Block filtering decomposition

Abstract

International audienceThis paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so-called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low-frequency modes on convergence and so decrease or eliminate the plateau that is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process. We show the efficiency of our preconditioner through a set of numerical experiments on symmetric and nonsymmetric matrices

Parallel design and performance of nested filtering factorization preconditioner

“…Given a set of vectors T which represent the directions to be preserved, the preconditioner M satisfies a right filtering property M T = AT . This is a property which has been exploited in different contexts, as multigrid methods [6], semiseparable matrices [14], incomplete factorizations [21,1,8], or nested factorization [4]. It is well known that for difficult problems with heterogeneities or multiscale physics, the iterative methods can converge very slowly, and this is often due to the presence of several low frequency modes.…”

“…Different preconditioners can be obtained by using different approximations, and more details can be found in [10]. The specific approximation used in NFF that allows to satisfy the right filtering property was introduced in [12] and is similar to the approach used in block filtering decomposition [11,8], but adapted to a recursive computation. A multithreaded implementation based on Cilk is reported in [16] and also in [17].…”