三维复合介质波动方程多尺度辛几何算法

本文针对三维复合介质波动方程,提出了一类多尺度辛几何算法.其主要内容有:1.快速振荡系数三维波动方程的多尺度渐近分析与收敛性估计;2.均匀化波动方程的辛几何算法;3.多尺度辛几何算法与数值实验结果.     

多尺度多重向量值双正交小波的构建算法与性质

研究多尺度多重向量值双正交小波的构建算法与性质.运用向量细分格式、矩阵理论和多重向量值多分辨分析,证明了与一对给定的多尺度多重向量值双正交尺度函数对应的多尺度多重向量值双正交小波函数的存在性.提出了紧支撑多尺度多重向量值双正交小波的构造算法.讨论了多尺度多重向量值小波包的性质,得到了多重向量值小波包的双正交公式与向量值小波包基.     

多尺度双向向量值小波的构造与性质

本文研究多尺度双向向量值正交小波的存在性、构造算法与性质.利用多分辨分析理论,时频分析方法与矩阵理论,给出紧支撑多尺度双向向量值正交小波的构造算法,得到多尺度双向向量值小波包的正交公式与向量值小波包基.推广了向量值正交小波的概念.     

多孔结构热传导问题的并行多尺度分析

多孔材料在航空航天、汽车、机械领域应用广泛,由于其微结构的多孔性,需要发展多尺度算法用于其性能预测.本文先应用Fourier变换将多孔区域的热传导问题转换为频域空间的复值问题,然后对频域问题做多尺度渐近分析,通过构造边界层证明了频域方程的多尺度截断误差估计.进一步,本文利用孔洞填充思想提出了一套在无孔区域上研究多孔区域的统一的多尺度方法,构造了一套预测多孔材料热性能的新的并行多尺度算法,结合逆积分变换给出了整个算法的误差估计.     

交错排列结构的多尺度渐近分析

针对一类交错排列结构上的具有快速振荡系数的椭圆问题进行了多尺度渐近分析.证明了多尺度渐近展开方法的相关基础定理和多尺度解的误差估计.数值算例验证了所提出的多尺度有限元算法的有效性.进一步地,讨论了不同交错排列方式对材料等效性能的影响.     

区间[-1,1]上的a尺度双正交多小波的构造

本文研究了一元a尺度紧支撑、双正交多小波的构造.在区间[-1,1],给出了利用a尺度双正交尺度向量构造a尺度双正交多小波的推导过程得到了一种有效的小波构造算法,并给出了数值算例.     

分数阶尺度函数的分解与重构算法

引入分数阶多分辨分析与分数阶尺度函数的概念.运用时频分析方法与分数阶小波变换,研究了分数阶正交小波的构造方法,得到分数阶正交小波存在的充要条件.给出分数阶尺度函数与小波的分解与重构算法,算法比经典的尺度函数与小波的分解与重构算法更具有一般性.     

周期振荡系数热传导方程的并行多尺度有限元算法

基于复合材料中具有周期震荡系数的热传导问题的并行多尺度有限元算法,通过构造边界层证明了有界多边形凸区域上的频域方程的多尺度截断误差估计;推导了算法过程中的累积有限元误差;结合逆积分变换给出了整个算法的误差估计.通过三维数值算例验证了算法的正确性与有效性.     

广义基插值的正交多尺度函数和多小波

给出一类具有广义插值的正交多尺度函数的构造方法, 并给出对应多小波的显示构造公式. 证明了该文构造的多小波拥有与多尺度函数相同的广义基插值性.从而建立了多小波子空间上的采样定理. 最后基于该文提供的算法构造出若干具有广义基插值的正交多尺度函数和多小波.     

多尺度优势模糊目标决策系统的粗糙集、最优尺度选择及约简

多尺度决策系统的知识获取是当今的研究热点之一。然而,在处理实际数据时,多尺度决策系统中的条件属性值之间可能存在优劣关系,决策属性取值可能为模糊数。针对这一类多尺度决策系统的知识获取问题,本文构建了多尺度优势模糊目标粗糙集模型,给出了该模型的最优尺度选择算法,并讨论了获取所有最优尺度约简的分辨矩阵法和获取一个最优尺度约简的简便算法。最后将本文提出的多尺度优势模糊粗糙集模型、最优尺度选择和规则获取算法应用于计算机审计风险评估,得到较为合理的评估规则。     

Analysis of Multiscale Methods

The heterogeneous multiscale method gives a general framework for the analysis of multiscale methods. In this paper, we demonstrate this by applying this framework to two canonical problems: The elliptic problem with multiscale coefficients and the quasicontinuum method.     

Multiscale KAM theorem for Hamiltonian systems

In this paper, we study the persistence of invariant tori in nearly integrable multiscale Hamiltonian systems with highorder degeneracy in the integrable part. Such Hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems. We introduce multiscale nondegenerate condition and multiscale Diophantine condition, comparable to the usual Diophantine condition. Using quasilinear KAM method, we prove a multiscale KAM theorem.     

