Optimal variable shape parameter for multiquadric based RBF-FD method

In this follow up paper to our previous study in Bayona et al. (2011) [2], we present a new technique to compute the solution of PDEs with the multiquadric based RBF finite difference method (RBF-FD) using an optimal node dependent variable value of the shape parameter. This optimal value is chosen so that, to leading order, the local approximation error of the RBF-FD formulas is zero. In our previous paper (Bayona et al., 2011) [2] we considered the case of an optimal (constant) value of the shape parameter for all the nodes. Our new results show that, if one allows the shape parameter to be different at each grid point of the domain, one may obtain very significant accuracy improvements with a simple and inexpensive numerical technique. We analyze the same examples studied in Bayona et al. (2011) [2], both with structured and unstructured grids, and compare our new results with those obtained previously. We also find that, if there are a significant number of nodes for which no optimal value of the shape parameter exists, then the improvement in accuracy deteriorates significantly. In those cases, we use generalized multiquadrics as RBFs and choose the exponent of the multiquadric at each node to assure the existence of an optimal variable shape parameter.     

Numerical Study on an RBF-FD Tangent Plane Based Method for Convection–Diffusion Equations on Anisotropic Evolving Surfaces

In this paper, we present a fully Lagrangian method based on the radial basis function (RBF) finite difference (FD) method for solving convection–diffusion partial differential equations (PDEs) on evolving surfaces. Surface differential operators are discretized by the tangent plane approach using Gaussian RBFs augmented with two-dimensional (2D) polynomials. The main advantage of our method is the simplicity of calculating differentiation weights. Additionally, we couple the method with anisotropic RBFs (ARBFs) to obtain more accurate numerical solutions for the anisotropic growth of surfaces. In the ARBF interpolation, the Euclidean distance is replaced with a suitable metric that matches the anisotropic surface geometry. Therefore, it will lead to a good result on the aspects of stability and accuracy of the RBF-FD method for this type of problem. The performance of this method is shown for various convection–diffusion equations on evolving surfaces, which include the anisotropic growth of surfaces and growth coupled with the solutions of PDEs.     

Propagation of premixed laminar flames in 3D narrow open ducts using RBF-generated finite differences

Laminar flame propagation is an important problem in combustion modelling for which great advances have been achieved both in its theoretical understanding and in the numerical solution of the governing equations in 2D and 3D. Most of these numerical simulations use finite difference techniques on simple geometries (channels, ducts, ...) with equispaced nodes. The objective of this work is to explore the applicability of the radial basis function generated finite difference (RBF-FD) method to laminar flame propagation modelling. This method is specially well suited for the solution of problems with complex geometries and irregular boundaries. Another important advantage is that the method is independent of the dimension of the problem and, therefore, it is very easy to apply in 3D problems with complex geometries. In this work we use the RBF-FD method to compute 2D and 3D numerical results that simulate premixed laminar flames with different Lewis numbers propagating in open ducts.     

Stabilization of RBF-generated finite difference methods for convective PDEs

Radial basis functions (RBFs) are receiving much attention as a tool for solving PDEs because of their ability to achieve spectral accuracy also with unstructured node layouts. Such node sets provide both geometric flexibility and opportunities for local node refinement. In spite of requiring a somewhat larger total number of nodes for the same accuracy, RBF-generated finite difference (RBF-FD) methods can offer significant savings in computer resources (time and memory). This study presents a new filter mechanism, allowing such gains to be realized also for purely convective PDEs that do not naturally feature any stabilizing dissipation.     

A class of discontinuous Petrov–Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D

The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L²-norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed.     

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming （SQP） algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.     

A modified tetrahedron shape measure and its application for 3D trilateration localization in mobile cluster networks

In the error analysis of 3D trilateration localization, we constructed a new tetrahedron shape measurement method based on the condition number of the tetrahedron. This method uses algebraic operations, which is simpler than the previous methods based on complex geometric operations, and is also suitable for the shape measurement of triangle. For the trilateration localization problem in 3D space, based on tetrahedron shape measurement (TSM), we designed an algorithm of selecting anchor nodes on the hollow sphere centered at the unknown node. Extensive simulation experiments show that the tetrahedron shape measurement method proposed in this paper is effective. The anchor node selection algorithm based on tetrahedron shape measurement (TSM) can effectively suppress the iterative error problem in trilateration localization. Furthermore, the calculation of the tetrahedron condition number can be used for the deployment of anchor nodes in trilateration localization.     

