基于小波算法的二阶线性定常分布参数系统的辨识

本文基于正交函数逼近方法,借助于小波变换,并利用其运算矩阵及其运算性质,研究了分布参数系统的辨识问题。将Haar小波正交基应用于分布参数系统的辨识中,经正交小波逼近变换,将原偏微分描述的分布参数系统转化为代数矩阵方程,并且,考虑了初始条件和边界条件,获得了算法简单、计算方便、具有较高精度的辨识算法,简化了分布参数系统辨识的求解过程,应用在分布参数系统辨识中不失为一种有效的分析方法。仿真实例表明了本文所提出的算法的有效性。     

Multilevel preconditioning of graph-Laplacians: Polynomial approximation of the pivot blocks inverses

We consider the discrete system resulting from mixed finite element approximation of a second-order elliptic boundary value problem with Crouzeix–Raviart non-conforming elements for the vector valued unknown function and piece-wise constants for the scalar valued unknown function. Since the mass matrix corresponding to the vector valued variables is diagonal, these unknowns can be eliminated exactly. Thus, the problem of designing an efficient algorithm for the solution of the resulting algebraic system is reduced to one of constructing an efficient algorithm for a system whose matrix is a graph-Laplacian (or weighted graph-Laplacian).     

Computing Green's function of the initial-boundary value problem for the wave equation in a layered cylinder

A new analytical method for the approximate computation of the time-dependent Green's function for the initial-boundary value problem of the three-dimensional wave equation on multi-layered bounded cylinder is suggested in this paper. The method is based on the derivation of eigenvalues and eigenfunctions for an ordinary differential equation with piecewise constant coefficients, and an approximate computation of Green's function in the form of the Fourier series with a finite number of terms relative to the orthogonal set of the derived eigenfunctions. The computational experiment confirms the robustness of the method.     

A novel finite element method for two-point boundary value problems

A finite element method is presented for solving boundary value problems for ordinary differential equations in which the general solution of the differential equation is computed first, followed by a selection procedure for the particular solution of the boundary value problem from the general solution. In this method, the discrete representation of the differential equation is a singular matrix equation, which is solved by using generalized matrix inversion. The technique is applied to both linear and nonlinear boundary value problems and to boundary value problems requiring eigenvalue evaluation. The solution of several examples involving different types of two-point boundary value problems is presented.     

Transient analysis of structural members by the CSDT riccati transfer matrix method

A method for the direct integration of the dynamic governing partial differential equations of motion for structural members is developed. This technique is called the continuous-space discrete-time (CSDT) Riccati transfer matrix method. This formulation transforms a boundary value problem of governing partial differential equations of motion into a boundary value problem of ordinary differential equations. First, a standard procedure such as finite differences is employed to discretize the time derivatives. Then, a line solution technique such as the Riccati transfer matrix method is utilized to integrate the spatial derivatives. The stability and accuracy of the CSDT Riccati transfer matrix method using the Newmark generalized acceleration formulation for time discretization is studied. For a particular class of governing equations, it is shown that the method is unconditionally stable without amplitude decay error for particular parameter values in the Newmark formulation. The method, however, exhibits period elongation error as a function of the time step. Numerical results for bar and beam example problems indicate that this may well be a viable method for calculating the dynamic response of linear structural members.     

Symbolic computation in studying stability of solutions of linear differential equations with periodic coefficients

An algorithm for symbolic computation of characteristic exponents of a linear system of differential equations with a periodic matrix represented by a power series in terms of a small parameter is discussed. The algorithm is based on the infinite determinant method. The corresponding procedures implemented in the Mathematica system and computation results related to the elliptic restricted four-body problem are presented.     

