In , we consider particular types of additive decompositions of homogeneous polynomials. The classical decomposition is the Waring decomposition, where we decompose polynomials as sums of powers of linear forms. Another well studied decomposition is the sometimes-called Chow decomposition, where we decompose polynomials as sums of products of linear forms. These are the extremal cases of the additive decompositions considered in this work. For a fixed partition $(d_1,...,d_s)⊢d$ of the degree of the polynomial, we consider decompositions as sums of degree forms of the form ${\ell }_1^{d_1}\cdots {\ell }_s^{d_s}$, where the $\ell$'s are linear forms. The homogeneous polynomials of the form ${\ell }_1^{d_1}\cdots {\ell }_s^{d_s}$ are parametrized by particular linear projections of certain Segre-Veronese varieties. The main results of this work concerns the dimension of the secant varieties to these projections of Segre-Veronese varieties. In particular, we compute their dimensions in the binary case (forms in two variables) and the case of secant lines varieties for any partition and any number of variables. From these results, we deduce the dimension of higher secant varieties in some particular cases.

This is a joint work with M. V. Catalisano, Luca Chiantini, and A. V. Geramita.

In , we introduce the notion of the Waring locus of a homogeneous polynomial. A Waring decomposition is an expression of a polynomial as sum of powers of linear forms. The smallest length of such a decomposition is called the Waring rank of the polynomial. A very difficult challenge is to compute the rank and a minimal decomposition of a given form. The Waring locus of a polynomial is the locus of linear forms that appear in a minimal decomposition of it. The idea behind this construction is to find an iterative approach to construct Waring decompositions step-by-step, by adding one power at the time. Moreover, we give a version of the famous Strassen conjecture on the additivity of rank for sums of polynomials in independent sets of variables. We compute the Waring loci in several cases as binary forms, quadrics, monomials and plane cubics and for some other particular families of polynomials.

This is a joint work with E. Carlini, and M. V. Catalisano.

In , we study two particular geometric constructions. Given two affine algebraic varieties, we define their Minkowski sum as the (Zariski) closure of the set of coefficient-wise sums of pairs of points in the two varieties. Given two projective varieties, we define their Hadamard product as the (Zariski) closure of the set of coefficient-wise multiplications of pairs of points in the two varieties. In particular, we focus on computing their dimensions and degrees in terms of the ones of the original varieties. Hadamard products are of particular interests as they can be used to parametrize particular families of tensors which rise naturally by studying Restricted Boltzmann Machines, which are particular structures used in Statistics and Machine Learning.

This is a joint work with N. Friedenberg, and R. Williams.

In , we analyze the decomposition problem of multivariate polynomial-exponential functions from truncated series and present new algorithms to compute their decomposition. Using the duality between polynomials and formal power series, we first show how the elements in the dual of an Artinian algebra correspond to polynomial-exponential functions. They are also the solutions of systems of partial differential equations with constant coefficients. We relate their representation to the inverse system of the roots of the characteristic variety. Using the properties of Hankel operators, we establish a correspondence between polynomial exponential series and Artinian Gorenstein algebras. We generalize Kronecker theorem to the multivariate case, by showing that the symbol of a Hankel operator of finite rank is a polynomial-exponential series and by connecting the rank of the Hankel operator with the decomposition of the symbol. A generalization of Prony's approach to multivariate decomposition problems is presented, exploiting eigenvector methods for solving polynomial equations. We show how to compute the frequencies and weights of a minimal polynomial-exponential decomposition, using the first coefficients of the series. A key ingredient of the approach is the flat extension criteria, which leads to a multivariate generalization of a rank condition for a Carathéodory-Fejér decomposition of multivariate Hankel matrices. A new algorithm is given to compute a basis of the Artinian Gorenstein algebra, based on a Gram-Schmidt orthogonalization process and to decompose polynomial-exponential series. A general framework for the applications of this approach is described and illustrated in different problems. We provide Kronecker-type theorems for convolution operators, showing that a convolution operator (or a cross-correlation operator) is of finite rank, if and only if, its symbol is a polynomial-exponential function, and we relate its rank to the decomposition of its symbol. We also present Kronecker-type theorems for the reconstruction of measures as weighted sums of Dirac measures from moments and for the decomposition of polynomial-exponential functions from values. Finally, we describe an application of this method for the sparse interpolation of polylog functions from values.

