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.     

Non-linear transformations of stochastic processes associated with Fourier transforms of band-limited positive functions†

The passage of Gaussian, processes (both stationary and non-stationary) through a device characterized by a convolution transform is studied. The output of such a device is a non-stationary stochastic process of harmonizable type. This process along with its finite Fourier transform are examined in detail. Some sampling theorems are also stated. A stochastic series representation in terms of prolate spheroidal wavefunctions is derived.     

Approximation of stochastic partial differential equations by a kernel-based collocation method

In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations (SPDEs). Using an implicit time-stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on the numerical solution of the elliptic equations at each time step. The estimator of the solution of the elliptic equations is given as a linear combination of reproducing kernels derived from the differential and boundary operators of the SPDE centred at collocation points to be chosen by the user. The random expansion coefficients are computed by solving a random system of linear equations. Numerical experiments demonstrate the feasibility of the method.     

Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators

This paper deals with the construction of a class of non-Gaussian positive-definite matrix-valued random fields whose mathematical properties allow elliptic stochastic partial differential operators to be modeled. The properties of this class is studied in details and the numerical procedure for constructing numerical realizations of the trajectories is explicitly given. Such a matrix-valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters. The class of random fields which can be approximated is presented and their experimental identification is analyzed. An example is given in three-dimensional linear elasticity for which the fourth-order elasticity tensor-valued random field is constructed for a random non-homogeneous anisotropic elastic material.     

基于局部熵双子空间的多模态过程故障检测

为了提高非高斯工业过程的检测性能, 提出局部熵双子空间(LEDS)的多模态过程故障检测方法. 运用局部 概率密度估计构建数据的局部熵矩阵, 消除数据的多模态特性. 用Kolmogorov-Smirnov (KS)检验局部熵数据中变 量的正态分布特性, 对高斯分布和非高斯分布的数据分别建立基于PCA的高斯子空间和ICA的非高斯子空间故障 检测模型. 利用Bayesian决策将检测结果转化成发生故障概率的形式, 将检测结果组合成最终的统计信息, 进行故 障检测. 将该方法应用于数值例子和田纳西–伊斯曼多模态过程, 仿真结果表明, 该方法在误报率较低的情况下, 故 障检测率最高, 优于PCA、局部熵PCA(LEPCA)和局部熵ICA(LEICA)方法.     

非高斯随机粗糙表面计算机仿真的研究

在摩擦学、光学等工程领域中,对于表面粗糙度的研究,总是以生成的随机粗糙表面为研究对象,且大多数研究都是建立在高斯随机表面的基础上,而实际的工程表面大多是非高斯随机表面.为此提出了一种基于快速傅里叶变换(FFT)、Johnson转换系统和自相关函数等理论仿真生成非高斯随机粗糙表面的方法,它可以生成具有给定偏斜度和峰度的随机粗糙表面.为了说明该方法的可行性和正确性,给出了不同偏斜度、峰度和自相关长度下的计算机仿真结果.结果表明:在一定的条件下用该方法仿真生成的非高斯随机粗糙表面,其输入的随机表面的统计参数与输出的统计参数吻合较好.     

An efficient approach for calculating default probabilities for cash-flow based project finance with reserve account

The quantitative assessment of risks associated with several types, eg, rating methods for cash-flow driven projects, can be reduced to determining the probability that a random variable, for instance representing a cash-flow, drops below a given threshold. That probability can be derived in an analytic closed form, if the underlying distribution is not too complex. However, in practice there is often a reserve account in place, which saves excess cash to reduce the volatility of the cash-flow available for debt service. Due to the reserve account, the derivation of a solution in an analytic closed form is even in the case of rather simple underlying distributions, eg, independent Gaussian distribution, not feasible. In this paper, we present two very efficient approximation methodologies for calculating the probability that a random variable falls under a threshold allowing the presence of a reserve account. The first proposed approach is derived using transition probabilities. The resulting recursive scheme can be implemented easily and yields fast and stable results even in the case of dependent cash-flows. The second methodology uses the similarity of the considered stochastic processes with convection-diffusion processes and combines the stochastic transition probabilities with the finite volume method, which is well known for solving partial differential equations. We present numerical results for some realistic test problems demonstrating convergence of order h for the transition probability based approach and \(h^2\) for the combination with the finite volume method for sufficiently smooth probability distributions.     

State estimation by orthogonal expansion of probability distributions

A recursive estimation scheme suitable for real-time implementation is derived for a class of nolinear systems and observations expressed as nonlinear functions in discrete time, corrupted by a non-Gaussian mutually correlated random white noise sequence. The probability densities are expanded as a Gram-Charlier series and a Gauss-Hermite quadrature formula is used for computing the expectations. In the multidimensional case an expansion about a density of mutually independent Gaussian variables is used instead of a general multidimensional Gaussian density, which may result in a poorer performance in linear systems with Gaussian noise. However, in the case of nonlinear systems and non-Gaussian noise, the computational simplifications which result, outweigh the impairment in performance if any. A computational example is included.     

