Fast solver for band Toeplitz-block-band Toeplitz system

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 blocks vanish, and that outside of the central $2k_2+1$ central diagonals, the coefficients in each block vanish. In this article, we give three direct methods of resolution with a count of $O(N^{3/2})$ operations to resolve the problem $Tx=b$. We give also a statistical study which prouve that the matrix $T$ becomes increasingly ill conditioned when the band widths decrease

Axisymmetric Level Set model of Leidenfrost effect

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 capillary interface with respect to a moving interface with phase change. We perform some numerical tests in the axisymmetric case and show that our numerical method, while not avoiding well known numerical caveats of diffuse interface methods, behave quite well in the limit of numerical interface width going to zero in comparison to an analytical formula

Decomposition of Low Rank Multi-Symmetric Tensor

International audienceWe 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 $A_\tau$ to compute the decomposition of its dual $T^*$ which is defined via a formal power series $τ$. We use the low rank decomposition of the Hankel operator $H_\tau$ associated to the symbol $\tau$ into a sum of indecomposable operators of low rank. A basis of $A_\tau$ is chosen such that the multiplication by some variables is possible. We compute the sub-coordinates of the evaluation points and their weights using the eigen-structure of multiplication matrices. The new algorithm that we propose works for small rank. We give a theoretical generalized approach of the method in n dimensional space. We show a numerical example of the decomposition of a multi-linear tensor of rank 3 in 3 dimensional space

Structured low rank decomposition of multivariate Hankel matrices

International audienceWe study the decomposition of a multivariate Hankel matrix H_σ as a sum of Hankel matrices of small rank in correlation with the decomposition of its symbol σ 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 exploiting the properties of the associated Artinian Gorenstein quotient algebra A_σ. A basis of A_σ is computed from the Singular Value Decomposition of a sub-matrix of the Hankel matrix H_σ. The frequencies and the weights are deduced from the generalized eigenvectors of pencils of shifted sub-matrices of H σ. Explicit formula for the weights in terms of the eigenvectors avoid us to solve a Vandermonde system. This new method is a multivariate generalization of the so-called Pencil method for solving Prony-type decomposition problems. We analyse its numerical behaviour in the presence of noisy input moments, and describe a rescaling technique which improves the numerical quality of the reconstruction for frequencies of high amplitudes. We also present a new Newton iteration, which converges locally to the closest multivariate Hankel matrix of low rank and show its impact for correcting errors on input moments

Resultant-based methods for plane curves intersection problems

http://www.springeronline.com/3-540-28966-6We 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 experiments on tangential problems, which show the efficiency of the approach. An industrial application of the method is presented at the end of the paper. It consists in recovering cylinders from a large cloud of points and requires intensive resolution of polynomial equations

Toeplitz and Toeplitz-block-Toeplitz matrices and their correlation with syzygies of polynomials.

International audienceIn 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 module is generated by two elements of degree $n$ and the solution of $T\,u=g$ can be reinterpreted as the remainder of an explicit vector depending on $g$, by these two generators

Matrices structurées et matrices de Toeplitz par blocs de Toeplitz en calcul numérique et formel

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 is why we look for using of the structure to reduce the computation's time. The structure of Toeplitz, Hankel, Cauchy, Vandermonde and other more general structure are very well expoiled to reduce the complexity of solving a linear system to O (n log n ^ 2) arithmetic operations. The two levels structured matrices and especially Toeplitz block Toeplitz (TBT) matrices appear in many applications. The purpose of this work is to find fast algorithms for TBT systems. In this thesis, we describe the difficulties of this problem. We give three fast algorithms, of complexity O (n ^ 3 / 2) operations, for Toeplitz band block Toeplitz band systems. we also give a new method of solving Toeplitz systems by giving a relationship between the solution of a Toeplutz system and syzygies of polynomials in one variable. We generalize this method for TBT matrices and we give a relationship between the solution of such linear system and syzygies of polynomials in two variables.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 sera un handicap pour le calcul. C'est pour cela qu'on cherche à utiliser la structure pour réduire le temps de calcul. La structure de Toeplitz, de Hankel, de Cauchy, de Vandermonde et d'autre structure plus générales sont bien exploitées pour réduire la complexité de résolution d'un système linéaire à O(n log^2 n) opérations arithmétiques. Les matrices structurées en deux niveaux et surtout les matrices de Toeplitz par blocs de Toeplitz (TBT) apparaissent dans beaucoup des applications. Le but de ce travail est de trouver des algorithmes de résolution rapide pour des systèmes TBT de grande taille. Dans cette thèse, on décrit les difficultés de ce problème. On donne trois algorithmes rapide, en O(n^3/2) opérations, de résolution pour les systèmes de Toeplitz bande par blocs Toeplitz bande. On donne aussi une nouvelle méthode de résolution des systèmes de Toeplitz scalaires en donnant une relation entre la solution d'un système de Toeplitz scalaires et les syzygies des polynômes en une seule variable. On généralise cette méthode pour les matrices TBT et on donne une relation entre la solution d'un tel système linéaire et les syzygies des polynômes en deux variables

Toeplitz and Toeplitz-block-Toeplitz matrices and their correlation with syzygies of polynomials.

International audienceIn 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 module is generated by two elements of degree $n$ and the solution of $T\,u=g$ can be reinterpreted as the remainder of an explicit vector depending on $g$, by these two generators