RBF-FD formulas and convergence properties

The local RBF is becoming increasingly popular as an alternative to the global version that suffers from ill-conditioning. In this paper, we study analytically the convergence behavior of the local RBF method as a function of the number of nodes employed in the scheme, the nodal distance, and the shape parameter. We derive exact formulas for the first and second derivatives in one dimension, and for the Laplacian in two dimensions. Using these formulas we compute Taylor expansions for the error. From this analysis, we find that there is an optimal value of the shape parameter for which the error is minimum. This optimal parameter is independent of the nodal distance. Our theoretical results are corroborated by numerical experiments.     

荒漠绿洲区带状防护林遥感提取方法研究——以磴口为例

防护林是我国荒漠绿洲区主要植被类型，可为该地区防风固沙、水盐调控、水热平衡提供有力保障，调查防护林空间分布信息十分重要。然而荒漠绿洲防护林条带较窄、斑块面积小、分布广且零散，不易大尺度准确提取。为解决此难点，以磴口县荒漠绿洲为研究区，基于GF-2号遥感影像，采用面向对象分类技术提取防护林空间分布信息。分类前，首先基于局部方差（LV）和LV变化率（ROC）曲线，选取分割防护林的最优分割尺度。然后采用随机森林（RF）算法的袋外误差率（OOB error）及基尼系数（Gini）对包含光谱、形状和纹理的分类特征进行重要性评估并筛选特征、优化模型参数；最后，基于随机森林、CART决策树、支持向量机（SVM）、K近邻（KNN）四种分类器提取防护林空间分布信息并对比验证。结果表明：(1）采用ROC-LV曲线方法相比于遍历分割参数，可更客观更高效地筛选最优分割参数的可能值；(2）基于RF算法计算的袋外误分率和基尼系数可以有效筛除冗余特征，配合分类器参数优化，在保证分类精度的同时，有效提高分类器性能，大幅提升数据处理速度；(3）基于实测数据集对分类结果进行验证，结果显示基于随机森林算法的特征优化结合SVM分类器提取的防护林空间分布信息精度最高，生产者精度达到97.14%，总体防护林面积为208.99 km2，与实际210 km2接近，在小区块中，SVM分类器的分类效果优于其他三种分类器；(4）因GF-2影像分辨率高，并且含有近红外波段，通过波段合成得到亚米级信息，故基于面向对象的方法能够以单条林带为基本单位研究防护林林网属性，例如提取断带信息等林网结构特征。研究结论可为荒漠绿洲区带状防护林提取提供重要技术支撑。     

基于压缩感知理论的大规模MIMO系统下行信道估计中的导频优化理论分析与算法设计

针对大规模多输入多输出(multiple input multiple output, MIMO)系统信道估计中的导频设计问题,在压缩感知理论框架下,提出了一种基于信道重构错误率最小化的自适应自相关矩阵缩减参数导频优化算法.首先以信道重构错误率最小化为目标,推导了正交匹配追踪(orthogonal matching pursuit, OMP)算法下信道重构错误率与导频矩阵列相关性之间的关系,并得出优化导频矩阵的两点准则,即导频矩阵列相关性期望和方差最小化;然后研究了优化导频矩阵的方法,并提出相应的自适应自相关矩阵缩减参数导频矩阵优化算法,即在每次迭代过程中,以待优化矩阵平均列相关程度是否减小作为判断条件,调整自相关矩阵缩减参数值,使参数不断趋近于理论最优.仿真结果表明,与采用Gaussian矩阵、Elad方法、低幂平均列相关方法得到的导频矩阵相比,本文所提方法具有更好的列相关性,且具有更低的信道重构错误率.     

