Houssam Khalil

Lebanese University · Faculty of Science

The Symmetric Tensor Approximation problem (STA) consists of approximating a symmetric tensor or a homogeneous polynomial by a linear combination of symmetric rank-1 tensors or powers of linear forms of low symmetric rank. We present two new Riemannian Newton-type methods for low rank approximation of symmetric tensor with complex coefficients. The...

The symmetric tensor rank approximation problem (STA) consists in computing the best low rank approximation of a symmetric tensor. We describe a Riemannian Newton iteration with trust region scheme for the STA problem. We formulate this problem as a Riemannian optimization problem by parameterizing the constraint set as the Cartesian product of Ver...

We study the decomposition of a multi-symmetric tensor T as a sum of powers of product of linear forms in correlation with the decomposition of its dual T ∗ as a weighted sum of evaluations. We use the properties of the associated Artinian Gorenstein Algebra Aτ to compute the decomposition of its dual T ∗ which is defined via a formal power series...

We study the decomposition of a multivariate Hankel matrix H\_$\sigma$ as a sum of Hankel matrices of small rank in correlation with the decomposition of its symbol $\sigma$ as a sum of polynomial-exponential series. We present a new algorithm to compute the low rank decomposition of the Hankel operator and the decomposition of its symbol exploitin...

We prove the existence and the uniqueness of entire solutions of elliptic Hessian equations in \(\mathbb {C}^{n}\) with rotational invariance in an appropriate weighted Hölder spaces.

We present a resultant-based method to calculate the overdetermined strata for degree 4 hyperbolic polynomials in one variable. It is a new method to calculate overdetermined strata. We present also the complete study of the overdetermined strata for degree 4 hyperbolic polynomials by the geometric method.

In this paper we calculate the Witt groups of P 1. It's a known result, but we calculate it by another method: we use the localisation theorem of Balmer and the excision theorem of S. Gille. Let X be a scheme which contains 1 2 and V BX be the category of locally free coherent OX-modules, i.e. vector bundles. Let L be a line bundle over X. We defin...

In the present paper, we consider degree 5 hyperbolic polynomials (HPs) in one variable (i.e., real and with all roots real). We are interested in such HPs whose number of equalities between roots of the polynomial and its derivatives is greater or equal to 4. We give a new method, based on resultants and subresultants, to calculate the overdetermi...

We propose a level-set model of phase change and apply it to the study of the Leidenfrost effect. The new ingredients used in this model are twofold: first we enforce by penalization the droplet temperature to the saturation temperature in order to ensure a correct mass transfer at interface, and second we propose a careful differentiation of the c...

We present a new superfast algorithm for solving Toeplitz systems. This algorithm is based on a relation between the solution of such problems and syzygies of polynomials or moving lines. We show an explicit connection between the generators of a Toeplitz matrix and the generators of the corresponding module of syzygies. We show that this module is...

Nous présentons une méthode directe pour résoudre un système de Toeplitz bande par blocs de Toeplitz bandes avec une complexité de opérations arithmétiques. L'idée de cet algorithme est de plonger une matrice de Toeplitz bande biniveaux dans une matrice circulante biniveaux. La technique de plonger (resp. de transformer) une matrice de Toeplitz ban...

In this paper, we re-investigate the resolution of Toeplitz systems $T u =g$, from a new point of view, by correlating the solution of such problems with syzygies of polynomials or moving lines. We show an explicit connection between the generators of a Toeplitz matrix and the generators of the corresponding module of syzygies. We show that this mo...

Several problems in applied mathematics require the solving of linear systems with very large sizes, and sometimes these systems must be solved multiple times. In such cases, the standard algorithms based on the Gauss elimination require O (n ^ 3) arithmetic operations to solve a system of size n, and it will be a handicap for the computation. That...

Let $T$ be a Toeplitz block Toeplitz matrix with coefficients in a field $K$ of size $N$. Assume that the blocks are of size $n\times n$ and that there are $n\times n$ blocks. Assume moreover that the matrix is block banded and that the blocks themselves have a block structure. This means that outside of the $2k_1+1$ central block diagonals, the bl...

Plusieurs problèmes en mathématiques appliquées requièrent la résolution de systèmes linéaires de très grandes tailles, et parfois ces systèmes doivent être résolus de multiples fois. Dans de tels cas, les algorithmes standards basés sur l'élimination de Gauss demandent O(n^3) opérations arithmétiques pour résoudre un système de taille n, et ce ser...

We present an algorithm for solving polynomial equations, which uses generalized eigenvalues and eigenvectors of resultant matrices. We give special attention to the case of two bivariate polynomials and the Sylvester or Bezout resultant constructions. We propose a new method to treat multiple roots, detail its numerical aspects and describe experi...