On sparse lu factorization procedures for the solution of parabolic differential equations in three space dimensions

The numerical solution of partial differential equations in 3 dimensions by finite difference methods leads to the problem of solving large order sparse structured linear systems.

In this paper, a factorization procedure in algorithmic form is derived yielding direct and iterative methods of solution of some interesting boundary value problems in physics and engineering.     

Iterative modified approximate factorization

An implicit procedure for minimizing the error introduced by approximate factorization in solving scalar or block multidiagonal matrix equations is presented with applications to fluid dynamics. High convergence rates of approximately 0.8 are achieved upon application to a wide range of problems. The implicit procedure can be combined with a multigrid procedure for further convergence acceleration.     

A difference scheme with high accuracy in time for fourth-order parabolic equations

A novel finite difference method is developed for the numerical solution of fourth-order parabolic partial differential equations in one and two space variables. The method is seen to evolve from a multiderivative method for second-order ordinary differential equations.The method is tested on three model problems, with constant coefficients and variable coefficients, which have appeared in the literature.     

Algorithms and applications for approximate nonnegative matrix factorization

The development and use of low-rank approximate nonnegative matrix factorization (NMF) algorithms for feature extraction and identification in the fields of text mining and spectral data analysis are presented. The evolution and convergence properties of hybrid methods based on both sparsity and smoothness constraints for the resulting nonnegative matrix factors are discussed. The interpretability of NMF outputs in specific contexts are provided along with opportunities for future work in the modification of NMF algorithms for large-scale and time-varying data sets.     

Block methods for parabolic equations

Block methods for the finite difference solution of linear one dimensional parabolic partial differential equations are considered. These schemes use two linear multistep formulae which, when applied simultaneously, advance the numerical solution by two time steps. No special starting procedure is required for their implementation. By careful choice of the coefficients in these formulae, all of the block methods derived in this paper are unconditionally stable and have high order accuracy. In addition, some of these schemes are suitable for problems involving a discontinuity between the initial and boundary conditions. The results of numerical experiments on two test problems are presented.     

The use of the generalized extended to the limit sparse factorization techniques for the solution of non-linear elliptic and parabolic difference equations

Generalized Extended to the Limit LU sparse factorization procedures for the solution of large sparse unsymmetric linear systems of irregular and unsymmetric structure are presented. Composite “inner-outer” iterative schemes incorporating these procedures are introduced for solving non-linear elliptic and parabolic difference equations. Applications of the methods on non-linear boundary-value problems are discussed and numerical results are given.     

Hartley series approximations for the parabolic equations

An approximate method for solving parabolic equations with a periodic boundary condition is proposed. The method is based upon using the Legendre series and the Hartley series to approximate the required solution. The parabolic equations are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. A numerical example is included to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

Parallel iterative methods for parabolic equations

For the problems of the parabolic equations in one- and two-dimensional space, the parallel iterative methods are presented to solve the fully implicit difference schemes. The methods presented are based on the idea of domain decomposition in which we divide the linear system of equations into some non-overlapping sub-systems, which are easy to solve in different processors at the same time. The iterative value is proved to be convergent to the difference solution resulted from the implicit difference schemes. Numerical experiments for both one- and two-dimensional problems show that the methods are convergent and may reach the linear speed-up.     

Generalized trapezoidal formulas for parabolic equations

We investigate the application of the one-parameter family of generalized trapezoidal formulas (GTFs) introduced in Chawla et al. [2] for the time-integration of parabolic equations. The resulting GTF finite-difference schemes (GTF-FDS) are, in general, second order in both time and space and unconditionally stable. Interestingly, there exists a method of the family which is third order in time. Unlike the popular Crank -Nicolson scheme, our present GTF-FDS can cope with discontinuities in the boundary conditions and the initial conditions. We consider extensions of the GTF-FDS for equations with derivative boundary conditions and to a nonlinear problem. Numerical experiments demonstrate the superiority of the present GTF-FDS, especially for the case of problems with discontinuities in the boundary and the initial conditions.     

从运动中恢复目标结构的改进因子分解法

因子分解法是从图像序列中恢复刚体目标几何结构的重要方法。针对传统因子分解法基本过程中存在的不足,及其容易失效的缺点,提出一种改进的因子分解法。该方法避开传统方法中求解修正矩阵的复杂过程,利用旋转矩阵的特性,直接修正由传统方法奇异值分解(SVD)得到的每帧图像的旋转矩阵,然后根据观测矩阵和得到的旋转矩阵,直接利用线性最小二乘法求解目标的结构矩阵。仿真和实测数据的实验结果表明,本文方法能够有效地从序列图像中恢复目标的几何结构,相比传统因子分解法,在稳定性上有较大的提升。     

An incomplete factorization iterative technique for elliptic equations

This paper describes efficient iterative techniques for solving the large sparse symmetric linear systems that arise from application of finite difference approximations to self-adjoint elliptic equations. We use an incomplete factorization technique with the method of D'Yakonov type, generalized conjugate gradient and Chebyshev semi-iterative methods. We compare these methods with numerical examples. Bounds for the 4-norm of the error vector of the Chebyshev semi-iterative method in terms of the spectral radius of the iteration matrix are derived.     

One-step splitting methods for semi-discrete parabolic equations

The main purpose of the paper is to discuss splitting methods for parabolic equations via the method of lines. Firstly, we deal with the formulation of these methods for autonomous semi-discrete equations $$\frac{{dy}}{{dt}} = f(y),{\rm E}f{\rm E}non - linear,$$ f satisfying a linear splitting relation \(f(y) = \sum\limits_{i = 1}^k {f_i (y)} \) . A class of one-step integration formulas is defined, which is shown to contain all known splitting methods, provided the functionsf _i are defined appropriately. For a number of methods stability results are given. Secondly, attention is paid to alternating direction methods for problems with an arbitrary non-linear coupling between space derivatives.     

Approximate controllability for fractional semilinear parabolic equations

In this paper, we deal with the control systems described by a large class of fractional semilinear parabolic equations. Firstly, we reformulate the fractional parabolic equations into abstract fractional differential equations associated with a semigroup on an appropriate Banach space. Secondly, we introduce a suitable concept on a mild solution for this kind of fractional parabolic equations and present the existence and uniqueness of mild solution by utilizing the theory of semigroup of linear operator, nonlinear analysis method and fixed point theorem. Then, the approximate controllability of the fractional semilinear parabolic equations is formulated and proved. At the end of the paper, an example is given to illustrate our main results.     

Controllability for parabolic equations with nonlinear memory

This paper is concerned with the controllability of a parabolic system with nonlinear memory. Based on the localized estimate of the solution, we prove that the system with a superlinear growth memory is not controllable. Furthermore, two controllability results for some initial data and targets are given as well.     

Explicit methods of numerical integration for parabolic equations

The principles are considered for the construction of numerical time integration schemes of parabolic equations. An approach is presented based on explicit iterations with Chebyshev parameters, which can be used to construct the schemes of the first and second orders of accuracy. The basic information has been systematized on the developed schemes and conditions of their applicability, including the application for the computation of high-temperature processes in thermonuclear targets.     

Non-polynomial cubic spline methods for the solution of parabolic equations

Second-order parabolic partial differential equations are solved by using a new three level method based on non-polynomial cubic spline in the space direction and finite difference in the time direction. Stability analysis of the method has been carried out and we have shown that our method is unconditionally stable. It has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and non-homogeneous cases can be obtained from our method. We also obtain a new high accuracy scheme of O(k ⁴+h ⁴). Numerical examples are given to illustrate the applicability and efficiency of the new method.