A Hybrid Method for Dynamic Mesh Generation Based on Radial Basis Functions and Delaunay Graph Mapping

Aiming at complex configuration and large deformation, an efficient hybrid method for dynamic mesh generation is presented in this paper, which is based on Radial Basis Functions (RBFs) and Delaunay graph mapping. Based on the computational mesh, a set of very coarse grid named as background grid is generated firstly, and then the computational mesh can be located at the background grid by Delaunay graph mapping technique. After that, the RBFs method is applied to deform the background grid by choosing partial mesh points on the boundary as the control points. Finally, Delaunay graph mapping method is used to relocate the computational mesh by employing area or volume weight coefficients. By applying different dynamic mesh methods to a moving NACA0012 airfoil, it can be found that the RBFs-Delaunay graph mapping hybrid method is as accurate as RBFs and is as efficient as Delaunay graph mapping technique. Numerical results show that the dynamic meshes for all test cases including one two-dimensional (2D) and two three-dimensional (3D) problems with different complexities, can be generated in an accurate and efficient manner by using the present hybrid method.     

On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases

We present a method to maximize the separation of two adjacent eigenfrequencies in structures with two material components. The method is based on finite element analysis and topology optimization in which an iterative algorithm is used to find the optimal distribution of the materials. Results are presented for eigenvalue problems based on the 1D and 2D scalar wave equations. Two different objectives are used in the optimization, the difference between two adjacent eigenfrequencies and the ratio between the squared eigenfrequencies. In the 1D case, we use simple interpolation of material parameters but in the 2D case the use of a more involved interpolation is needed, and results obtained with a new interpolation function are shown. In the 2D case, the objective is reformulated into a double-bound formulation due to the complication from multiple eigenfrequencies. It is shown that some general conclusions can be drawn that relate the material parameters to the obtainable objective values and the optimized designs.     

柱透镜组在小角度测量中的应用

为了实现三维角度的同时测量,提出了一种基于柱透镜组的小角度测量方法。首先,在传统激光准直法的基础上,采用柱透镜组和特殊的四面体反射镜代替平面反射镜作为合作目标,用于表征三维角度的变化。然后,利用矩阵分析光束在传播过程中的形状、位置参数变化,说明了采用柱透镜组进行小角度测量的算法。最后,在实验室条件下对扭转角的测量进行了实验验证。实验结果表明:在工作距离1.2 m,光束直径5 mm时,扭转角测量范围20',测量误差RMS值优于8",基本满足非接触小角度测量的要求,具有一定的工程实用价值。     

An Efficient Dynamic Mesh Generation Method for Complex Multi-Block Structured Grid

Aiming at a complex multi-block structured grid, an efficient dynamic mesh generation method is presented in this paper, which is based on radial basis functions (RBFs) and transfinite interpolation (TFI). When the object is moving, the multi-block structured grid would be changed. The fast mesh deformation is critical for numerical simulation. In this work, the dynamic mesh deformation is completed in two steps. At first, we select all block vertexes with known deformation as center points, and apply RBFs interpolation to get the grid deformation on block edges. Then, an arc-lengthbased TFI is employed to efficiently calculate the grid deformation on block faces and inside each block. The present approach can be well applied to both two-dimensional (2D) and three-dimensional (3D) problems. Numerical results show that the dynamic meshes for all test cases can be generated in an accurate and efficient manner.     

Statistical Inference of the Generalized Inverted Exponential Distribution under Joint Progressively Type-II Censoring

In this paper, we study the statistical inference of the generalized inverted exponential distribution with the same scale parameter and various shape parameters based on joint progressively type-II censored data. The expectation maximization (EM) algorithm is applied to calculate the maximum likelihood estimates (MLEs) of the parameters. We obtain the observed information matrix based on the missing value principle. Interval estimations are computed by the bootstrap method. We provide Bayesian inference for the informative prior and the non-informative prior. The importance sampling technique is performed to derive the Bayesian estimates and credible intervals under the squared error loss function and the linex loss function, respectively. Eventually, we conduct the Monte Carlo simulation and real data analysis. Moreover, we consider the parameters that have order restrictions and provide the maximum likelihood estimates and Bayesian inference.     

Rotational transport on a sphere: Local node refinement with radial basis functions

This paper develops an algorithm for radial basis function (RBF) local node refinement and implements it for vortex roll-up and transport on a sphere. A heuristic based on an electrostatic repulsion type principle is used to re-distribute the nodes, clustering in areas where higher resolution is needed. It is then important to have a scheme that varies the shape of the RBFs over the domain so as to counteract the effects of Runge phenomena where the nodes are sparse. The roll-up of two diametrically opposed moving vortices are studied. The performance differences between near-uniform and refined nodes are addressed in terms of convergence, time stability, and computational cost. RBF results are put into context by comparison with published results for methods such as finite volume and discontinuous Galerkin.     

