tau [\tilde x(\tau )} \) denotes any approximate solution of (1a) obtained by any method] can be approximated by means of a sinusoidal function. In section 9 is derived another kind of bound, for constant barriers, by using the approximate Kernel of section 7 and classical results.     

Gradient methods exploiting spectral properties

We propose a new stepsize for the gradient method. It is shown that this new stepsize will converge to the reciprocal of the largest eigenvalue of the Hessian, when Dai-Yang's asymptotic optimal gradient method (Computational Optimization and Applications, 2006, 33(1): 73–88) is applied for minimizing quadratic objective functions. Based on this spectral property, we develop a monotone gradient method that takes a certain number of steps using the asymptotically optimal stepsize by Dai and Yang, and then follows by some short steps associated with this new stepsize. By employing one step retard of the asymptotic optimal stepsize, a nonmonotone variant of this method is also proposed. Under mild conditions, R-linear convergence of the proposed methods is established for minimizing quadratic functions. In addition, by combining gradient projection techniques and adaptive nonmonotone line search, we further extend those methods for general bound constrained optimization. Two variants of gradient projection methods combining with the Barzilai-Borwein stepsizes are also proposed. Our numerical experiments on both quadratic and bound constrained optimization indicate that the new proposed strategies and methods are very effective.     

Bounds on the optimal quantization performance of dynamically quantized linear systems with bounded noise

This paper studies a class of quantized linear control systems with diagonalizable system matrices and perturbed by bounded noise. The quantization performance of such systems is measured with the supremum of the quantization error sequence. Our goal is to improve this performance (reduce the quantization error) through designing appropriate quantization policies. Due to their efficiency, the dynamic quantization policies are considered in this paper. A lower bound on the optimal (minimum) quantization error is provided. We also propose a new quantization policy, whose quantization error is an upper bound on the optimal one. A more tractable upper quantization error bound is derived from the new policy. It is shown through simulations that the new policy's quantization error is very close to the lower bound, which confirms both the tightness of the lower bound and the efficiency of the new policy. The achieved lower and upper quantization error bounds, together with the quantization error of the new quantization policy, may provide a good indication on the optimal quantization performance. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Nonlinear dynamic analysis of frame structures

A geometrically nonlinear dynamic analysis method is presented for frames which may be subjected to finite rotations in three-dimensional space. The proposed method is based on the static geometrically nonlinear analysis method reported by Yoshida et al., in which the governing incremental equilibrium equation is represented by the coordinates after the deformation themselves rather than conventional displacements. The governing dynamic equilibrium equation for each element is obtained from the static equation by adding the inertia term. In the solution procedure, a modified Steffensen's iteration process is introduced and combined with the two-step approximation and iterative correction solution procedure developed for static analysis. A numerical example of a curved cantilever beam under lateral loads indicates the effectiveness of the proposed method in cases with three-dimensional finite rotations. Forced vibration analyses of a two-hinged shallow arch are conducted under centrally concentrated loading with several loading amplitudes. The resulting dynamic buckling load is compared with that given by Gregory and Plaut in 1982, who used Galerkin method, and shows good agreement.