Given a multi-index sequence $\sigma$, we present in a new efficient algorithm to compute generators of the linear recurrence relations between the terms of $\sigma$. We transform this problem into an algebraic one, by identifying multi-index sequences, multivariate formal power series and linear functionals on the ring of multivariate polynomials. In this setting, the recurrence relations are the elements of the kernel $I_{\sigma }$ of the Hankel operator $H_{\sigma }$ associated to $\sigma$. We describe the correspondence between multi-index sequences with a Hankel operator of finite rank and Artinian Gorenstein Algebras. We show how the algebraic structure of the Artinian Gorenstein algebra $A_{\sigma }$ associated to the sequence $\sigma$ yields the structure of the terms ${\sigma }_{\alpha }$ for all $\alpha \in {\mathbb{N}}^n$. This structure is explicitly given by a border basis of $A_{\sigma }$, which is presented as a quotient of the polynomial ring $K[x_1,...,x_n]$ by the kernel $I_{\sigma }$ of the Hankel operator $H_{\sigma }$. The algorithm provides generators of $I_{\sigma }$ constituting a border basis, pairwise orthogonal bases of $A_{\sigma }$ and the tables of multiplication by the variables in these bases. It is an extension of Berlekamp-Massey-Sakata (BMS) algorithm, with improved complexity bounds. We present applications of the method to different problems such as the decomposition of functions into weighted sums of exponential functions, sparse interpolation, fast decoding of algebraic codes, computing the vanishing ideal of points, and tensor decomposition. Some benchmarks illustrate the practical behavior of the algorithm.

In , 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 exploiting the properties of the associated Artinian Gorenstein quotient algebra $A_{\sigma }$. A basis of $A_{\sigma }$ is computed from the Singular Value Decomposition of a sub-matrix of the Hankel matrix $H_{\sigma }$. The frequencies and the weights are deduced from the generalized eigenvectors of pencils of shifted sub-matrices of $H_{\sigma }$. 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 analyze its numerical behavior 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.

This is a joint work with Houssam Khalil.

In , 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 $A_{\tau }$ 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 Hτ 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.

This is a joint work with Houssam Khalil.

A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop in computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projective varieties $X_1,...,X_k\subset {\mathbb{P}}^N$ defined over $ℂ$. After computing ranks over $ℂ$, we also explore computing real ranks. Various examples are included to demonstrate this numerical algebraic geometric approach.

This is a joint work with Alessandra Bernardi, Noah S. Daleo, Jonathan D. Hauenstein.

In , we consider rational maps whose source is a product of two subvarieties, each one being embedded in a projective space. Our main objective is to investigate birationality criteria for such maps. First, a general criterion is given in terms of the rank of a couple of matrices that became to be known as Jacobian dual matrices. Then, we focus on rational maps from the product of two projective lines to the projective plane in very low bidegrees and provide new matrix-based birationality criteria by analyzing the syzygies of the defining equations of the map, in particular by looking at the dimension of certain bigraded parts of the syzygy module. Finally, applications of our results to the context of geometric modeling are discussed at the end of the paper.

This is a joint work with Nicolás Botbol (University of Buenos Aires), Marc Chardin (University of Paris VI), Hamid Seyed Hassanzadeh (University of Rio de Janeiro), Aron Simis (University of Pernambuci) and Quang Hoa Tran (University of Paris VI).

In , we develop a new approach to the discriminant of a complete intersection curve in the 3-dimensional projective space. By relying on the resultant theory, we first prove a new formula that allows us to define this discriminant without ambiguity and over any commutative ring, in particular in any characteristic. This formula also provides a new method for evaluating and computing this discriminant efficiently, without the need to introduce new variables as with the well-known Cayley trick. Then, we obtain new properties and computational rules such as the covariance and the invariance formulas. Finally, we show that our definition of the discriminant satisfies to the expected geometric property and hence yields an effective smoothness criterion for complete intersection space curves. Actually, we show that in the generic setting, it is the defining equation of the discriminant scheme if the ground ring is assumed to be a unique factorization domain.