FFT-based methods for nonlinear image restoration in confocal microscopy

Recently we developed a new method for attenuation correction in 3D imaging by a confocal scanning laser microscope (CSLM) in the (epi)fluorescence mode. The fundamental element in our approach consisted of multiplying the measured fluorescent intensity by a correction factor involving a convolution integral of this intensity, which can be computed efficiently by the fast Fourier transform (FFT). The resulting algorithm is one or two orders of magnitude faster than an existing iterative method, but it was found to have a somewhat smaller accuracy. In this paper we improve on this latter point by reformulating the problem as a statistical estimation problem. In particular, we derive first-order-moment and cumulant estimators leading to a nonlinear integral equation for the unknown fluorescent density, which is solved by an iterative method in which in each step a discrete convolution is performed by using the FFT. We find that only a few iterations are needed. It is shown that the estimators proposed here are more accurate than the existing iterative method, while they retain the advantage in computational efficiency of the FFT-based approach.     

Fourier encoding of closed planar boundaries

A circular Gaussian autoregressive (CGAR) source is used as a model for closed planar curves. A class of suboptimal encoding schemes is considered which separately quantize the Fourier coefficients of the boundary. Application of rate-distortion theoretic techniques leads to parametric equations describing the optimal encoding bound. Interpretation of these equations establishes a sampling criterion and a computationally efficient transform encoding scheme for the suboptimal class. Several variants of this transform encoding scheme are suggested and compared to the encoding bound.     

Stochastic analysis of the LMS algorithm for cyclostationary colored Gaussian and non-Gaussian inputs

This paper studies the stochastic behavior of the LMS algorithm in a system identification framework for a cyclostationary colored input without assuming a Gaussian distribution for the input. The input cyclostationary signal is modeled by a colored random process with periodically time-varying power. The generation of the colored non-Gaussian random process is parametrized in novel manner by passing a Gaussian random process through a coloring filter followed by a zero memory nonlinearity. The unknown system parameters are fixed in most of the cases studied here. Mathematical models are derived for the behavior of the mean and mean-square-deviation (MSD) and the excess mean-square error (EMSE) of the adaptive weights as a function of the input cyclostationarity. The models display the dependence of the algorithm upon the input nonlinearity and coloration. Three nonlinearities that are studied in detail with Monte Carlo simulations provide strong support for the theory.     

A cell technique for computing the failure probabilities of structural systems

Several methods for computing structural system reliability are reviewed. A discretization or cell technique for determining the failure probabilites of structural systems is proposed. The Gaussian numerical integration method is introduced to improve its computational accuracy. The method can apply to Gaussian or non-Gaussian variables with linear or nonlinear safety margin equations. The method is easy to realize and requires no iteration or partial differentiation of the safety margin equations. The theory and examples show that the computer run-time of this method is very low.     

A priori error estimates of a meshless method for optimal control problems of stochastic elliptic PDEs

In this paper, we study the optimal control problems of stochastic elliptic equations with random field in its coefficients. The main contributions of this work are two aspects. Firstly, a meshless method coupled with the stochastic Galerkin method is investigated to approximate the control problems, which is competitive for high-dimensional random inputs. Secondly, a priori error estimates are derived for the solutions to the control problems. Some numerical tests are carried out to confirm the theoretical results and to demonstrate the efficiency of the proposed method.     

基于二阶加权差分的湍流退化图像快速复原

针对航天飞行器成像系统大气湍流扰动条件下退化图像的快速复原需求,对二维卷积理论进行了探讨,构造了移位算子。提出了基于二维卷积理论和移位算子的复杂背景图像大气湍流退化图像的相邻帧快速复原算法,该算法主要将基于二阶的加权差分极小作为空间相关性约束应用在湍流退化图像的复原校正过程中,自定义几个二阶加权差分项,推导了二阶加权差分算子矩阵快速构造的理论方法。在此基础上,将点扩展函数的非线性估计转化为基于递归迭代的线性方程快速求解,为大气湍流干扰所引起的退化图像的复原校正提供了一种快速的有效方法。实际场景大气湍流退化图像的复原校正实验结果表明,该方法对湍流退化图像的复原效果好,且算法稳定,速度较快。     

Numerical Studies of Three-dimensional Stochastic Darcy’s Equation and Stochastic Advection-Diffusion-Dispersion Equation

Solute transport in randomly heterogeneous porous media is commonly described by stochastic flow and advection-dispersion equations with a random hydraulic conductivity field. The statistical distribution of conductivity of engineered and naturally occurring porous material can vary, depending on its origin. We describe solutions of a three-dimensional stochastic advection-dispersion equation using a probabilistic collocation method (PCM) on sparse grids for several distributions of hydraulic conductivity. Three random distributions of log hydraulic conductivity are considered: uniform, Gaussian, and truncated Gaussian (beta). Log hydraulic conductivity is represented by a Karhunen-Loève (K-L) decomposition as a second-order random process with an exponential covariance function. The convergence of PCM has been demonstrated. It appears that the accuracy in both the mean and the standard deviation of PCM solutions can be improved by using the Jacobi-chaos representing the truncated Gaussian distribution rather than the Hermite-chaos for the Gaussian distribution. The effect of type of distribution and parameters such as the variance and correlation length of log hydraulic conductivity and dispersion coefficient on leading moments of the advection velocity and solute concentration was investigated.     

