二维限定PEBI网格生成技术的研究

该文给出了二维限定PEBI网格的有关概念,对其生成技术进行了分析和研究,提出了一种简捷有效的生成算法-控制圆算法。最后给出了用于油藏数值模拟领域的PEBI网格例子,验证了该算法的正确性和有效性。     

三维PEBI网格生成的初步研究

论文给出了PEBI网格的有关概念,对三维PEBI网格生成进行了分析,提出一个定理并进行了证明。随之提出一种简捷有效的生成算法-控制球算法。最后给出了三维PEBI网格例子,验证了该算法的正确性和有效性。算法可以直接用于油藏模拟计算,并可推广到其它领域。     

Spark框架优化的大规模谱聚类并行算法

为解决谱聚类在大规模数据集上存在的计算耗时和无法聚类等性能瓶颈制约,提出了基于Spark技术的大规模数据集谱聚类的并行化算法。首先,通过单向循环迭代优化相似矩阵的构建,避免重复计算;然后,通过位置变换和标量乘法替换来优化Laplacian矩阵的构建与正规化,降低存储需求;最后,采用近似特征向量计算来进一步减少计算量。不同测试数据集上的实验结果表明：随着测试数据集的规模增加,所提算法的单向循环迭代和近似特征值计算的运行时间呈线性增长,增长缓慢,其近似特征向量计算与精确特征向量计算取得相近的聚类效果,并且算法在大规模数据集上表现出良好的可扩展性。在获得较好的谱聚类性能的基础上,改进算法提高了运行效率,有效缓解了谱聚类的计算耗时及无法聚类问题。     

三维网格模型特征向量水印嵌入

为了保证在一定鲁棒性的基础上提高三维网格模型水印算法的水印容量,提出一种基于网格拉普拉斯矩阵特征向量的三维网格模型半盲水印算法。在水印嵌入阶段,计算Tutte拉普拉斯矩阵,然后对其进行特征值分解进而得到特征向量,扰动拉普拉斯矩阵的特征向量以实现水印的嵌入。为了使水印引起的模型失真尽可能的小,在水印算法优化阶段,设计了对应特征向量矩阵的选中矩阵,并启发式地计算出水印嵌入的具体特征向量分量。在水印提取阶段,用扰动后的特征向量与水印模型的特征向量相减以实现水印信息的提取。对于规模较大的模型,先用谱聚类算法分割成较小的子网格,然后在每个子网格中逐一嵌入水印。该算法在水印提取阶段不需要原始网格模型,但需要记录更改后的特征向量,实现了水印算法的半盲检测。实验结果表明,该算法能抵抗仿射变换、随机噪声、平滑、均匀量化、裁剪等常见攻击,具有较强的鲁棒性,同时极大提升了水印负载容量。     

基于均值的谱聚类特征向量选择算法

在数据聚类当中,谱聚类是最流行的方法之一,其性能取决于所选取相关图的拉普拉斯(Laplacian)矩阵的特征向量。对于一个K类问题,Ng-Jordan-Weiss(NJW)谱聚类算法通常采用Laplacian矩阵的前K个最大特征值对应的特征向量作为数据的一种表示。然而,对于某些分类问题,这K个特征向量不一定能够很好地体现原始数据的信息。本文提出一种基于均值的谱聚类特征向量选择算法。该算法首先得出图的Laplacian矩阵的前3K个最大特征值的均值,然后选取K个离均值最近的特征值所对应的特征向量。相比传统谱聚类算法,该算法在UCI数据集上获得了较好的聚类性能。     

奇异摄动反应扩散方程数值模拟的粒子群优化算法

针对Shishkin网格方法在数值求解奇异摄动反应扩散方程时,网格过度点参数的选取具有不确定性的缺陷,提出了一种用粒子群优化(PSO)算法估计Shishkin网格参数的方法。首先基于有限差分方法,构造了以误差范数最小为目标的无约束优化问题,并用PSO算法进行了求解。该方法克服了人为选择参数的缺陷。实验结果表明：与单纯形算法相比,PSO算法在优化Shishkin网格参数时能够收敛到全局最优解;而且在最优网格参数下,奇异摄动反应扩散方程的数值结果在边界层的精度也得到了明显提高,进一步说明了所提方法的有效性和可行性。     

