Using Chebyshev Polynomials to Approximate Partial Differential Equations

This paper suggests a simple method based on a Chebyshev approximation at Chebyshev nodes to approximate partial differential equations (PDEs). It consists in determining the value function by using a set of nodes and basis functions. We provide two examples: pricing a European option and determining the best policy for shutting down a machine. The suggested method is flexible, easy to programme and efficient. It is also applicable in other fields, providing efficient solutions to complex systems of PDEs.     

Partial Differential Equations and Bivariate Orthogonal Polynomials

In 1929, S. Bochner identified the families of polynomials which are eigenfunctions of a second-order linear differential operator. What is the appropriate generalization of this result to bivariate polynomials? One approach, due to Krall and Sheffer in 1967 and pursued by others, is to determine which linear partial differential operators have orthogonal polynomial solutions with all the polynomials in the family of the same degree sharing the same eigenvalue. In fact, such an operator only determines a multi-dimensional eigenspace associated with each eigenvalue; it does not determine the individual polynomials, even up to a multiplicative constant. In contrast, our approach is to seek pairs of linear differential operators which have joint eigenfunctions that comprise a family of bivariate orthogonal polynomials. This approach entails the addition of some “normalizing" or “regularity" conditions which allow determination of a unique family of orthogonal polynomials. In this article we formulate and solve such a problem and show with the help of Mathematica that the only solutions are disk polynomials. Applications are given to product formulas and hypergroup measure algebras.     

Single Layer Chebyshev Neural Network Model for Solving Elliptic Partial Differential Equations

The purpose of the present study is to solve partial differential equations (PDEs) using single layer functional link artificial neural network method. Numerical solution of elliptic PDEs have been obtained here by applying Chebyshev neural network (ChNN) model for the first time. Computations become efficient because the hidden layer is eliminated by expanding the input pattern by Chebyshev polynomials. Feed forward neural network model with unsupervised error back propagation principle is used for modifying the network parameters and to minimize the computed error function. Numerical efficiency and accuracy of the ChNN model are investigated by three test problems of elliptic partial differential equations. The results obtained by this method are compared with the existing methods and are found to be in good agreement.     

Haar小波求解线性分数阶微积分方程

分数阶微积分方程的应用逐渐受到广大研究者的重视.论文以Haar小波基函数来逼近线性分数阶微积分方程,通过数值算例的数值解与精确解进行比较,结果表明论文方法是有效的且具有较高的精度.     

一种基于偏微分方程的信号增强模型

在峰检测中,由于重叠峰和弱峰的存在,峰的漏检和错检率较高。为此本文在峰检测环节之前增加一个峰增强环节,使重叠峰的分离度以及弱峰幅度均增大。增强环节中采用的方法是将经典非线性扩散与导数谱相结合,即将导数谱增强后的信号作为经典非线性扩散的初始信号,经过一定时间的扩散得到增强后的信号。作为效果检验,首先对比了信号经过所提模型增强前后的效果,之后对比了其他信号增强模型的效果,结果表明本文所提模型有效。最后将本文模型应用于MALDI质谱峰增强。     

对称四阶偏微分方程去噪算法

通过分析整体变分模型的去噪原理和效果,提出一个新的四阶偏微分方程去噪模型,用于克服二阶偏微分方程去噪后使图像分块的缺点,同时保持去噪后图像的高保真性,并发展一个基于四邻域系统的对称离散化算法用于求解新模型,应用中值滤波去除四阶偏微分方程去噪所引起的亮点。实验结果表明,与传统方法相比,以该算法去噪后的图像具有更好的质量和视觉效果。     

Factoring and Solving Linear Partial Differential Equations

The problem of factoring a linear partial differential operator is studied. An algorithm is designed which allows one to factor an operator when its symbol is separable, and if in addition the operator has enough right factors then it is completely reducible. Since finding the space of solutions of a completely reducible operator reduces to the same for its right factors, we apply this approach and execute a complete analysis of factoring and solving a second-order operator in two independent variables. Some results on factoring third-order operators are exhibited.AMS Subject Classifications: 35A25, 35C05, 35G05.     

