Bernard Philippe
Title :
A Parallel GMRES Method Preconditioned by a Multiplicative Schwarz Iteration
Abstract :
Domain decomposition provides a class of methods suitable for the solutions of linear or
nonlinear systems of equations arising from the discretization of partial differential
equations. For linear problems, domain decomposition methods are often used as
preconditioner for Krylov subspace iterations.
Traditionally, there are two classes of iterative method which derive from domain
decomposition with overlap: say Additive Schwarz and Multiplicative Schwarz. When using
those two methods as solvers, the convergence rates are very slow and the convergence is
just proved for symmetric positive definite matrices and M-matrix. For that reason, the
particular interest of Schwarz methods is as preconditioner of Krylov subspace methods.
The additive version is usually preferred because it is easily implemented on a parallel
computer although it is usually a less efficient preconditioner than its multiplicative
version.
The challenge of this work was to derive a fully automatic parallel GMRES method
preconditioned through a Multiplicative Schwarz iteration based on algebraic domain
decomposition.
For that purpose the following results have been obtained:
- Construction of an automatic 1D partitionner of a sparse matrix,
- Derivation of an explicit expression of the preconditioner,
- Parallel pipeline to build a basis of the Krylov subspace which is then orthogonalized,
- Control of the dimension of the basis trough an estimation of the involved roundoff errors,
- Design of a code in the PETSc format.
Numerical experiments illustrate the results.