Optimal redundancy resolution of a kinematically redundant manipulator for a cyclic task

This article proposes a method for the global optimization of redundancy over the whole task period in a kinematically redundant manipulator. The necessary conditions based on the calculus of variations for integral-type criteria result in a second-order differential equation. For a cyclic task, the boundary conditions for conservative joint motions are discussed. Then, we reformulate a two-point boundary value problem to an initial value adjustment problem and suggest a numerical search method based on the iterative optimization for providing a globally optimal solution using the gradient projection method. Since the initial joint velocity is parameterized with the number of redundancy, we only search parameter values in the parameterized space using the configuration error between the initial and final time. We show through numerical examples that multiple nonhomotopic extremal solutions satisfying periodic boundary conditions exist according to initial joint velocities for the same initial configuration. Finally, we discuss an algorithm for topological liftings of the paths and demonstrate the generality of the proposed method by considering the dynamics of a manipulator.     

Quasi-trefftz spectral method for separable linear elliptic equations

A new numerical method QTSM which was earlier suggested for solving boundary value problems with elliptic operators with constant coefficients is applied to separable, second-order elliptic equations with varying coefficients. The problems in irregular domains are considered. The method combines the properties of the boundary methods with the spectral representation of the solution in the form of expansion over the eigenfunctions of some Sturm-Liouville problem. The method is tested on several one- and two-dimensional problems with exact analytic solution. The possibilities of further developments of the method are discussed.     

Numerical analysis of coupled hydromagnetic wave equations with a finite difference scheme

A finite difference scheme offering second-order accuracy is introduced to solve numerically a system of two mixed-type coupled partial differential equations with variable coefficients. The stability conditions of the scheme have been examined by both the Fourier method and the matrix method. The Fourier method via the local transform is first used to investigate parametrically the stability conditions of the proposed scheme. The stability conditions are checked point by point for the entire domain of interest without involving the convolution of the Fourier transform. These conditions are further verified by the matrix method. Since two different methods are employed, one can ensure that the stability conditions are achieved consistently. Moreover, the optimum parameters increasing the accuracy of the numerical solutions can be determined during the stability analysis. The proposed numerical algorithm has been demonstrated by a boundary value problem which considers the coupling and propagation of hydromagnetic waves in the magnetosphere.     

层次式直接边界元计算VLSI三维互连电容

文中将Ａｐｐｅｌ处理多体问题的层次式算法思想实现于直接边界元法,用以计算ＶＬＳＩ三维互连寄生电容。直接边界积分方程同时含有边界上的电势与法向电场强度,能比间接边界元法更方便地处理多介质及有限介质结构,直接边界元法的层次式计算涉及对三种边界（强加边界、自然边界与介质交界面）及两种积分核（１／ｒ与１／ｒ＾３）的处理,显著区别于基于间接边界元法、仅处理强加边界与一种分核的层次式算法。文中以边界元的层次划     

Some symbolic computation algorithms in cosmic dynamics problems

An algorithm for computing fundamental solutions to a linear system of differential equations with a periodic matrix represented by a power series in terms of a small parameter is discussed. An algorithm based on the infinite determinant method for determining boundaries between regions of stability and instability for such a system in the parameter space is also discussed. The corresponding procedures implemented in the Mathematica system and computation results related to the elliptic restricted five-body problem are presented.     

二阶中心差分粒子滤波FastSLAM算法

为改善SLAM算法中非线性系统状态估计精度不高,计算繁杂的问题,本文创新性地提出了基于二阶中心差分滤波并融合最新观测数据来产生建议分布函数的新算法。新算法基于二阶sterling插值公式处理SLAM中的非线性系统问题,无须计算雅可比矩阵,容易实现。此外,该算法使用Cholesky分解技术,在SLAM概率估计中直接依据协方差平方根因子进行传播,保证协方差矩阵正定性的同时减小了局部线性化的截断误差。仿真试验表明,在粒子数相同的情况下,二阶中心差分FastSLAM(SOFastSLAM)在不同噪声条件下的估计精度均优于FastSLAM2.0、UFastSLAM算法,且用时最少,证实了SOFastSLAM算法的优越性。     

Uniformly convergent second-order difference scheme for a singularly perturbed periodical boundary value problem

A periodic boundary value problem with a small parameter multiplying the first- and second-order derivatives is considered. The problem is discretized using a hybrid difference scheme on a Shishkin mesh. We show that the scheme is almost second-order convergent in the maximum norm, which is independent of a singular perturbation parameter. Numerical experiment supports these theoretical results.     