A System for Performance Evaluation of Partial Differential Equations Software

This paper describes a system to systematically compare the performance of various methods (software modules) for the numerical solution of partial differential equations. We discuss the general nature and large size of this performance evaluation problem and the data one obtains. The system meets certain design objectives that ensure a valid experiment: 1) precise definition of a particular measurement; 2) uniformity in defimition of variables entering the experiment; and 3) reproducibility of results. The ease of use of the system makes it possible to make the large sets of measurements necessary to obtain confidence in the results and its portability allows others to check or extend the measurements. The system has four parts: 1) semiautomatic generation of problems for experimental input; 2) the ELLPACK system for actually solving the equation; 3) a data management system to organize and access the experimental data; and 4) data analysis programs to extract graphical and statistical summaries from the data.     

Rational Solutions of Riccati-like Partial Differential Equations

When factoring linear partial differential systems with a finite-dimensional solution space or analysing symmetries of nonlinear ODEs, we need to look for rational solutions of certain nonlinear PDEs. The nonlinear PDEs are called Riccati-like because they arise in a similar way as Riccati ODEs. In this paper we describe the structure of rational solutions of a Riccati-like system, and an algorithm for computing them. The algorithm is also applicable to finding all rational solutions of Lie’s system { ∂_xu + u² + a₁u + a₂v + a₃, ∂_yu + uv + b₁u + b₂v + b₃, ∂_xv + uv + c₁u + c₂v + c₃, ∂_yv + v² + d₁u + d₂v + d₃},where a₁, . . . , d₃are rational functions of x and y.     

Multivariate Median Filters and Partial Differential Equations

Multivariate median filters have been proposed as generalizations of the well-established median filter for gray-value images to multichannel images. As multivariate median, most of the recent approaches use the \(L^1\) median, i.e., the minimizer of an objective function that is the sum of distances to all input points. Many properties of univariate median filters generalize to such a filter. However, the famous result by Guichard and Morel about approximation of the mean curvature motion PDE by median filtering does not have a comparably simple counterpart for \(L^1\) multivariate median filtering. We discuss the affine equivariant Oja median and the affine equivariant transformation–retransformation \(L^1\) median as alternatives to \(L^1\) median filtering. We analyze multivariate median filters in a space-continuous setting, including the formulation of a space-continuous version of the transformation–retransformation \(L^1\) median, and derive PDEs approximated by these filters in the cases of bivariate planar images, three-channel volume images, and three-channel planar images. The PDEs for the affine equivariant filters can be interpreted geometrically as combinations of a diffusion and a principal-component-wise curvature motion contribution with a cross-effect term based on torsions of principal components. Numerical experiments are presented, which demonstrate the validity of the approximation results.     

Solving nonlinear Partial Differential Equations on hypercubes

This paper presents an efficient parallel algorithm for solving nonlinear Partial Differential Equations (PDEs), occurring in heat transfer and fluid flow simulation, on hypercube machines. To evaluate its performance, an expression for the efficiency of the algorithm is derived. The results show that the hypercubes are well suited for solving nonlinear PDEs.     

图像放大的偏微分方程方法

在分析一些常见的图像放大方法的基础上,根据图像像素值特点及近期偏微分方程在图像处理中的应用,将图像的像素值看作是平面物体的温度;利用偏微分方程理论中的热传导数学模型,提出了基于一种新颖的热传导方程初边值问题的图像放大法;并根据其物理意义,设计相应的差分算法．实验证明,这是一种有效的图像放大方法．     

Image Sequence Analysis via Partial Differential Equations

This article deals with the problem of restoring and motion segmenting noisy image sequences with a static background. Usually, motion segmentation and image restoration are considered separately in image sequence restoration. Moreover, motion segmentation is often noise sensitive. In this article, the motion segmentation and the image restoration parts are performed in a coupled way, allowing the motion segmentation part to positively influence the restoration part and vice-versa. This is the key of our approach that allows to deal simultaneously with the problem of restoration and motion segmentation. To this end, we propose a theoretically justified optimization problem that permits to take into account both requirements. The model is theoretically justified. Existence and unicity are proved in the space of bounded variations. A suitable numerical scheme based on half quadratic minimization is then proposed and its convergence and stability demonstrated. Experimental results obtained on noisy synthetic data and real images will illustrate the capabilities of this original and promising approach.     

