Multi-scale Discontinuous Galerkin Method for Solving Elliptic Problems with Curvilinear Unidirectional Rough Coefficients

In this paper, we propose a multi-scale discontinuous Galerkin (DG) method for second-order elliptic problems with curvilinear unidirectional rough coefficients by choosing a special non-polynomial approximation space. The key ingredient of the method lies in the incorporation of the local oscillatory features of the differential operators into the approximation space so as to capture the multi-scale solutions without having to resolve the finest scales. The unidirectional feature of the rough coefficients allows us to construct the basis functions of the DG non-polynomial approximation space explicitly, thereby greatly increasing the algorithm efficiency. Detailed error estimates for two-dimensional second-order DG methods are derived, and a general guidance on how to construct such non-polynomial basis is discussed. Numerical examples are also presented to validate and demonstrate the effectiveness of the algorithm.     

Two-Mode Adaptive Fuzzy Control With Approximation Error Estimator

In this paper, we propose a two-mode adaptive fuzzy controller with approximation error estimator. In the learning mode, the controller employs some modified adaptive laws to tune the fuzzy system parameters and an approximation error estimator to compensate for the inherent approximation error. In the operating mode, the fuzzy system parameters are fixed, only the estimator is updated online. Mathematically, we show that the closed-loop system is stable in the sense that all the variables are bounded in both modes. We also establish mathematical bounds on the tracking error, state vector, control signal and the RMS error. Using these bounds, we show that controller's design parameters can be chosen to achieve desired control performance. After that, an algorithm to automatically switch the controller between two modes is presented. Finally, simulation studies of an inverted pendulum system and a Chua's chaotic circuit demonstrate the usefulness of the proposed controller.     

基于自适应鲸鱼优化算法和容错邻域粗糙集的特征选择算法

针对传统鲸鱼优化算法(WOA)不能有效处理连续型数据、邻域粗糙集对噪声数据的容错性较差等问题,文中提出基于自适应WOA和容错邻域粗糙集的特征选择算法.首先,为了避免WOA过早陷入局部最优,基于迭代周期构建分段式动态惯性权重,改进WOA的收缩包围和螺旋捕食行为,设计自适应WOA.然后,为了解决邻域粗糙集对噪声数据缺乏容错性的问题,引入邻域内相同决策特征所占的比例,定义容错邻域上下近似集、容错近似精度和近似粗糙度、容错依赖度及容错近似条件熵.最后,基于容错邻域粗糙集构造适应度函数,使用自适应WOA,不断迭代以获取最优子群.高维数据集上采用费雪评分算法进行初步降维,降低算法的时间复杂度.在8个低维UCI数据集和6个高维基因数据集上的实验表明,文中算法可有效选择特征个数较少且分类精度较高的特征子集.     

Shape‐adaptive 3‐D mesh simplification based on local optimality measurement

Mesh simplification is the process of reducing the number of triangles in a mesh representation of object surface. For a given level of detail or error tolerance, the conventional mesh simplification algorithms maximize the edge length globally, without explicitly considering local object shape. In this paper, we present a shape‐adaptive mesh simplification algorithm that locally maximizes edge length, depending on local shape. The proposed algorithm achieves shape‐adaptive simplification by iteratively maximizing edges between vertices, based on comparison with the ‘optimal’ edge lengths derived from local directional curvatures for a given error tolerance. Edge‐based processing facilitates the local shape adaptation and preserves sharp features. Experimental results demonstrate the efficacy of the proposed algorithm, by showing good visual quality and extremely small approximation error. Copyright © 2003 John Wiley & Sons, Ltd.     

Stochastic grid bundling method for backward stochastic differential equations

ABSTRACT

In this work, we apply the Stochastic Grid Bundling Method (SGBM) to numerically solve backward stochastic differential equations (BSDEs). The SGBM algorithm is based on conditional expectations approximation by means of bundling of Monte Carlo sample paths and a local regress-later regression within each bundle. The basic algorithm for solving the backward stochastic differential equations will be introduced and an upper error bound is established for the local regression. A full error analysis is also conducted for the explicit version of our algorithm and numerical experiments are performed to demonstrate various properties of our algorithm.     