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.     

基于微分算子小波显式表示的分布参数系统最优逼近控制

基于微分算子在紧支撑正交小波基下的精确显式表示，给出了一种分布参数系统最优控制的逼近计算方法．将微分算子投影到小波空间，利用其矩阵表示形式，将分布参数系统的最优控制转化为集中参数系统最优控制问题．该方法不需要为边界条件重新构造基函数，在将偏微分方程转化为其常微分方程近似形式的过程中，不需要考虑边界条件的影响，因此计算方便、适用范围广，同时具有很高的精度和计算效率,可以对计算误差进行预测．利用该方法进行了基于Daubechies (db1)小波的仿真计算，并对计算结果进行了验证．     

Calcolo degli autovalori di un parametro nei problemi differenziali ordinari con condizioni ai limiti contenenti il parametro stesso

In this paper we present a numerical method for the computation of eigenvalues (real or complex) of one parameter contained in a 2^nd order linear homogeneous differential equation, with Sturm-Liouville boundary conditions. This method, particularly apt for a computer, shortens considerably the calculation time thanks to an algorithm which quickly transforms a determinant, containing the eigenvalue in all its elements, into an equivalent determinant which contains the eingenvalue only in the elements of the principal diagonal; we think that the suggested transformation has some new points and advantages in comparison with the method based on the inversion of a matrix. The method, used to computer the roots of an algebraic equation with real coefficients, shortens further the calculation time when we have to find the eigenvalues of smaller modulus. If initially the eigenvalue is also found in the boundary conditions, the calculation are carried out in a similar way. This paper ends with some examples and notes about the program in «FORTRAN» language.     

互耦效应下多组相干源的波达方位估计算法

对于实际环境中存在的多径现象和阵元间的互耦效应,提出一种互耦效应下针对相干源的波达方向估计算法。首先,通过波达方向矩阵法利用二阶矩求出互耦效应下的广义导向矢量;然后对广义导向矢量进行 子空间平滑,通过矩阵变换得到一个线性约束下的规划问题,实现相干源方位和互耦系数的级联估计。该算法只需利用二阶矩求得广义导向矢量,相比常规的四阶累积量方法,减少了计算量;本文算法在解互耦和解相干过程中都没有损失阵列孔径,极大提高了阵元利用效率。计算机仿真结果验证了该算法的有效性。     

Some simple boundary value problems with corner singularities and boundary layers

Some boundary value problems for a second-order elliptic partial differential equation in a polygonal domain are considered. The highest order terms in the equation are multiplied by a small parameter, leading to a singularly perturbed problem. The singular perturbation causes boundary layers and interior layers in the solution, and the corners of the polygon cause corner singularities in the solution. The paper considers pointwise bounds for derivatives of the solution that show the influence of these layers and corner singularities. Several recent results on this problem are surveyed, and some open problems are stated and discussed.     

Numerical solution of nonlinear system of second-order boundary value problems using cubic B-spline scaling functions

A numerical technique is presented for the solution of nonlinear system of second-order boundary value problems. This method uses the cubic B-spline scaling functions. The method consists of expanding the required approximate solution as the elements of cubic B-spline scaling function. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results.     

基于小波逼近变换的非线性分布参数系统辨识

利用Haar小波正交规范基的微分运算矩阵及其运算性质,将描述一类非线性分布参数系统的偏微分方程转化为代数矩阵方程,结合最小二乘法,确定出待辨识的系统参数,避免了对偏微分方程进行多重积分运算的繁琐;并且,可以不考虑初始条件和边界条件,较其他采用积分运算矩阵的辨识方法要简单得多,简化了分布参数系统辨识的求解过程。该方法简单,计算量小,辨识精度高。仿真结果表明了该算法应用在非线性分布参数系统辨识中的有效性。