This is a joint work with Ibrahim Nonkané (University of Ouaga II).

In we examine implicit representations of parametric or point cloud models, based on interpolation matrices, which are not sensitive to base points. We show how interpolation matrices can be used for ray shooting of a parametric ray with a surface patch, including the case of high-multiplicity intersections. Most matrix operations are executed during pre-processing since they solely depend on the surface. For a given ray, the bottleneck is equation solving. Our Maple code handles bicubic patches within 1 second, though numerical issues might arise. Our second contribution is to extend the method to parametric space curves and, generally, to codimension > 1, by computing the equations of (hyper)surfaces intersecting precisely at the given object. By means of Chow forms, we propose a new, practical, randomized algorithm that always produces correct output but possibly with a non-minimal number of surfaces. For space curves, we obtain 3 surfaces whose polynomials are of near-optimal degree; in this case, computation reduces to a Sylvester resultant. We illustrate our algorithm through a series of examples and compare our Maple prototype with other methods implemented in Maple, i.e., Gröbner basis and implicit matrix representations. Our Maple prototype is not faster but yields fewer equations and seems more robust than Maple's implicitize; it is also comparable with the other methods for degrees up to 6.

Joint work with I.S. Kotsireas.

In , we present an efficient algorithm for the certication of numeric solutions to eigenproblems. The algorithm relies on a mixture of ball arithmetic, a suitable Newton iteration, and clustering of eigenvalues that are close.

This is a joint work with Joris Van Der Hoeven.

In we consider the approximation of two NP-hard problems: Minkowski Decomposition (MinkDecomp) of lattice polygons in the plane and the closely related problem of Multidimensional Subset Sum (kD-SS) in arbitrary dimension. In kD-SS we are given an input set $S$ of k-dimensional vectors, a target vector $t$ and we ask if there exists a subset of $S$ that sums up to $t$. We prove, through a gap-preserving reduction, that, for general dimension k, kD-SS is not in APX although the classic 1D-SS is in PTAS. On the positive side, we present an $O(n^3/{ϵ}^2)$ approximation grid based algorithm for 2D-SS, where $n$ is the cardinality of the set and $ϵ>0$ bounds the difference of some measure of the input polygon and the sum of the output polygons. We also describe two approximation algorithms with a better experimental ratio. Applying one of these algorithms, and a transformation from MinkDecomp to 2D-SS, we can approximate Mink-Decomp. For an input polygon $Q$ and parameter $ϵ$, we return two summands $A$ and $B$ such that $A+B=Q^{\text{'}}$ with $Q^{\text{'}}$ being bounded in relation to $Q$ in terms of volume, perimeter, or number of internal lattice points, an additive error linear in and up to quadratic in the diameter of $Q$. A similar function bounds the Hausdorff distance between $Q$ and $Q^{\text{'}}$. We offer experimental results based on our implementation.

Joint with C. Tzovas.

The construction of $r$-nets offers a powerful tool in computational and metric geometry. In , we focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate $r$-nets with respect to Euclidean distance. For any fixed $ϵ>0$, the approximation factor is $1+ϵ$ and the complexity is polynomial in the dimension and subquadratic in the number of points. The algorithm succeeds with high probability. More specifically, the best previously known LSH-based construction is improved in terms of complexity by reducing the dependence on $ϵ$, provided that $ϵ$ is sufficiently small. Our method does not require LSH but, instead, follows Valiant's (2015) approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which $r$-nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the $(1+ϵ)$-approximate kth nearest neighbor distance in time subquadratic in the size of the input.

Joint with G. Avarikioti, L. Kavouras.

In , a new algebraic method for extracting tori from a minimal point set, made of two oriented points and a simple point, is proposed. We prove a degree bound on the number of such tori; this bound is reached on examples, even when we restrict to smooth tori. Our method is based on pre-computed closed formulae well suited for numerical computations with approximate input data.

