Three areas of study in computational and applied mathematics will be studied.
The first deals with asynchronous parallel methods. In these iterative methods the computational effort is distributed among the processors.
The processors proceed with their computation and exchange information only when their own computational step is completed. This is done without waiting for all processors to complete their tasks, that is, without synchronization points. In this way, processor idle time is minimized, and the overall computational time can be reduced.
The second topic relates to the linear algebra representation of multiplicative Schwarz methods for the iterative solution of discretized partial differential equations. A weighted max norm will be used to study the convergence of these methods in several circumstances, including cases when there is no underlying grid, and the case with a "coarse grid" correction.
The third problem to be addressed is the comparison of the rate of convergence of two different iterative stationary methods for the solution of singular linear systems of algebraic equations. Several authors have shown that the usual hypothesis in similar comparisons for the nonsingular case do not apply here. A different partial order will be used to try to obtain similar comparison theorems.
Under the High Performance Computing and Communication initiative, computers with several thousands of processors are being designed. Problems requiring a large computational effort could be solved in these machines by distributing the work among the processors and exchanging information as the computation proceeds. In order to increase the computational efficiency, this exchange of information can be done asynchronously, that is, without waiting for all processors to reach a certain predetermined point. In this project, several aspects of these asynchronous methods will be studied. For example, it is important to know for which kind of problems this approach will be advantageous. One needs to know how to formulate the problems so they conform to the conditions needed for these methods to work. Asynchronous methods are possibly the kind of methods which will allow the next generation of parallel machines to attain their expected potential. Another part of the project relates to singular linear systems. These systems arise in numerous applications, such as queuing models of telecommunication networks. Given two methods to solve the equations representing these models, one wants to know which is faster. Tools to that effect will be developed.
