Section: New Results

Algebraic representations for geometric modeling

Multihomogeneous Polynomial Decomposition using Moment Matrices

Participants : Alessandra Bernardi, Jérôme Brachat, Bernard Mourrain.

In [33] , we address the important problem of tensor decomposition which can be seen as a generalisation of Singular Value Decomposition for matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and we give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterization of border bases. A new algorithm is described: it applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester on binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.

This is a joint work with Pierre Comon (I3S, CNRS).

On the variety parametrizing completely decomposable polynomials.

Participant : Alessandra Bernardi.

The purpose of the paper [15] is to relate the variety parameterizing completely decomposable homogeneous polynomials of degree $d$ in $n+1$ variables on an algebraically closed field, called ${\mathrm{Split}}_d\left({\mathbb{P}}^n\right)$, with the Grassmannian of $n-1$ dimensional projective subspaces of ${\mathbb{P}}^{n+d-1}$. We compute the dimension of some secant varieties to ${\mathrm{Split}}_d\left({\mathbb{P}}^n\right)$ and find a counterexample to a conjecture that wanted its dimension related to the one of the secant variety to $𝔾(n-1,n+d-1)$. Moreover by using an invariant embedding of the Veronse variety into the Plücker space, then we are able to compute the intersection of $𝔾(n-1,n+d-1)$ with ${\mathrm{Split}}_d\left({\mathbb{P}}^n\right)$, some of its secant variety, the tangential variety and the second osculating space to the Veronese variety.

This is a joint work with Enrique Arrondo (Universidad Complutense de Madrid, Spain)

Computing symmetric rank for symmetric tensors.

Participant : Alessandra Bernardi.

In [21] we consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for $2\times ...\times 2$ tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetric rank strata for some secant varieties of Veronese varieties.

This is a joint work with Alessandro Gimigliano and Monica Idà (Univesrità di Bologna, Italy).

Higher secant varieties of ${\mathbb{P}}^n\times {\mathbb{P}}^m$ embedded in bi-degree $(1,d)$.

Participant : Alessandra Bernardi.

Let $X_{(1,d)}^{(n,m)}$ denote the Segre-Veronese embedding of ${\mathbb{P}}^n\times {\mathbb{P}}^m$ via the sections of the sheaf $𝒪(1,d)$. In [20] we study the dimensions of higher secant varieties of $X_{(1,d)}^{(n,m)}$ and we prove that there is no defective $s^{th}$ secant variety, except possibly for $n$ values of $s$. Moreover when $\left(\genfrac{}{}{0pt}{}{m+d}{d}\right)$ is multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X_{(1,d)}^{(n,m)}$ has the expected dimension for every $s$.

This is a joint work with Enrico Carlini (Politecnico di Torino, Italy, Maria Virginia Catalisano (Università di Genova, Italy).

On the $X$-rank with respect to linear projections of projective varieties.

Participant : Alessandra Bernardi.

In [17] we improve the known bound for the $X$-rank $R_X\left(P\right)$ of an element $P\in {\mathbb{P}}^N$ in the case in which $X\subset {\mathbb{P}}^n$ is a projective variety obtained as a linear projection from a general $v$-dimensional subspace $V\subset {\mathbb{P}}^{n+v}$. Then, if $X\subset {\mathbb{P}}^n$ is a curve obtained from a projection of a rational normal curve $C\subset {\mathbb{P}}^{n+1}$ from a point $O\subset {\mathbb{P}}^{n+1}$, we are able to describe the precise value of the $X$-rank for those points $P\in {\mathbb{P}}^n$ such that $R_X\left(P\right)\le R_C\left(O\right)-1$ and to improve the general result. Moreover we give a stratification, via the $X$-rank, of the osculating spaces to projective cuspidal projective curves $X$. Finally we give a description and a new bound of the $X$-rank of subspaces both in the general case and with respect to integral non-degenerate projective curves.

This is a joint work with Edoardo Ballico (Università di Trento, Italy).

Decomposition of homogeneous polynomials with low rank.

Participant : Alessandra Bernardi.

Let $F$ be a homogeneous polynomial of degree $d$ in $m+1$ variables defined over an algebraically closed field of characteristic zero and suppose that $F$ belongs to the $s$-th secant varieties of the standard Veronese variety $X_{m,d}\subset {\mathbb{P}}^{\left(\genfrac{}{}{0pt}{}{m+d}{d}\right)-1}$ but that its minimal decomposition as a sum of $d$-th powers of linear forms $M_1,...,M_r$ is $F=M_1^d+\cdots +M_r^d$ with $r>s$. In [16] we show that if $s+r\le 2d+1$ then such a decomposition of $F$ can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of $F$ if the rank is at most $d$ and a mild condition is satisfied.

This is a joint work with Edoardo Ballico (Università di Trento, Italy).

On the $X$-rank with respect to linearly normal curves.

Participant : Alessandra Bernardi.

In [18] we study the $X$-rank of points with respect to smooth linearly normal curves $X\subset {\mathbb{P}}^n$ of genus $g$ and degree $n+g$.

