Sophie Moufawad

Noura Al Helwani, Sophie Moufawad, Georges Sakr

Many physical and engineering systems require solving direct problems to predict behavior and inverse problems to determine unknown parameters from measurement. In this work, we study both aspects for systems governed by differential equations, contrasting well-established numerical methods with new AI-based techniques, specifically Physics-Informed Neural Networks (PINNs). We first analyze the logistic differential equation, using its closed-form solution to verify numerical schemes and validate PINN performance. We then address the Porous Medium Equation (PME), a nonlinear partial differential equation with no general closed-form solution, building strong solvers of the direct problem and testing techniques for parameter estimation in the inverse problem. Our results suggest that PINNs can closely estimate solutions at competitive computational cost, and thus propose an effective tool for solving both direct and inverse problems for complex systems.

Sophie Moufawad

Recently, enlarged Krylov subspace methods, that consists of enlarging the Krylov subspace by a maximum of t vectors per iteration based on the domain decomposition of the graph of A, were introduced in the aim of reducing communication when solving systems of linear equations Ax=b. In this paper, the s-step enlarged Krylov subspace Conjugate Gradient methods are introduced, whereby s iterations of the enlarged Conjugate Gradient methods are merged in one iteration. The numerical stability of these s-step methods is studied, and several numerically stable versions are proposed. Similarly to the enlarged Krylov subspace methods, the s-step enlarged Krylov subspace methods have a faster convergence than Krylov methods, in terms of iterations. Moreover, by computing st basis vectors of the enlarged Krylov subspace $\mathscr{K}_{k,t}(A,r_0)$ at the beginning of each s-step iteration, communication is further reduced. It is shown in this paper that the introduced methods are parallelizable with less communication, with respect to their corresponding enlarged versions and to Conjugate Gradient.

Sophie Moufawad, Nabil Nassif, Faouzi Triki

We consider the mathematical model of gas trapping in deep polar ice (firns), which consists of a parabolic partial differential equation, that can degenerate at one boundary extreme. In [1], we considered all the coefficients to be constants, except the diffusion coefficient D(z) that is to be reconstructed. In this paper, we assume both the diffusion coefficient D(z) and the volume fraction f(z) are functions. The difficulty in this problem, both theoretically and computationally, arises from the fact that D(z) and f(z) may be zero at bottom of the firn. To handle such degeneracy, we defined appropriate weighted Sobolev spaces and used Lion's theorem to prove existence and uniqueness of the semi-variational formulation of the Firn PDE. A full discrete system is obtained through a P1 Finite element Galerkin procedure in space and an Euler-Implicit scheme in time. Sufficient conditions for the existence and uniqueness of the solution for the discrete system are obtained.