On efficient simulation of non-Newtonian flow in saturated porous media with a multigrid adaptive refinement solver

Flow of non-Newtonian fluid in saturated porous media can be described by the continuity equation and the generalized Darcy law. Here we discuss the efficient solution of the resulting second order nonlinear elliptic equation. The equation is discretized by the finite volume method on a cell-centered grid. Local adaptive refinement of the grid is introduced in order to reduce the number of unknowns. We develop a special implementation, that allows us to perform unstructured local refinement in conjunction with the finite volume discretization. Two residual based error indicators are exploited in the adaptive refinement criterion. Second order accurate discretization of the fluxes on the interfaces between refined and non-refined subdomains, as well as on the boundaries with Dirichlet boundary condition, are presented here as an essential part of an accurate and efficient algorithm. A nonlinear full approximation storage multigrid algorithm is developed especially for the above described composite (coarse plus locally refined) grid approach. In particular, second order approximation of the fluxes around interfaces is a result of a quadratic approximation of slave nodes in the multigrid-adaptive refinement (MG-AR) algorithm. Results from numerical solution of various academic and practice-induced problems are presented and the performance of the solver is discussed.     

Adaptive finite volume methods for displacement problems in porous media

In this paper we consider adaptive numerical simulation of miscible and immiscible displacement problems in porous media, which are modeled by single and two phase flow equations. Using the IMPES formulation of the two phase flow equation both problems can be treated in the same numerical framework. We discretise the equations by an operator splitting technique where the flow equation is approximated by Raviart-Thomas mixed finite elements and the saturation or concentration equation by vertex centered finite volume methods. Using a posteriori error estimates for both approximation schemes we deduce an adaptive solution algorithm for the system of equations and show the applicability in several examples.     

Approximation of high-degree and procedural curves

The objective of this paper is to present an efficient adaptive algorithm to approximate high-degree and procedural continuous parametric curves by integral B-splines. This approximation algorithm covers nonperiodic and periodic curves. The approximation algorithm is motivated and accompanied by an extensive discussion on approximation errors for position and derivatives accuracies. This discussion includes the derivation of local error bounds for position and derivative errors. The practicality and efficiency of the algorithm is demonstrated for a variety of examples in geometric modeling of engineering structures. The approximation algorithm for the curve forms the basis of a similar surface approximation procedure.     

An Adaptive Finite Element Method for the Wave Scattering with Transparent Boundary Condition

Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions. The model is formulated as a boundary value problem for the Helmholtz equation with a transparent boundary condition. Based on a duality argument technique, an a posteriori error estimate is derived for the finite element method with the truncated Dirichlet-to-Neumann boundary operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of boundary operator which decays exponentially with respect to the truncation parameter. A new adaptive finite element algorithm is proposed for solving the acoustic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are marked through the finite element discretization error. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method.     

基于三支决策的自适应粗糙集近似

讨论基于三支决策的自适应粗糙集近似方法,给出自适应粗糙集近似属性评价函数的算法。为清楚表达在粗糙集近似过程中对象和属性之间的关系,引入描述集的概念,在描述集的意义下根据属性评价函数对目标概念进行自适应粗糙集近似。最后,把自适应粗糙集近似过程与认知过程进行比较,并分析它们的相似性,借助该近似过程的思想,在一定程度上提高人们的认知效率。     

Some Recent Developments in Superconvergence of Discontinuous Galerkin Methods for Time-Dependent Partial Differential Equations

In this paper, we briefly review some recent developments in the superconvergence of three types of discontinuous Galerkin (DG) methods for time-dependent partial differential equations: the standard DG method, the local discontinuous Galerkin method, and the direct discontinuous Galerkin method. A survey of our own results for various time-dependent partial differential equations is presented and the superconvergence phenomena of the aforementioned three types of DG solutions are studied for: (i) the function value and derivative approximation at some special points, (ii) cell average error and supercloseness.     

