Parallelization of Krylov Methods
Frédéric Magoulès, François‐Xavier Roux, Guillaume Houzeaux · 2015
The Krylov methods require three types of operations: matrix-vector products, scalar products and linear vector combinations. This point is discussed in detail in this chapter, particularly in the context of substructuring methods for sparse matrices. The chapter then compares the matrix-vector products obtained with the parallelizations based on node and element sets. The decomposition method based on elements requires duplicating the vector components associated with the equations of the interfaces. In the case of division into several subdomains, it is necessary to find an algorithm that does not duplicate the contributions of the interface nodes to the scalar product. This issue can be solved in different ways. This chapter presents three of them, referred to by weight, distributivity and ownership. The exchange of data on the interfaces is carried out using point-to-point functions of MPI. These exchanges are local as a subdomain only exchanges data with its neighbors.