高维数据非参数密度估计的低维流形代表点法     

王树良  李英  耿晶 《武汉大学学报(信息科学版)》2021,46(1):65-70

非参数核方法由于采用统一的度量标准,在大数据中利用高维样本数据学习时容易遭遇维数灾难问题。挖掘高维空间中的低维几何特性,有助于揭示数据分布的流形结构,进而利用有限样本的高维数据在低维子空间逼近数据的真实分布。基于此,提出一种新的高维数据密度非参数估计的低维流形代表点法,通过从高维空间中挖掘数据分布的几何结构来估计密度。首先,通过寻找局部区域内能够代表流形结构主方向的点,计算局部协方差矩阵,描述局部的数据分布;然后,考虑流形结构中附近数据点不同的影响,根据每个样本数据点对密度的贡献进行加权。与传统的核密度估计方法和流形核密度方法进行了对比实验,结果表明,该方法能够快速稳健地进行密度估计,反映数据的真实分布。  相似文献   

Energy‐work‐based numerical manifold seepage analysis with an efficient scheme to locate the phreatic surface     

Yuan Wang  Mengsu Hu  Quanlin Zhou  Jonny Rutqvist 《国际地质力学数值与分析法杂志》2014,38(15):1633-1650

A major challenge in seepage analysis is to locate the phreatic surface in an unconfined aquifer. The phreatic surface is unknown and assumed as a discontinuity separating the seepage domain into dry and wet parts, thus should be determined iteratively with special schemes. In this study, we systematically developed a new numerical manifold method (NMM) model for unconfined seepage analysis. The NMM is a general numerical method for modeling continuous and discontinuous deformation in a unified mathematical form. The novelty of our NMM model is rooted in the NMM two‐cover‐mesh system: the mathematical covers are fixed and the physical covers are adjusted with iterations to account for the discontinuity feature of the phreatic surface. We developed an energy‐work seepage model, which accommodates flexible approaches for boundary conditions and provides a form consistent with that in mechanical analysis with clarified physical meaning of the potential energy. In the framework of this energy‐work seepage model, we proposed a physical concept model (a pipe model) for constructing the penalty function used in the penalty method to uniformly deal with Dirichlet, Neumann, and material boundaries. The new NMM model was applied to study four example problems of unconfined seepage with varying geometric shape, boundary conditions, and material domains. The comparison of our simulation results to those of existing numerical models for these examples indicates that our NMM model can achieve a high accuracy and faster convergence speed with relatively coarse meshes. This NMM seepage model will be a key component of our future coupled hydro‐mechanical NMM model. Copyright © 2014 John Wiley & Sons, Ltd.  相似文献   

Breakdown of the slow manifold in the Shallow-Water equations     

Irad Yavneh  James C. Mcwilliams 《地球物理与天体物理流体动力学》2013,107(2-4):131-161

Abstract

Numerical solutions are obtained by implicit multigrid solvers for initial-value problems in the rotating Shallow-Water Equations (SWE) with spatially complex initial conditions. Companion solutions are also obtained with the Shallow-Water Balance Equations (SWBE), both to determine the initial conditions for the SWE and to provide a comparison solution that lies entirely on the slow, advective manifold. We make use of a control parameter (here the Rossby number, R) to regulate the degree of slowness and balance. While there are measurable discrepancies between the evolving SWE and SWBE solutions for all R, there is a distinct, spatially local breakdown both of the slow manifold in the SWE solution and in the closeness of correspondence between the SWE and SWBE solutions. This critical value for breakdown is only slightly smaller than the R values at which, first, the SWE evolution becomes singular (i.e., the fluid depth vanishes), or second, a consistent initial condition for the SWBE cannot be defined. This breakdown is most clearly evident in a sudden increase in vertical velocity near the center of a strong, cyclonic vortex; its behavior is primarily associated with an enhanced dissipation rather than an initiation of gravity-wave propagation. The numerical performance of the multigrid solvers is satisfactory even in the difficult circumstances near solution breakdown or singularity.  相似文献   

利用偏最小二乘方法修复高光谱影像等距映射降维中遗失点的坐标     

孙伟伟  刘春  施蓓琦  李巍岳 《武汉大学学报(信息科学版)》2012,37(5):550-554

将Isomap流形学习方法应用于高光谱影像非线性降维时,在构建最短路径过程中,其边界点往往被忽略而没有低维流形坐标。对此,引入偏最小二乘方法来模拟修复遗失点的流形坐标,并从两个方面进行了综合评价。实验结果表明,模拟流形坐标与实际坐标吻合很好。  相似文献   

用流形坐标差异图提取高光谱影像潜在特征     

孙伟伟  刘春  施蓓琦  李巍岳 《遥感学报》2013,17(6):1327-1443

等距映射和局部切空间排列降维后,低维流形坐标能够保留原始高光谱影像中地物光谱信息,用于提取原始影像的潜在特征。然而这两种流形方法的理论差异导致其低维坐标继承光谱信息的能力不同,对比这两种流形坐标可凸显出原始影像内部的潜在特征。因此,本文基于等距映射和局部切空间排列非线性降维,提出两种流形坐标的差异图法来提取高光谱影像内部的潜在特征。首先,根据流形坐标的光谱解释确定两种坐标的每一维代表相同的光谱信息。其次,根据相同的光谱特征,归一化两种流形坐标并调整坐标轴方向,统一两种坐标到相同的坐标框架。最后,通过加权流形图相减得到坐标差异图,采用经典的图像处理方法提取潜在特征。采用两个实验并对比等距映射和局部切空间排列方法的降维结果来验证本文方法。结果表明,流形坐标差异图能够成功提取单一流形结果无法得到的潜在特征,如靠河岸的浅水区域和大场景沼泽地中的低分辨率道路。这为高光谱影像的潜在特征提取研究提供了一种新方法。  相似文献   

