Jean-Paul Chehab

Jean-Paul Chehab, Marcos Raydan

We extend the geometrical inverse approximation approach for solving linear least-squares problems. For that we focus on the minimization of $1-\cos(X(A^TA),I)$, where $A$ is a given rectangular coefficient matrix and $X$ is the approximate inverse. In particular, we adapt the recently published simplified gradient-type iterative scheme MinCos to the least-squares scenario. In addition, we combine the generated convergent sequence of matrices with well-known acceleration strategies based on recently developed matrix extrapolation methods, and also with some deterministic and heuristic acceleration schemes which are based on affecting, in a convenient way, the steplength at each iteration. A set of numerical experiments, including large-scale problems, are presented to illustrate the performance of the different accelerations strategies.

Jean-Paul Chehab, Marcos Raydan

We focus on inverse preconditioners based on minimizing $F(X) = 1-\cos(XA,I)$, where $XA$ is the preconditioned matrix and $A$ is symmetric and positive definite. We present and analyze gradient-type methods to minimize $F(X)$ on a suitable compact set. For that we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of $F(X)$ on the feasible set. Preliminary and encouraging numerical results are also presented in which dense and sparse approximations are included.

Hyam Abboud, Clara Al Kosseifi, Jean-Paul Chehab

We introduce a family of bi-grid schemes in finite elements for solving 2D incompressible Navier-Stokes equations in velocity and pressure $(u,p)$. The new schemes are based on projection methods and use two pairs of FEM spaces, a sparse and a fine one. The main computational effort is done on the coarsest velocity space with an implicit and unconditionally time scheme while its correction on the finer velocity space is realized with a simple stabilized semi-implicit scheme whose the lack of stability is compensated by a high mode stabilization procedure; the pressure is updated using the free divergence property. The new schemes are tested on the lid driven cavity up to $Re=7500$. An enhanced stability is observed as respect to classical semi-implicit methods and an important gain of CPU time is obtained as compared to implicit projection schemes.

Hyam Abboud, Clara Al Kosseifi, Jean-Paul Chehab

In this work, we propose a bi-grid scheme framework for the Allen-Cahn equation in Finite Element Method. The new methods are based on the use of two FEM spaces, a coarse one and a fine one, and on a decomposition of the solution into mean and fluctuant parts. This separation of the scales, in both space and frequency, allows to build a stabilization on the high modes components: the main computational effort is concentrated on the coarse space on which an implicit scheme is used while the fluctuant components of the fine space are updated with a simple semi-implicit scheme, they are smoothed without damaging the consistency. The numerical examples we give show the good stability and the robustness of the new methods. An important reduction of the computation time is also obtained when comparing our methods with fully implicit ones.

Jean-Paul Chehab, Madalina Petcu

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. To this end, 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 evolution equation obtained. 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.