Preconditioning parallel multisplittings for solving systems of equations

This dissertation concerns parallel methods for solving systems of equations and unconstrained optimization problems.;There has been increased interest in large scale computation, arising from problems in discretization of partial differential equations, mathematical programming, and optimization problems. The preconditioned conjugate gradient algorithm is one of the major methods to deal with these problems. For example, large scale problems can often be solved quite efficiently by using the conjugate gradient algorithm with a polynomial preconditioner on vector computers.;We propose a parallel algorithm to solve large scale problems for parallel machines with MIMD (Multiple instruction stream-multiple data stream) architecture. There are two kinds of tasks in this algorithm. The first provides a subspace over which the second will minimize an error function. The subspace generating task is further subdivided into two or more tasks.;We use multisplittings of the matrix A on the first kind tasks. The matrix A is split in several different ways, and each processor is responsible for computing a piece of the solution corresponding to a particular splitting. This effectively produces a preconditioner for A so that in the second kind task the solution to the original problem is well-approximated on a low dimensional subspace. In this dissertation we study a concrete case with three tasks.;The second part of the dissertation discusses the unconstrained optimization problem. This is the problem of finding the minimizing point of a nonlinear function of n variables. It is assumed that the nonlinear function f is at least twice continuously differentiable and bounded below.;For finding the minimizing point of a nonlinear function, the local model is the quadratic model. The fundamental method for solving this model is Newton method.;Applying our algorithm to the approximate Newton equation, and combining with a line search method, we have a parallel truncated-Newton algorithm. In this algorithm we use finite difference approximation to the product of the Hessian matrix with a vector.