On target for Venus – set oriented computation of energy efficient low thrust trajectories     

Michael Dellnitz  Oliver Junge  Marcus Post  Bianca Thiere 《Celestial Mechanics and Dynamical Astronomy》2006,95(1-4):357-370

Recently new techniques for the design of energy efficient trajectories for space missions have been proposed that are based on the circular restricted three body problem as the underlying mathematical model. These techniques exploit the structure and geometry of certain invariant sets and associated invariant manifolds in phase space to systematically construct energy efficient flight paths. In this paper, we extend this model in order to account for a continuously applied control force on the spacecraft as realized by certain low thrust propulsion systems. We show how the techniques for the trajectory design can be suitably augmented and compute approximations to trajectories for a mission to Venus.  相似文献   

数值流形方法及在川滇地区地壳运动研究中的应用(英文)     

Caijun Xu  Changrong Yi  Ting Chen 《地球空间信息科学学报》2008,11(3):157-161

The numerical manifold method （NMM） can calculate the movements and deformations of structures or materials. Both the finite element method （FEM） for continua and the discontinuous deformation analysis （DDA） for block systems are special cases of NMM. NMM has separate mathematical covers and physical meshes： the mathematical covers define only fine or rough approximations; as the real material boundary, the physical mesh defines the integration fields. The mathematical covers are triangle units; the physical mesh includes the fault boundaries, joints, blocks and interfaces of different crust zones on the basis of a geological tectonic background. Aiming at the complex problem of continuous and discontinuous deformation across the Chinese continent, the numerical manifold method （NMM） is brought in to study crustal movement of the Stchuan-Yunnan area. Based on the GPS velocity field in the Sichuan-Yunnan area, a crustal strain and stress field is simulated and analyzed. Moreover, results show that the NMM is a more suitable method than DDA in simulating the movement of the Sichuan-Yunnan area. Finally, a kind of mechanism of crustal motion in the Sichuan-Yunnan area is discussed in the paper.  相似文献   

数值流形方法及其与耦合方法的比较     

张勇慧  平扬  刘明智 《岩土力学》2004,25(Z2):93-96

介绍了数值流形方法的基本原理,讨论了数学覆盖、物理覆盖、权函数和位移函数的概念,比较了该方法与DDA和FEM的耦合方法在对一维特性函数逼近的精确度,并用这两种方法模拟了地下放顶煤开挖的过程,最后的结果显示数值流形方法具有较好的相容性和精确度.  相似文献   

数值流形方法中覆盖位移函数的改进     

周小义  邓安福  黄润英 《岩土工程技术》2007,21(1):1-3

通过对数值流形方法刚度矩阵形成的研究,提出一种改进的覆盖位移函数,以改善刚度矩阵计算。分析表明,改进的覆盖位移函数可使刚度矩阵的局部大数数量级明显的降低,提高计算效率,给矩阵求解提供适当的意见,并给出算例验证方法的正确性。  相似文献   

Barrier destruction and Lagrangian predictability at depth in a meandering jet     

G. -C. Yuan  L. J. Pratt  C. K. R. T. Jones 《Dynamics of Atmospheres and Oceans》2002,35(1)

Numerical simulations of a jet with large amplitude meanders are used to explore chaotic advection processes and underlying geometry changes as functions of the ambient potential vorticity gradient β. Variations in β in the 2D model qualitatively simulate changes in depth in 3D, surface-intensified jets such as the Gulf Stream. As β is reduced, corresponding to motion on increasingly deep isopycnal surfaces, a number of geometrical transitions take place in the flanges and across the core of the jet. The most important is a joining (or separatrix reconnection) of heteroclinic cat’s eyes structures lying to the north and south of the jet core. The jet core acts as a barrier to transport, but this barrier is breached when the cat’s eyes merge. The subsequent chaotic transport across the jet is demonstrated by calculations of effective invariant manifolds (EIMs) originating in hyperbolic regions to the north and south of the core. Destruction of the central barrier occurs as β is lowered through a narrow window W about β=0 and is marked by transitions form a meandering jet through a vortex street with no central meandering flow to a vortex street with a retrograde meander. Such small values of β are deemed reasonable in view of measurements of low potential vorticity gradients in the deep Gulf Stream. The strength of the central barrier for β outside W is tested by varying β about a mean value β₀ and detecting the minimum amplitude of fluctuation necessary for destruction of the barrier. It is found that the barrier is stronger for β₀>0, at least by this measure. A striking difference is that, for β<0, some disturbances may destroy the barrier without oscillating across W; whereas for β>0, destruction of the barrier may only occur when β passes through W. Changes in underlying geometry also occur in the flanges of the jet and these changes alter the locations in which fluid is preferentially stirred and mixed. Float trajectories can be regular or irregular depending upon where the instrument is launched and this is demonstrated by plotting trajectories from inside and outside regions of chaotic advection.  相似文献