Applied Numerical Analysis. By C.F. Gerald. Pp. xii, 340. £3·60. 1970. (Addison-Wesley.)

We survey recent trends in practical algorithms for balanced graph partitioning, point to applications and discuss future research directions.

/pdf/recent-advances-in-graph-partitioning-4eois71ejs.pdf

Recent Advances in Graph Partitioning

In this paper, we compute and visualize solutions of several major types of semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in 2D. We present the mountain–pass algorithm (MPA), the scaling iterative algorithm (SIA), the monotone iteration and the direct iteration algorithms (MIA and DIA). Semilinear elliptic equations are well known to be rich in their multiplicity of solutions. Many such physically significant solutions are also known to lack stability and, thus, are elusive to capture numerically. We will compute and visualize the profiles of such multiple solutions, thereby exhibiting the geometrical effects of the domains on the multiplicity. Special emphasis is placed on SIA and MPA, by which multiple unstable solutions are computed. The domains include the disk, symmetric or nonsymmetric annuli, dumbbells, and dumbbells with cavities. The nonlinear partial differential equations include the Lane–Emden equation, Chandrasekhar's equation, Henon's equation, a singularly perturbed equation, and equations with sublinear growth. Relevant numerical data of solutions are listed as possible benchmarks for other researchers. Commentaries from the existing literature concerning solution behavior will be made, wherever appropriate. Some further theoretical properties of the solutions obtained from visualization will also be presented.

https://www.math.tamu.edu/~j.zhou/CNZ.pdf

Algorithms and visualization for solutions of nonlinear elliptic equations

A class of numerical methods is developed for second order Volterra integro- differential equations by using a Legendre spectral approach. We provide a rigorous error analysis for the proposed methods, which shows that the numerical errors decay exponentially in the L ∞ -norm and L 2 -norm. Numerical examples illustrate the conver- gence and effectiveness of the numerical methods. AMS subject classifications: 45L10, 65R20, 65D15

Convergence Analysis of the Legendre Spectral Collocation Methods for Second Order Volterra Integro-Differential Equations

We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the Chebyshev polynomial of order N (El-gendi's method). The solutions are exact at these points, apart from round-off computer errors and the convergence of other numerical methods used in solving the linear system of equations. Applications to initial value problems in time-dependent quantum field theory, and second-order boundary value problems in fluid dynamics are presented.

Numerical approximations using Chebyshev polynomial expansions: El-gendi's method revisited

Using finite difference methods to solve nonlinear Volterra integro-differential equations with two point boundary conditions give rise to a symmetric banded coefficient matrix. A typical method for solving systems of this form involves the LU method. In this paper the original system is modified to allow the implementation of a special fast algorithm for solving tridiagonal systems. Numerical examples are given to compare an efficient form of the LU method with the new approach.

A fast method for solving second Order boundary value volterra Integro-differential equations

There are many articles on symmetric tridiagonal Toeplitz and circulant systems. Such systems in areas including numerical methods for solving boundary value differential equations and in graph theory. These matrices can often be written as the product of bidiagonal matrices. In this article, nonsymmetric Toepliz systems and nonsymmetric circulant systems are examined. The coefficient matrix is split into two bidiagonal matrices and the efficient solution of the resulting systems is considered.

A parallel method for linear equations with tridiagonal Toeplitz coefficient matrices

In 1994, Yan and Chung produced a fast algorithm for solving a diagonally dominant symmetric Toeplitz tridiagonal system of linear equations Ax = b. In this work a method will be presented which will allow for problems of the above nature to be split into two separate systems which can be solved in parallel, and then combined and corrected to obtain a solution to the original system. An error analysis will be provided along with example cases and time comparison results.

R. E. Shaw

Papers

A fast method for solving second Order boundary value volterra Integro-differential equations

A parallel method for linear equations with tridiagonal Toeplitz coefficient matrices

A split-correct parallel algorithm for solving tridiagonal symmetric toeplitz systems

A parallel algorithm for solving Toeplitz linear systems

A communication-less parallel algorithm for tridiagonal Toeplitz systems