矩阵形式二次修正Maxwell-Dirac系统的多尺度算法

带二次修正项的Dirac方程在拓扑绝缘体、石墨烯、超导等新材料电磁光特性分析中有着十分广泛的应用.本文工作的创新点有：一是首次提出了矩阵形式带有二次修正项的Dirac方程,它是比较一般的数学框架,涵盖了上述材料体系很多重要的物理模型,具体见附录A;二是针对上述材料体系的电磁响应问题,提出了有界区域Weyl规范下具有周期间断系数矩阵形式带二次修正项Maxwell-Dirac系统的多尺度渐近方法,结合Crank-Nicolson有限差分方法和自适应棱单元方法,发展了一类多尺度算法.数值试验结果验证了多尺度渐近方法的正确性和算法的有效性.     

Multiscale asymptotic expansions and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients

In this paper, we discuss the multiscale analysis and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients. The formal multiscale asymptotic expansions of the solutions for these problems in four specific cases are presented. Higher order corrector methods are constructed and associated explicit convergence rates are obtained in some cases. A multiscale numerical method and a symplectic geometric scheme are introduced. Finally, some numerical results and unsolved problems are presented, and these numerical results support strongly the convergence theorem of this paper.     

Multiscale modeling and analysis for some special additive noises driven stochastic partial differential equations

This paper is concerned with some special additive noises driven stochastic partial differential equations with multiscale parameters. Then, the constraint energy minimizing generalized multiscale finite element method with a novel multiscale spectral representation of the noise is constructed to solve the multiscale models. The corresponding convergence analysis and error estimates are derived, and the effects of noises on the accuracy of the multiscale computation are demonstrated. Some numerical examples are provided to validate our theoretic analysis, and numerical results show the highly efficient computational performance of our method, which is a beneficial attempt to deal with the noises in the complex multiscale stochastic physical system.     

A fast multiscale Kantorovich method for weakly singular integral equations

In this paper, we use the idea of Kantorovich regularization to develop the fast multiscale Kantorovich method and the fast iterated multiscale Kantorovich method. For some kinds of weakly singular integral equations with nonsmooth inhomogeneous terms, we show that our two proposed methods can still obtain the optimal order of convergence and superconvergence order, respectively. Numerical examples are given to demonstrate the efficiency of the methods.     

Multiscale discontinuous Petrov–Galerkin method for the multiscale elliptic problems

In this article, we present a new multiscale discontinuous Petrov–Galerkin method (MsDPGM) for multiscale elliptic problems. This method utilizes the classical oversampling multiscale basis in the framework of a Petrov–Galerkin version of the discontinuous Galerkin method, allowing us to better cope with multiscale features in the solution. MsDPGM takes advantage of the multiscale Petrov–Galerkin method (MsPGM) and the discontinuous Galerkin method (DGM). It can eliminate the resonance error completely and decrease the computational costs of assembling the stiffness matrix, thus, allowing for more efficient solution algorithms. On the basis of a new H² norm error estimate between the multiscale solution and the homogenized solution with the first‐order corrector, we give a detailed convergence analysis of the MsDPGM under the assumption of periodic oscillating coefficients. We also investigate a multiscale discontinuous Galerkin method (MsDGM) whose bilinear form is the same as that of the DGM but the approximation space is constructed from the classical oversampling multiscale basis functions. This method has not been analyzed theoretically or numerically in the literature yet. Numerical experiments are carried out on the multiscale elliptic problems with periodic and randomly generated log‐normal coefficients. Their results demonstrate the efficiency of the proposed method.     

Multiscale support vector regression method in Sobolev spaces on bounded domains

In this paper, we investigate the multiscale support vector regression (SVR) method for approximation of functions in Sobolev spaces on bounded domains. The Vapnik ?-intensive loss function, which has been developed well in learning theory, is introduced to replace the standard l² loss function in multiscale least squares methods. Convergence analysis is presented to verify the validity of the multiscale SVR method with scaled versions of compactly supported radial basis functions. Error estimates on noisy observation data are also derived to show the robustness of our proposed algorithm. Numerical simulations support the theoretical predictions.     

Multiscale methods for elliptic homogenization problems

In this article we study two families of multiscale methods for numerically solving elliptic homogenization problems. The recently developed multiscale finite element method [Hou and Wu, J Comp Phys 134 (1997), 169–189] captures the effect of microscales on macroscales through modification of finite element basis functions. Here we reformulate this method that captures the same effect through modification of bilinear forms in the finite element formulation. This new formulation is a general approach that can handle a large variety of differential problems and numerical methods. It can be easily extended to nonlinear problems and mixed finite element methods, for example. The latter extension is carried out in this article. The recently introduced heterogeneous multiscale method [Engquist and Engquist, Comm Math Sci 1 (2003), 87–132] is designed for efficient numerical solution of problems with multiscales and multiphysics. In the second part of this article, we study this method in mixed form (we call it the mixed heterogeneous multiscale method). We present a detailed analysis for stability and convergence of this new method. Estimates are obtained for the error between the homogenized and numerical multiscale solutions. Strategies for retrieving the microstructural information from the numerical solution are provided and analyzed. Relationship between the multiscale finite element and heterogeneous multiscale methods is discussed. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006