Approximate Solutions of Initial Value Problems for Ordinary Differential Equations Using Radial Basis Function Networks

We present a numerical approach for the approximate solutions of first order initial value problems (IVP) by using unsupervised radial basis function networks. The proposed unsupervised method is able to solve IVPs with high accuracy. In order to demonstrate the efficiency of the proposed approach, we also compare its solutions with the solutions obtained by a previously proposed neural network method for representative examples.     

The Relationships Between Chebyshev,Legendre and Jacobi Polynomials: The Generic Superiority of Chebyshev Polynomials and Three Important Exceptions

We analyze the asymptotic rates of convergence of Chebyshev, Legendre and Jacobi polynomials. One complication is that there are many reasonable measures of optimality as enumerated here. Another is that there are at least three exceptions to the general principle that Chebyshev polynomials give the fastest rate of convergence from the larger family of Jacobi polynomials. When $f(x)$ is singular at one or both endpoints, all Gegenbauer polynomials (including Legendre and Chebyshev) converge equally fast at the endpoints, but Gegenbauer polynomials converge more rapidly on the interior with increasing order $m$ . For functions on the surface of the sphere, associated Legendre functions, which are proportional to Gegenbauer polynomials, are best for the latitudinal dependence. Similarly, for functions on the unit disk, Zernike polynomials, which are Jacobi polynomials in radius, are superior in rate-of-convergence to a Chebyshev–Fourier series. It is true, as was conjectured by Lanczos 60 years ago, that excluding these exceptions, the Chebyshev coefficients $a_{n}$ usually decrease faster than the Legendre coefficients $b_{n}$ by a factor of $\sqrt{n}$ . We calculate the proportionality constant for a few examples and restrictive classes of functions. The more precise claim that $b_{n} \sim \sqrt{\pi /2} \sqrt{n} a_{n}$ , made by Lanczos and later Fox and Parker, is true only for rather special functions. However, individual terms in the large $n$ asymptotics of Chebyshev and Legendre coefficients usually do display this proportionality.     

Using Distributed Computers to Deterministically Approximate Higher Dimensional Convection Diffusion Equations

A new approach to solving D> 3 spatial dimensional convection-diffusion equation on clusters of workstations is derived by exploiting the stability and scalability of the combination of a generalized D dimensional high-order compact (HOC) implicit finite difference scheme and parallelized GMRES(m). We then consider its application to multifactor Option pricing using the Black–Scholes equation and further show that an isotropic fourth order compact difference scheme is numerically stable and determine conditions under which its coefficient matrix is positive definite. The performance of GMRES(m) on distributed computers is limited by the inter-processor communication required by the matrix-vector multiplication. It is shown that the compact scheme requires approximately half the number of communications as a non-compact difference scheme of the same order of truncation error. As the dimensionality is increased, the ratio of computation that can be overlapped with communication also increases. CPU times and parallel efficiency graphs for single time step approximation of up to a 7D HOC scheme on 16 processors confirm the numerical stability constraint and demonstrate improved parallel scalability over non-compact difference schemes.     

一种解偏微分方程逆问题的并行算法

论述一种偏微分方程逆问题的数值解法和阵列机与 PVM平台上实现的并行算法。     

一种基于四阶偏微分方程的噪声消除方法

介绍了一种基于四阶偏微分方程(PDE)噪声消除方法.采用分片平面图像逼近被处理图像.在噪声消除同时保持良好边缘,避免了使用二阶偏微分方程处理图像常常出现的块效应.实验结果表明该方法是行之有效的.     

一种基于四阶偏微分方程的噪声消除方法

介绍了一种基于四阶偏微分方程(PDE)噪声消除方法。采用分片平面图像逼近被处理图像。在噪声消除同时保持良好边缘,避免了使用二阶偏微分方程处理图像常常出现的块效应。实验结果表明该方法是行之有效的。