## Section: New Results

### Preconditioning techniques for solving large systems of equations

Participants : Laura Grigori, Riadh Fezzanni, Sophie Moufawad.

A different direction of research is related to preconditioning large sparse linear systems of equations. This research is performed in the context of ANR PETALh project (2011-2012), which follows the ANR PETAL project (2008-2009). It is conducted in collaboration with Frederic Nataf from University Paris 6.

Several highly used preconditioners are for example the incomplete LU factorizations and Schwarz based approaches as used in domain decomposition. Most of these preconditioners are known to have scalability problems. The number of iterations can increase significantly when the size of the problem increases or when the number of independent domains is increased. This is often due to the presence of several low frequency modes that hinder the convergence of the iterative method. To address this problem, we study a different class of preconditioners, called direction preserving or filtering preconditioners. These preconditioners have the property of being identical to the input matrix on a given filtering vector. A judicious choice of the vector allows to alleviate the effect of low frequency modes on the convergence.

We consider in particular two classes of preconditioners. The first preconditoner is an incomplete decomposition that satisfies the filtering property [13] . The nested preconditioner has the same property for a specific vector of all ones. However the construction is different and takes advantage of a nested structure of the input matrix. The previous research on these methods considered only matrices arising from the discretization of PDEs on structured grids, where the matrix has a block tridiagonal structure. This structure imposes a sequential computation of the preconditioner and it is not suitable for the more general case of unsructured grids. Hence, while very efficient, the usage of these preconditioners was very limited. At the beginning of this research we have obtained several theoretical results for these methods that demonstrate their numerical behavior and convergence properties for cases arising from the discretization of PDEs on structured grids [13] . But the main result is the development of a generalized method [10] , [11] that has two important properties: it allows the filtering property to be satisfied for any input matrix; the matrix can be reordered such that its computation is highly parallel. Experimental results show that the method is very efficient for certain classes of matrices, and shows good scalability results in terms of both problem size and number of processors. In addition to finalizing this work, our research also focused on extending the block filtering factorization to include other approximation techniques that allowed us to introduce a parameter whose tuning permits to solve very difficult problems.