基于谱方法的无向赋权图剖分算法*

在多水平方法初始剖分阶段提出了一种基于谱方法的无向赋权图剖分算法SPWUG,给出了基于Lanczos迭代计算Laplacian矩阵次小特征值及特征向量的实现细节。SPWUG算法借助Laplacian矩阵次小特征值对应的特征向量,刻画了节点间相对距离,将基于非赋权无向图的Laplacian谱理论在图的剖分应用方面扩展到无向赋权图上,实现了对最小图的初始剖分。基于ISPD98电路测试基准的实验表明,SPWUG算法取得了一定性能的改进。实验分析反映了在多水平方法中,最小图上的全局近似最优剖分可能是初始图的局部最     

谱聚类算法对输入数据顺序的敏感性*

结合矩阵分析知识,还原了实施谱聚类算法过程中的矩阵表示.发现了不同数据输入顺序使得相应的Affinity矩阵及Laplacian矩阵是相似的.这样,Laplacian矩阵的特征向量生成的矩阵Y也是相似的;而以Y的行向量作为输入数据的K-平均算法依赖于初始的k个对象的选择.由此给出了导致谱聚类算法对数据输入顺序敏感的原因.     

基于拟Laplacian谱和点对拓扑特征的点模式匹配算法

针对谱匹配方法对噪声和出格点的鲁棒性较差的问题,提出了一种基于拟Laplacian谱和点对拓扑特征的点模式匹配算法。首先,用赋权图的最小生成树构造无符号Laplacian矩阵,通过对矩阵谱分解得到的特征值和特征向量表示点的特征,进而计算点的初始匹配概率;其次,利用点对拓扑特征的相似性测度来定义点对间的局部相容性,然后借助概率松弛的方法更新由拟Laplacian谱得到的匹配概率,得出匹配结果。对比实验结果表明,该方法在处理存在噪声和出格点的点集匹配上具有较高的鲁棒性。     

融合局部结构学习的大规模子空间聚类算法

常规的大规模子空间聚类算法在计算锚点亲和矩阵时忽略了数据之间普遍存在的局部结构，且在计算拉普拉斯（Laplacian）矩阵的近似特征向量时存在较大误差，不利于数据聚类。针对上述问题，提出一种融合局部结构学习的大规模子空间聚类算法（LLSC）。所提算法将局部结构学习嵌入锚点亲和矩阵的学习，从而能够综合利用全局和局部信息挖掘数据的子空间结构；此外，受非负矩阵分解（NMF）的启发，设计一种迭代优化方法以简化锚点亲和矩阵的求解过程；其次，根据Nystr?m近似方法建立锚点亲和矩阵与Laplacian矩阵的数学联系，并改进Laplacian矩阵特征向量的计算方法以提升聚类性能。相较于LMVSC(Large-scale Multi-View Subspace Clustering)、SLSR(Scalable Least Square Regression)、LSC-k(Landmark-based Spectral Clustering using k-means)和k-FSC(k-Factorization Subspace Clustering),LLSC在4个广泛使用的大规模数据集上显示出...     

基于谱抠图的遥感图像滑坡半自动提取

为统计遥感图像中滑坡区域的有效数据,提出基于谱抠图的遥感图像滑坡半自动提取方法。建立抠图拉普拉斯矩阵,计算特征向量,自动确定聚类数,利用爬山算法对图像聚类,根据特征向量和用户交互数据得到抠图成分,去除平滑项,得到前景透明度。实验结果证明该方法能够有效提取滑坡信息,准确率高,稳定性强。     

A space-filling visualization technique for multivariate small-world graphs

We introduce an information visualization technique, known as GreenCurve, for large multivariate sparse graphs that exhibit small-world properties. Our fractal-based design approach uses spatial cues to approximate the node connections and thus eliminates the links between the nodes in the visualization. The paper describes a robust algorithm to order the neighboring nodes of a large sparse graph by solving the Fiedler vector of its graph Laplacian, and then fold the graph nodes into a space-filling fractal curve based on the Fiedler vector. The result is a highly compact visualization that gives a succinct overview of the graph with guaranteed visibility of every graph node. GreenCurve is designed with the power grid infrastructure in mind. It is intended for use in conjunction with other visualization techniques to support electric power grid operations. The research and development of GreenCurve was conducted in collaboration with domain experts who understand the challenges and possibilities intrinsic to the power grid infrastructure. The paper reports a case study on applying GreenCurve to a power grid problem and presents a usability study to evaluate the design claims that we set forth.     

