Krylov method algorithms adapted to the grid computing

Abstract

This chapter describes the modified algorithms for general Krylov methods that will be adapted to perform on the grid architectures. The main idea of this chapter is to modify the Krylov method algorithms, in order to delay as far as possible the dot products computation without destroying the Krylov method properties and convergence. For this purpose with symbolic calculus software, it defines the induction formula for a Krylov method in order to go directly from the kth iterate to the (k+p)th iterate. The secondary idea is to develop Krylov space projection techniques that can handle the powerful mass storage capability of the grid, once the Krylov vectors are computed efficiently on it. The chapter focuses on the Krylov methods where the mass storage of data is playing key role on the numerical performance and the global communications due to dot products are the main parallelism difficulty. It shows that projection techniques on Krylov spaces are a powerful numerical tool to know how to efficiently compute the Krylov space. The combination of numeric and symbolic calculus allows a delay in the dot product, and in performing them in one pass. The numerical results obtained are encouraging and can be improved with the mixing of the Krylov space building process and the minimizing process.