基于自适应空间刨分的网格简化算法

提出了一种基于自适应空间刨分的网格简化算法，算法首先对模型中的所有的顶点进行量化赋予一个二次误差阵，并将它们视为一个簇，然后沿坐标轴方向将它们刨分成八个子簇并不断迭代刨分生成新的子簇直至达到指定的精度，将最终的离散点集用适当的方法重新进行三角化，得到简化模型，该算法不仅速度快，能在任意限定的时间内产生一个可显示的结果，而且结果质量也很好.另外，本文还用给出的实例与其他相关算法进行了比较．     

图像阈值化的自适应粗糙熵方法

摘 要：目的：图像阈值化将灰度图像转换为二值图像,被广泛应用于多个领域。因实际工程应用中固有的不确定性,自动阈值选择仍然是一个极具挑战的课题。针对图像自动阈值化问题,提出了一种利用粗糙集的自适应方法。方法：该方法分析了基于粗糙集的图像表示框架,建立了图像粗糙粒度与局部灰度标准差的相互关系,通过最小化自适应粗糙粒度准则获得最优的划分粒度。进一步在该粒度下构造了图像目标和背景的上下近似集及其粗糙不确定度,通过搜索灰度级最大化粗糙熵获得图像最优灰度阈值,并将图像目标和背景的边界作为过渡区,利用其灰度均值作为阈值完成图像二值化。结果：对所提出的方法通过多个图像分三组进行了实验比较,包括三种经典阈值化方法和一种利用粗糙集的方法。其中,所提出的方法生成的可视化二值图像结果远远优于传统粗糙集阈值化方法。此外,也采用了误分率、平均结构相似性、假阴率和假阳率等指标进一步量化评估与比较相关实验结果。定性和定量的实验结果表明,所提出方法的图像分割质量较高、性能稳定。结论：所提出的方法适应能力较好,具有合理性和有效性,可以作为现有经典方法的有力补充。     

Least mean square algorithms with Markov regime-switching limit

We analyze the tracking performance of the least mean square (LMS) algorithm for adaptively estimating a time varying parameter that evolves according to a finite state Markov chain. We assume the Markov chain jumps infrequently between the finite states at the same rate of change as the LMS algorithm. We derive mean square estimation error bounds for the tracking error of the LMS algorithm using perturbed Lyapunov function methods. Then combining results in two-time-scale Markov chains with weak convergence methods for stochastic approximation, we derive the limit dynamics satisfied by continuous-time interpolation of the estimates. Unlike most previous analyzes of stochastic approximation algorithms, the limit we obtain is a system of ordinary differential equations with regime switching controlled by a continuous-time Markov chain. Next, to analyze the rate of convergence, we take a continuous-time interpolation of a scaled sequence of the error sequence and derive its diffusion limit. Somewhat remarkably, for correlated regression vectors we obtain a jump Markov diffusion. Finally, two novel examples of the analysis are given for state estimation of hidden Markov models (HMMs) and adaptive interference suppression in wireless code division multiple access (CDMA) networks.     

A posteriori error analysis of an augmented fully-mixed formulation for the stationary Boussinesq model

In this paper we undertake an a posteriori error analysis along with its adaptive computation of a new augmented fully-mixed finite element method that we have recently proposed to numerically simulate heat driven flows in the Boussinesq approximation setting. Our approach incorporates as additional unknowns a modified pseudostress tensor field and an auxiliary vector field in the fluid and heat equations, respectively, which possibilitates the elimination of the pressure. This unknown, however, can be easily recovered by a postprocessing formula. In turn, redundant Galerkin terms are included into the weak formulation to ensure well-posedness. In this way, the resulting variational formulation is a four-field augmented scheme, whose Galerkin discretization allows a Raviart–Thomas approximation for the auxiliary unknowns and a Lagrange approximation for the velocity and the temperature. In the present work, we propose a reliable and efficient, fully-local and computable, residual-based a posteriori error estimator in two and three dimensions for the aforementioned method. Standard arguments based on duality techniques, stable Helmholtz decompositions, and well-known results from previous works, are the main underlying tools used in our methodology. Several numerical experiments illustrate the properties of the estimator and further validate the expected behavior of the associated adaptive algorithm.     