A three-step system calibration procedure with error compensation for 3D shape measurement

System calibration, which usually involves complicated and time-consuming procedures, is crucial for any three-dimensional （3D） shape measurement system based on vision. A novel improved method is proposed for accurate calibration of such a measurement system. The system accuracy is improved with considering the nonlinear measurement error created by the difference between the system model and real measurement environment. We use Levenberg-Marquardt optimization algorithm to compensate the error and get a good result. The improved method has a 50% improvement of re-projection accuracy compared with our previous method. The measurement accuracy is maintained well within 1.5% of the overall measurement depth range.     

三维形貌柔性测量系统标定方法及验证

为了避免机器人模型误差对三维形貌柔性测量系统手眼标定的影响,对手眼关系的标定方法进行了研究。提出了一种融合特征点拟合的手眼标定方法。将三维形貌扫描仪安装在工业机器人末端搭建三维形貌柔性测量系统。标定时,首先利用激光跟踪仪对工业机器人末端法兰盘坐标系进行测量,得到两者转换关系;然后,利用三维形貌扫描仪和激光跟踪仪对空间固定的特征点组进行测量,利用特征点约束和基于罗德里格矩阵的算法求解两者转换关系即可间接地求解出手眼关系。基于ATOS三维扫描仪、安川HP20D机器人和API公司生产的激光跟踪仪进行了手眼标定实验,并进行了精度验证。结果表明:标定后的三维形貌柔性测量系统,其重复性测量精度(3σ)不超过0.1 mm,长度测量精度的均方根误差在0.2 mm以内,点云拼接精度优于±0.7 mm。该方法有效避免了传统手眼标定过程中会引入机器人模型误差的问题,在求解手眼关系解时采用了线性的解法,并且适用于三维形貌柔性测量系统。     

Reduced surface point selection options for efficient mesh deformation using radial basis functions

Previous work by the authors has developed an efficient method for using radial basis functions (RBFs) to achieve high quality mesh deformation for large meshes. For volume mesh deformation driven by surface motion, the RBF system can become impractical for large meshes due to the large number of surface (control) points, and so a particularly effective data reduction scheme has been developed to vastly reduce the number of surface points used. The method uses a chosen error function on the surface mesh to select a reduced subset of the surface points; this subset contains a sufficiently small number of points so as to make the volume deformation fast, and a correction function is used to correct those surface points not included. Hence, the scheme is split such that both parts are working on appropriate problems. RBFs are an excellent way of finding smooth orthogonality preserving global deformations, but are less suitable for enforcing an exact geometry for a large number of points, while a simpler approach is ideal for diffusing small changes evenly but has quality (and possibly expense) drawbacks if used for the entire volume. However, alternatives exist for the error function used to select the reduced data set, so here a comparison is made between three different options: the surface error function, the unit function and the power function. Tests run on structured and unstructured meshes show that the surface error function gives the lowest errors, but this also requires a deformed surface shape to be known in advance of the simulation. The unit and power functions both avoid the need for a deformed surface, and the unit function is shown to be superior.     

The extended Fourier transform for 2D spectral estimation.

We present a linear algebraic method, named the eXtended Fourier Transform (XFT), for spectral estimation from truncated time signals. The method is a hybrid of the discrete Fourier transform (DFT) and the regularized resolvent transform (RRT) (J. Chen et al., J. Magn. Reson. 147, 129-137 (2000)). Namely, it estimates the remainder of a finite DFT by RRT. The RRT estimation corresponds to solution of an ill-conditioned problem, which requires regularization. The regularization depends on a parameter, q, that essentially controls the resolution. By varying q from 0 to infinity one can "tune" the spectrum between a high-resolution spectral estimate and the finite DFT. The optimal value of q is chosen according to how well the data fits the form of a sum of complex sinusoids and, in particular, the signal-to-noise ratio. Both 1D and 2D XFT are presented with applications to experimental NMR signals.