Convolution theorems for the linear canonical transform and their applications

As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.     

Numerical methods for the solution of the degenerate nonlinear elliptic equations arising in optimal stochastic control theory

Three distinct but related results are obtained. First, an iterative method is derived for obtaining the solution of optimal control problems for Markov chains. The method usually converges much faster, and requires less computer storage space, than the methods of Howard or Eaton and Zadeh. Second, nonlinear finite difference equations, which "approximate" the nonlinear degenerate elliptic equation (2) arising out of the stochastic optimization problem (1), are found. The difference equations, and their solution, may have a meaning for the control problem even when it cannot be proved that (2) has a solution. The iterative methods for the iterative solution of these nonlinear systems are discussed and compared. Both converge to the solution, provided that the difference equations were derived using the method introduced in the paper; one, new to this paper, often much faster than the other (Theorem 2). In fact, the typical time required for the numerical solution is about the time required for a related linear problem. The method of obtaining the difference equations, and the proof of convergence of the associated iterative procedures, are illustrated by a detailed example. Finally, specific numerical results for a "minimum average time" type of optimization problem are presented and discussed.     

Preconditioning for a Class of Spectral Differentiation Matrices

We propose an efficient preconditioning technique for the numerical solution of first-order partial differential equations (PDEs). This study has been motivated by the computation of an invariant torus of a system of ordinary differential equations. We find the torus by discretizing a nonlinear first-order PDE with a full two-dimensional Fourier spectral method and by applying Newton’s method. This leads to large nonsymmetric linear algebraic systems. The sparsity pattern of these systems makes the use of direct solvers prohibitively expensive. Commonly used iterative methods, e.g., GMRes, BiCGStab and CGNR (Conjugate Gradient applied to the normal equations), are quite slow to converge. Our preconditioner is derived from the solution of a PDE with constant coefficients; it has a fast implementation based on the Fast Fourier Transform (FFT). It effectively increases the clustering of the spectrum, and speeds up convergence significantly. We demonstrate the performance of the preconditioner in a number of linear PDEs and the nonlinear PDE arising from the Van der Pol oscillator     

L1 Fourier spectral methods for a class of generalized two-dimensional time fractional nonlinear anomalous diffusion equations

In this paper, L1 Fourier spectral methods are derived to obtain the numerical solutions for a class of generalized two-dimensional time-fractional nonlinear anomalous diffusion equations involving Caputo fractional derivative. Firstly, we establish the L1 Fourier Galerkin full discrete and L1 Fourier collocation schemes with Fourier spectral discretization in spatial direction and L1 difference method in temporal direction. Secondly, stability and convergence for both Galerkin and collocation approximations are proved. It is shown that the proposed methods are convergent with spectral accuracy in space and $(2?\alpha )$ order accuracy in time. For implementation, the equivalence between pseudospectral method and collocation method is discussed. Furthermore, a numerical algorithm of Fourier pseudospectral method is developed based on two-dimensional fast Fourier transform (FFT2) technique. Finally, numerical examples are provided to test the theoretical claims. As is shown in the numerical experiments, Fourier spectral methods are powerful enough with excellent efficiency and accuracy.     

Stochastic simulation of non-Gaussian/non-stationary properties in a functionally graded plate

Composites with a graded profile between two distinct phases, titled functionally graded materials (FGM’s), have been proposed for a variety of applications. Many current numerical models of FGM’s overlook the inherent randomness observed in actual specimens, which arises due to the complexity and difficulty in tailoring the gradients to exact specifications during manufacturing processes. The effect of random variations, in both the component volume fractions and the porosity, on the response of a functionally graded metal/ceramic plate subjected to constant thermal boundary conditions is investigated in this work. These random variations are introduced into the FGM model via stochastic simulation. Of interest is the fact that the volume fraction and the porosity are both non-Gaussian and non-stationary. The non-stationarity is present due to the fact that the non-Gaussian probability density function describing both properties changes with location. Sample simulations are performed using a translation mapping from a stationary, standard Gaussian process to a non-stationary, non-Gaussian process. The results are used to quantify the effects of variations in the micromechanical configuration on the temperature, thermal stresses and safety factor within a functionally graded plate.