Approximate polynomial preconditionings applied to biharmonic equations

Applying a finite difference approximation to a biharmonic equation results in a very ill conditioned system of equations. This paper examines the conjugate gradient method used with polynomial preconditioning techniques for solving such linear systems. A new approach using an approximate polynomial preconditioner is described. The preconditioner is constructed from a series approximation based on the Laplacian finite difference matrix. A particularly attractive feature of this approach is that the Laplacian matrix consists of far fewer non-zero entries than the biharmonic finite difference matrix. Moreover, analytical estimates and computational results show that this preconditioner is more effective (in terms of the rate of convergence and the computational work required per iteration) than the polynomial preconditioner based on the original biharmonic matrix operator. The conjugate gradient algorithm and the preconditioning step can be efficiently implemented on a vector super-computer such as the CDC CYBER 205.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada Grant U0375; and in part by NASA (funded under the Space Act Agreement C99066G) while the author was visiting ICOMP, NASA Lewis Research Center.The work of this author was supported by an Izaak Walton Killam Memorial Scholarship.     

一类基于谱方法的强化学习混合迁移算法

在状态空间比例放大的迁移任务中, 原型值函数方法只能有效迁移较小特征值对应的基函数, 用于目标任务的值函数逼近时会使部分状态的值函数出现错误. 针对该问题, 利用拉普拉斯特征映射能保持状态空间局部拓扑结构不变的特点, 对基于谱图理论的层次分解技术进行了改进, 提出一种基函数与子任务最优策略相结合的混合迁移方法. 首先, 在源任务中利用谱方法求取基函数, 再采用线性插值技术将其扩展为目标任务的基函数; 然后, 用插值得到的次级基函数(目标任务的近似Fiedler特征向量)实现任务分解, 并借助改进的层次分解技术求取相关子任务的最优策略; 最后, 将扩展的基函数和获取的子任务策略一起用于目标任务学习中. 所提的混合迁移方法可直接确定目标任务部分状态空间的最优策略, 减少了值函数逼近所需的最少基函数数目, 降低了策略迭代次数, 适用于状态空间比例放大且具有层次结构的迁移任务. 格子世界的仿真结果验证了新方法的有效性.     

MultiGrid Preconditioners for Mixed Finite Element Methods of the Vector Laplacian

Due to the indefiniteness and poor spectral properties, the discretized linear algebraic system of the vector Laplacian by mixed finite element methods is hard to solve. A block diagonal preconditioner has been developed and shown to be an effective preconditioner by Arnold et al. (Acta Numer 15:1–155, 2006). The purpose of this paper is to propose alternative and effective block diagonal and approximate block factorization preconditioners for solving these saddle point systems. A variable V-cycle multigrid method with the standard point-wise Gauss–Seidel smoother is proved to be a good preconditioner for the discrete vector Laplacian operator. The major benefit of our approach is that the point-wise Gauss–Seidel smoother is more algebraic and can be easily implemented as a black-box smoother. This multigrid solver will be further used to build preconditioners for the saddle point systems of the vector Laplacian. Furthermore it is shown that Maxwell’s equations with the divergent free constraint can be decoupled into one vector Laplacian and one scalar Laplacian equation.     

Random-Walk Computation of Similarities between Nodes of a Graph with Application to Collaborative Recommendation

This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted and undirected graph. It is based on a Markov-chain model of random walk through the database. More precisely, we compute quantities (the average commute time, the pseudoinverse of the Laplacian matrix of the graph, etc.) that provide similarities between any pair of nodes, having the nice property of increasing when the number of paths connecting those elements increases and when the "length" of paths decreases. It turns out that the square root of the average commute time is a Euclidean distance and that the pseudoinverse of the Laplacian matrix is a kernel matrix (its elements are inner products closely related to commute times). A principal component analysis (PCA) of the graph is introduced for computing the subspace projection of the node vectors in a manner that preserves as much variance as possible in terms of the Euclidean commute-time distance. This graph PCA provides a nice interpretation to the "Fiedler vector," widely used for graph partitioning. The model is evaluated on a collaborative-recommendation task where suggestions are made about which movies people should watch based upon what they watched in the past. Experimental results on the MovieLens database show that the Laplacian-based similarities perform well in comparison with other methods. The model, which nicely fits into the so-called "statistical relational learning" framework, could also be used to compute document or word similarities, and, more generally, it could be applied to machine-learning and pattern-recognition tasks involving a relational database