模糊系统逼近理论：现状与展望

近年来,各类模糊系统已被证明是万能函数逼 近器,它们能实现任意的非线性连续控制规律和动态模型．本文综述与分析了模糊系统逼近 理论研究的最新成果,并指出了进一步研究工作的方向．     

Neural Network Method for Solving Partial Differential Equations

A method is presented to solve partial differential equations (pde's) and its boundary and/or initial conditions by using neural networks. It uses the fact that multiple input, single output, single hidden layer feedforward networks with a linear output layer with no bias are capable of arbitrarily well approximating arbitrary functions and its derivatives, which is proven by a number of authors and well known in literature. Knowledge about the pde and its boundary and/or initial conditions is incorporated into the structures and the training sets of several neural networks. In this way we obtain networks of which some are specifically structured. To find the solution of the pde and its boundary and/or initial conditions we have to train all obtained networks simultaneously. Therefore we use an evolutionary algorithm to train the networks. We demonstrate the working of our method by applying it to two problems.     

Numerically Invariant Signature Curves

Corrected versions of the numerically invariant expressions for the affine and Euclidean signature of a planar curve introduced by Calabi et al. in (Int. J. Comput. Vision, 26: 107–135, 1998) are presented. The new formulas are valid for fine but otherwise arbitrary partitions of the curve. We also give numerically invariant expressions for the four differential invariants parameterizing the three dimensional version of the Euclidean signature curve, namely the curvature, the torsion and their derivatives with respect to arc length.     

图象重建中的有理逼近方法

提出一个由不完备投影数据重建图象的有理逼近方法。该方法首先引入图象象素间连接的假设,然后导出满足最优解的线性方程组,再通过在此方程组中引入一个人工参数将其变形,并用摄动展开方法解新形成的线性方程组,再利用向量值函数的有理逼近来构造原线性方程组的解。该方法还避免了原方程组直接求解计算量非常大的问题,因为使用该有理逼近方法,只需展开几项,便可获得较满意的重建图象。     

Convergence analysis of convex incremental neural networks

Recently, a convex incremental algorithm (CI-ELM) has been proposed in Huang and Chen (Neurocomputing 70:3056–3062, 2007), which randomly chooses hidden neurons and then analytically determines the output weights connecting with the hidden layer and the output layer. Though hidden neurons are generated randomly, the network constructed by CI-ELM is still based on the principle of universal approximation. The random approximation theory breaks through the limitation of most conventional theories, eliminating the need for tuning hidden neurons. However, due to the random characteristic, some of the neurons contribute little to decrease the residual error, which eventually increase the complexity and computation of neural networks. Thus, CI-ELM cannot precisely give out its convergence rate. Based on Lee’s results (Lee et al., IEEE Trans Inf Theory 42(6):2118–2132, 1996), we first show the convergence rate of a maximum CI-ELM, and then systematically analyze the convergence rate of an enhanced CI-ELM. Different from CI-ELM, the hidden neurons of the two algorithms are chosen by following the maximum or optimality principle under the same structure as CI-ELM. Further, the proof process also demonstrates that our algorithms achieve smaller residual errors than CI-ELM. Since the proposed neural networks remove these “useless” neurons, they improve the efficiency of neural networks. The experimental results on benchmark regression problems will support our conclusions. The work is under the funding of Singapore MOE AcRF Tier 1 grant WBS No: R 252-000-221-112.     

Convergence analysis of geometrical multigrid methods for solving data-sparse boundary element equations

The convergence analysis of multigrid methods for boundary element equations arising from negative-order pseudo-differential operators is quite different from the usual finite element multigrid analysis for elliptic partial differential equations. In this paper, we study the convergence of geometrical multigrid methods for solving large-scale, data-sparse boundary element equations. In particular, we investigate multigrid methods for \(\mathcal{H}\)-matrices arising from the adaptive cross approximation to the single layer potential operator.     

Interactive classification for pre-integrated volume rendering of high-precision volume data

The pre-integrated volume rendering technique is widely used for creating high quality images. It produces good images even though the transfer function is nonlinear. Because the size of the pre-integration lookup table is proportional to the square of data precision, the required storage and computation load steeply increase for rendering of high-precision volume data. In this paper, we propose a method that approximates the pre-integration function proportional to the data precision. Using the arithmetic mean instead of the geometric mean and storing opacity instead of extinction density, this technique reduces the size and the update time of the pre-integration lookup table so that it classifies high-precision volume data interactively. We demonstrate performance gains for typical renderings of volume datasets.     

On invariant ellipsoids for discrete-time systems by saturated optimal controls

Analytical approximation of the maximal invariant ellipsoid for discrete-time linear systems with saturated optimal control is established, which is less conservative than existing computationally un-intensive results. Simultaneously, necessary and sufficient conditions for such approximation being equal to the real maximal invariant ellipsoid is presented. All results are given analytically and can easily be implemented in practice. An illustrative example is given to show the effectiveness of the proposed approach.     

Espaces métriques rationnellement présentés et complexité, le cas de l'espace des fonctions réelles uniformément continues sur un intervalle compact

We define the notion of rational presentation of a complete metric space, in order to study metric spaces from the algorithmic complexity point of view. In this setting, we study some representations of the space C[0,1] of uniformly continuous real functions over [0,1] with the usual norm: ||f||_∞ = Sup{|f(x)|; 0x1}. This allows us to have a comparison of global kind between complexity notions attached to these presentations. In particular, we get a generalization of Hoover's results concerning the Weierstrass approximation theorem in polynomial time. We get also a generalization of previous results on analytic functions which are computable in polynomial time.     

Error Estimates for Semidiscrete Finite Element Approximations of the Stokes Equations Under Minimal Regularity Assumptions

Semidiscrete (spatially discrete) finite element approximations of the Stokes equations are studied in this paper. Properties of L ², H ¹ and H ^–1 projections onto discretely divergence-free spaces are discussed and error estimates are derived under minimal regularity assumptions on the solution.