On stochastic gradient and subgradient methods with adaptive steplength sequences

Traditionally, stochastic approximation (SA) schemes have been popular choices for solving stochastic optimization problems. However, the performance of standard SA implementations can vary significantly based on the choice of the steplength sequence, and in general, little guidance is provided about good choices. Motivated by this gap, we present two adaptive steplength schemes for strongly convex differentiable stochastic optimization problems, equipped with convergence theory, that aim to overcome some of the reliance on user-specific parameters. The first scheme, referred to as a recursive steplength stochastic approximation (RSA) scheme, optimizes the error bounds to derive a rule that expresses the steplength at a given iteration as a simple function of the steplength at the previous iteration and certain problem parameters. The second scheme, termed as a cascading steplength stochastic approximation (CSA) scheme, maintains the steplength sequence as a piecewise-constant decreasing function with the reduction in the steplength occurring when a suitable error threshold is met. Then, we allow for nondifferentiable objectives but with bounded subgradients over a certain domain. In such a regime, we propose a local smoothing technique, based on random local perturbations of the objective function, that leads to a differentiable approximation of the function. Assuming a uniform distribution on the local randomness, we establish a Lipschitzian property for the gradient of the approximation and prove that the obtained Lipschitz bound grows at a modest rate with problem size. This facilitates the development of an adaptive steplength stochastic approximation framework, which now requires sampling in the product space of the original measure and the artificially introduced distribution.     

基于局部梯度特征的自适应多结点样条图像插值

为了获得质量更好的插值图像,提出了一种新的C^2。连续的支撑区间为（-2,2）的三次多结点样条插值核函数．通过增加结点带来的自由度构造了多结点样条插值公式;分析了在适当的边界条件和约束下三次多结点样条插值的逼近阶;将一维多结点样条插值算法推广到二维,建立了用于图像数据的插值公式;如果忽视图像的局部特征,通常双三次多结点样条插值图像的边缘会有模糊的现象,为此。对多结点样条插值应用逆梯度,得到了自适应多结点样条插值算法;实验所得误差图像和实验所得图像的峰值信噪比也证实了用自适应多结点样条插值算法重建的图像具有更高的质量．     

Localized adaptive bounds for approximation-based backstepping

Recent research has established the utility of adaptive bounds on model uncertainty in adaptive approximation-based control. Such bounds have utility both for robust control law design and for self-organizing approximators that could adjust the number of basis elements N by adding additional approximation resources in the regions where the approximation error bound is large. Existing adaptive bounding methods utilize algorithms with global forgetting. In this article, we investigate methods to develop bounds on approximation accuracy that involve local forgetting. The importance of local versus global forgetting is motivated in the text and illustrated with an example.     

Adaptive dimension splitting algorithm for three-dimensional elliptic equations

An adaptive dimension splitting algorithm for three-dimensional (3D) elliptic equations is presented in this paper. We propose residual and recovery-based error estimators with respect to X?Y plane direction and Z direction, respectively, and construct the corresponding adaptive algorithm. Two-sided bounds of the estimators guarantee the efficiency and reliability of such error estimators. Numerical examples verify their efficiency both in estimating the error and in refining the mesh adaptively. This algorithm can be compared with or even better than the 3D adaptive finite element method with tetrahedral elements in some cases. What is more, our new algorithm involves only two-dimensional mesh and one-dimensional mesh in the process of refining mesh adaptively, and it can be implemented in parallel.     

A modified trust region algorithm for nonlinear equations with new updating rule of trust region radius

In this article, we present a trust region algorithm for the nonlinear equations with a new updating rule of the trust region radius, which takes some function of the residual. We show that under the local error bound condition which is weaker than the non-singularity, the new algorithm converges quadratically to some solution of the nonlinear equations. Numerical results show that the new algorithm performs very well for some singular nonlinear equations.