Pierluigi Amodio

Parallel factorizations for tridiagonal matrices

The authors analyze the problem of solving tridiagonal linear systems on parallel computers. A wide class of efficient parallel solvers is derived by considering different parallel factorizations of partitioned matrices. These solvers have a minimum requirement of data transmission. In fact, communication is only needed for solving a “reduced system,” whose dimension depends on the number of parallel processors used. Moreover, for a given partitioned tridiagonal matrix, the reduced system (which is again tridiagonal) is the same, and represents the only sequential part of the corresponding parallel solver.Three examples are discussed in more detail; one of them derives a very efficient parallel method based on the cyclic reduction algorithm.

Stability of some boundary value methods for the solution of initial value problems

The stability properties of three particular boundary value methods (BVMs) for the solution of initial value problems are considered. Our attention is focused on the BVMs based on the midpoint rule, on the Simpson method and on an Adams method of order 3. We investigate their BV-stability regions by considering the scalar test problem and constant stepsize. The study of the conditioning of the coefficient matrix of the discrete problem is extended to the case of variable stepsize and block ODE problems. We also analyse an appropriate choice for the stepsize for stiff problems. Numerical tests are reported to evidentiate the effectiveness of the BVMs and the differences among the BVMs considered.

A Cyclic Reduction Approach to the Numerical Solution of Boundary Value ODEs

A parallel algorithm for solving linear systems that arise from the discretization of boundary value ODEs is described. It is a modification of a cyclic reduction algorithm that takes advantage of the almost block diagonal structure of the linear system. A stable modification of the original algorithm is also proposed. Numerical results on a distributed memory parallel computer with 32 processors are presented and discussed.

Boundary value methods based on Adams-type methods

Abstract The aim of this paper is to derive Boundary Value Methods (BVMs) based on k-step Adams-type methods for the solution of initial value problems. BVMs lead to a discrete boundary value problem which needs one initial and k − 1 final conditions. We prove that the choice of boundary conditions, instead of the usual initial conditions, improves the stability properties of the classical Adams methods. For example, methods of order up to 6 are almost BV-A-stable, and those of order up to 9 are BV-A0-stable.

A boundary value approach to the numerical solution of initial value problems by multistep methos

A boundary value appraoch to the numerical solution of initial value problems by means of linear multistep methods is presented. This theory is based on the study of linear difference equations when their general solution is computed by imposing boundary conditions. All the main stability and convergence properties of the obtained methods are investigated abd compared to those of the classical multistep methods. Then, as an example, new itegration formulas, called extended trapezoidal rules, are derived. For any order they have the same stability properties (in the sense of the definitions given in this paper) of the trapezoidal rule, which is the first method in this class. Some numerical examples are presented to confirm the theoretical expectations and to allow us to trust a future code based on boundary value methods.

Parallel implementation of block boundary value methods for ODEs

The parallel solution of initial value problems for ODEs has been the subject of much research in the last thirty years, and different approaches to the problem have been devised. In this paper we examine the parallel methods derived by block boundary value methods (BVMs), recently introduced for approximating Hamiltonian problems. Here we restrict the analysis of the methods when applied to linear problems, since their nonlinear parallel implementation deserves further study. However, for linear problems, the methods can reach a high parallel efficiency. Some of these solvers can also be adapted for approximating continuous two-point boundary value problems. Numerical tests carried out on a distributed memory parallel computer are reported.

Parallel factorizations and parallel solvers for tridiagonal linear systems

Abstract We formalize the concept of parallel factorization as a set of scalar factorizations. By means of this concept we are able to give a unified approach to the problem of solving tridiagonal linear systems on parallel computers. A parallel tridiagonal solver is associated with each parallel factorization, but a parallel factorization can be associated with many parallel tridiagonal solvers. As an example, some parallel factorizations are obtained by simple extension of well-known scalar ones. Numerical tests, obtained on a network of transputers, are reported for comparison.

Parallel block preconditioning for the solution of boundary value methods

Abstract The discrete problem associated with a two-step boundary value method (BVM) for the solution of initial value problems is a non-symmetric block tridiagonal system. This system may be efficiently solved on a parallel computer by using a conjugate gradient type method with a suitable preconditioning. In this paper we consider a BVM based on an Adams method of order three and the trapezoidal method. The structure of the coefficient matrix allows us to derive good stability properties and an efficient preconditioning. Both the theoretical properties and the parallel implementation are discussed in more detail. In the numerical tests section, the preconditioning has been associated with the Bi-CGSTAB algorithm. The parallel algorithm has been tested on a network of transputers.

A note on the efficient implementation of implicit methods for ODEs

The use of implicit methods for ODEs, e.g. implicit Runge-Kutta schemes, requires the solution of nonlinear systems of algebraic equations of dimension sm, where m is the size of the continuous differential problem to be approximated. Usually, the solution of this system represents the most time-consuming section in the implementation of such methods. Consequently, the efficient solution of this section would improve their performance. In this paper, we propose a new iterative procedure to solve such equations on sequential computers.

Numerical solution of differential algebraic equations and computation of consistent initial/boundary conditions

Boundary value methods for the solution of differential-algebraic equations are described. We consider both initial and boundary value problems and derive an algorithm that does not require additional information from the user, but only the initial or boundary conditions needed in theory to obtain a unique solution.