Studies and systematic exploration of parallel iterative processes
Submitted: September 2005
Estimated date of evaluation results: January 2006
The project proposed is focused on the study, using various methods and techniques, of parallel iterative processes, as they appear in the context of discrete dynamical systems. Two case studies are considered: that of nonlinear equation systems and that of P systems. Each of these case studies are investigated using specific methods, complemented by techniques of systematic exploration developed in the frame of Mathematical Knowledge Management (MKM). In the case of nonlinear equation systems, the purpose of our project is the analysis of the asynchronous algorithms of block-Jacobi type for solving nonlinear equations. Some particular implementations are considered, by using classical serial algorithms: Newton, Gradient (Fridmann variant), conjugate gradient, quasi-Newton. The sparsity of the systems plays an important role in our development and it is motivated by a number of practical applications which have these characteristics. Also, we partly deal with some mathematical aspects of the parallel solving of nonlinear systems, such as convergence, rate of convergence and numerical stability. We investigate: (1) Preconditioning of large, sparse nonlinear systems; (2) Stopping criteria; (3) Design of a strategy for the partitioning of systems in blocks, using load balancing algorithms. For P systems, we propose the development of a behavior analysis based on bigraphs. This represents a new approach in P system behavior modelling. It will facilitate the comparative analysis and the design of optimal P systems. In addition, this project proposes the use of the techniques for systematic mathematical theory exploration introduced recently by Bruno Buchberger. These techniques, based on knowledge schemes, are being implemented in the mathematical assistant system Theorema, which will be used in the case studies mentioned above.