Jean-Paul Chehab

Incremental incomplete LU factorizations with applications

This paper addresses the problem of computing preconditioners for solving linear systems of equations with a sequence of slowly varying matrices. This problem arises in many important applications. For example, a common situation in computational fluid dynamics, is when the equations change only slightly, possibly in some parts of the physical domain. In such situations it is wasteful to recompute entirely any LU or ILU factorizations computed for the previous coefficient matrix. A number of techniques for computing incremental ILU factorizations are examined. For example we consider methods based on approximate inverses as well as alternating techniques for updating the factors L and U of the factorization.

An implicit preconditioning strategy for large-scale generalized Sylvester equations

Abstract Large-scale generalized Sylvester equations appear in several important applications. Although the involved operator is linear, solving them requires specialized techniques. Different numerical methods have been designed to solve them, including direct factorization methods suitable for small size problems, and Krylov-type iterative methods for large-scale problems. For these iterative schemes, preconditioning is always a difficult task that deserves to be addressed. We present and analyze an implicit preconditioning strategy specially designed for solving generalized Sylvester equations that uses a preconditioned residual direction at every iteration. The advantage is that the preconditioned direction is built implicitly, avoiding the explicit knowledge of the given matrices. Only the effect of the matrix–vector product with the given matrices is required. We present encouraging numerical experiments for a set of different problems coming from several applications.

Stabilized Times Schemes for High Accurate Finite Differences Solutions of Nonlinear Parabolic Equations

The Residual Smoothing Scheme (RSS) have been introduced in Averbuch et al. (A fast and accurate multiscale scheme for parabolic equations, unpublished) as a backward Euler’s method with a simplified implicit part for the solution of parabolic problems. RSS have stability properties comparable to those of semi-implicit schemes while giving possibilities for reducing the computational cost. A similar approach was introduced independently in Costa (Time marching techniques for the nonlinear Galerkin method, 1998), Costa et al. (SIAM J Sci Comput 23(1):46–65, 2001) but from the Fourier point of view. We present here a unified framework for these schemes and propose practical implementations and extensions of the RSS schemes for the long time simulation of nonlinear parabolic problems when discretized by using high order finite differences compact schemes. Stability results are presented in the linear and the nonlinear case. Numerical simulations of 2D incompressible Navier–Stokes equations are given for illustrating the robustness of the method.

Parallel matrix function evaluation via initial value ODE modeling

The purpose of this article is to propose ODE based approaches for the numerical evaluation of matrix functions f ( A ) , a question of major interest in the numerical linear algebra. For that, we model f ( A ) as the solution at a finite time T of a time dependent equation. We use parallel algorithms, such as the parareal method, on the time interval 0 , T in order to solve the obtained evolution equation. When f ( A ) is reached as a stable steady state, it can be computed by combining parareal algorithms and optimal control techniques. Numerical illustrations are given.