We prove that, for such a curve $X$, under certain circumstances, the $X$-rank of a general point of $X$-border rank equal to $s$ is less or equal than $n+1-s$.

In the particular case of $g=2$ we give a complete description of the $X$-rank if $n=3,4$; while if $n\ge 5$ we study the $X$-rank of points belonging to the tangential variety of $X$.

This is a joint work with Edoardo Ballico (Università di Trento, Italy).

Symmetric tensor rank with a tangent vector: a generic uniqueness theorem

Participant : Alessandra Bernardi.

Let $X_{m,d}\subset {\mathbb{P}}^N$, $N:=\left(\genfrac{}{}{0pt}{}{m+d}{m}\right)-1$, be the order $d$ Veronese embedding of ${\mathbb{P}}^m$. Let $\tau \left(X_{m,d}\right)\subset {\mathbb{P}}^N$, be the tangent developable of $X_{m,d}$. For each integer $t\ge 2$ let $\tau (X_{m,d},t)\subseteq {\mathbb{P}}^N$, be the join of $\tau \left(X_{m,d}\right)$ and $t-2$ copies of $X_{m,d}$. In [19] we prove that if $m\ge 2$, $d\ge 7$ and $t\le 1+\lfloor \left(\genfrac{}{}{0pt}{}{m+d-2}{m}\right)/(m+1)\rfloor$, then for a general $P\in \tau (X_{m,d},t)$ there are uniquely determined $P_1,\cdots ,P_{t-2}\in X_{m,d}$ and a unique tangent vector $\nu$ of $X_{m,d}$ such that $P$ is in the linear span of $\nu \cup \{P_1,\cdots ,P_{t-2}\}$, i.e. a degree $d$ linear form $f$ (a symmetric tensor $T$ of order $d$) associated to $P$ may be written as

with $L_i$ linear forms on ${\mathbb{P}}^m$ ($v_i$ vectors over a vector field of dimension $m+1$ respectively), $1\le i\le t$, that are uniquely determined (up to a constant).

This is a joint work with Edoardo Ballico (Università di Trento, Italy).

Parametrization of computational domain in isogeometric analysis: methods and comparison

Participants : André Galligo, Bernard Mourrain.

Parameterization of computational domain plays an important role in isogeometric analysis as mesh generation in finite element analysis. In this paper, we investigate this problem in the 2D case, i.e, how to parametrize the computational domains by planar B-spline surface from the given CAD objects (four boundary planar B-spline curves). Firstly, two kinds of sufficient conditions for injective B-spline parameterization are derived with respect to the control points. Then we show how to find good parameterization of computational domain by solving a constraint optimization problem, in which the constraint condition is the injectivity sufficient conditions of planar B-spline parametrization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of planar B-spline parameterization. By using this method, the resulted parameterization has no self-intersections, and the isoparametric net has good uniformity and orthogonality. After introducing a posteriori error estimation for isogeometric analysis, we propose $r$-refinement method to optimize the parameterization by repositioning the inner control points such that the estimated error is minimized. Several examples are tested on isogeometric heat conduction problem to show the effectiveness of the proposed methods and the impact of the parameterization on the quality of the approximation solution. Comparison examples with known exact solutions are also presented. This joint work with Régis Duvigneau (EPI OPALE) and Gang Xu (Hangzhou Dianzi University, China) is published in [31] .

Variational Harmonic Method for Parameterization of Computational Domain in 2D Isogeometric Analysis

Participants : André Galligo, Bernard Mourrain.

In isogeometric anlaysis, parameterization of computational domain has great effects as mesh generation in finite element analysis. In this paper, based on the concept of harmonic map from the computational domain to parametric domain, a variational approach is proposed to construct the parameterization of computational domain for 2D isogeometric analysis. Different from the previous elliptic mesh generation method in finite element analysis, the proposed method focus on isogeometric version, and converts the elliptic PDE into a nonlinear optimization problem. A regular term is integrated into the optimization formulation to achieve more uniform grid near convex (concave) parts of the boundary. Several examples are presented to show the efficiency of the proposed method.

This joint work with Régis Duvigneau (EPI OPALE) and Gang Xu (Hangzhou Dianzi University, China) is published in [36] .

Warp-based Helical Implicit Primitives

Participant : Evelyne Hubert.

Implicit modeling with skeleton-based primitives has been limited up to now to planar skeletons elements, since no closed-form solution was found for convolution along more complex curves. We show that warping techniques can be adapted to efficiently generate convolution-like implicit primitives of varying radius along helices, a useful 3D skeleton found in a number of natural shapes. Depending on a single parameter of the helix, we warp it onto an arc of circle or onto a line segment. For those latter skeletons closed form convolutions are known for entire families of kernels. The new warps introduced preserve the circular shape of the normal cross section to the primitive.

This is joint work with Cédric Zanni and Marie-Paule Cani from the project-team EVASION (INRIA Grenoble Rhône-Alpes / LJK Laboratoire Jean Kuntzmann) which is publiched in [37] .