Given a skeleton made of line segments we describe how to obtain a coarse mesh (or scaffold) of a surface surrounding it. We emphasize in the key result that allows us to complete a previous approach that could not treat skeletons with cycles.

We analyze the space of differentiable functions on a quad-mesh $ℳ$, which are composed of 4-split spline macro-patch elements on each quadrangular face. We describe explicit transition maps across shared edges, that satisfy conditions which ensure that the space of differentiable functions is ample on a quad-mesh of arbitrary topology. These transition maps define a finite dimensional vector space of $G^1$ spline functions of bi-degree $\le (k,k)$ on each quadrangular face of $ℳ$. We determine the dimension of this space of $G^1$ spline functions for $k$ big enough and provide explicit constructions of basis functions attached respectively to vertices, edges and faces. This construction requires the analysis of the module of syzygies of univariate b-spline functions with b-spline function coefficients. New results on their generators and dimensions are provided. Examples of bases of $G^1$ splines of small degree for simple topological surfaces are detailed and illustrated by parametric surface constructions.

This is a joint work with Nelly Villamizar

Motivated by the Magneto HydroDynamic (MHD) simulation for Tokamaks with Isogeometric analysis, we present in a new type of splines defined over a rectangular mesh with arbitrary topology, which are piecewise polynomial functions of bidegree $(d,d)$ and $C^r$ parameter continuity. In particular, we compute their dimension and exhibit basis functions called Hermite bases for bicubic spline spaces. We investigate their potential applications for solving partial differential equations (PDEs) over a complex physical domain in the framework of Isogeometric analysis. In particular, we analyze the property of approximation of these spline spaces for the $L^2$-norm. Despite the fact that the basis functions are singular at extraordinary vertices, we show that the optimal approximation order and numerical convergence rates are reached by setting a proper parameterization.

This is a joint work with Meng Wu, Bernard Mourrain, André Galligo, Boniface Nkonga

Isogeometric analysis (IGA) is a method for solving geometric partial differential equations (PDEs). Generating parameterizations of a PDE's physical domain is the basic and important issues within IGA framework. In , we present a global $H^1$-parameterization method for a planar physical domain with complex topology.

This is a joint work with Meng Wu, Boniface Nkonga.

In , we present convergence rates for solving elliptic boundary value problems with singular parameterizations in isogeometric analysis. First, the approximation errors with the $L^2\left(\Omega \right)$-norm and the $H^1\left(\Omega \right)$-seminorm are estimated locally. The impact of singularities is considered in this framework. Second, the convergence rates for solving PDEs with singular parameterizations are discussed. These results are based on a weak solution space that contains all of the weak solutions of elliptic boundary value problems with smooth coefficients. For the smooth weak solutions obtained by isogeometric analysis with singular parameterizations and the finite element method, both are shown to have the optimal convergence rates. For non-smooth weak solutions, the optimal convergence rates are reached by setting proper singularities of a controllable parameterization, even though convergence rates are not optimal by finite element method, and the convergence rates by isogeometric analysis with singular parameterizations are better than the ones by the finite element method.

This a joint work with Meng Wu, Yicao Wang, Boniface Nkonga, Changzheng Cheng.

The precise control of geometric models plays an important role in many domains such as computer-aided geometric design and numerical simulation. For shape optimization in computational fluid dynamics (CFD), the choice of control parameters and the way to deform a shape are critical. In , we describe a skeleton-based representation of shapes adapted for CFD simulation and automatic shape optimization. Instead of using the control points of a classic B-spline representation, we control the geometry in terms of architectural parameters. We assure valid shapes with a strong shape consistency control. Deformations of the geometry are performed by solving optimization problems on the skeleton. Finally, a surface reconstruction method is proposed to evaluate the shape's performances with CFD solvers. We illustrate the approach on two problems: the foil of an AC45 racing sail boat and the bulbous bow of a fishing trawler. For each case, we obtain a set of shape deformations and then we evaluate and analyzed the performances of the different shapes with CFD computations.

This is a joint work with Yann Roux, Matthieu Durand, Guillaume Fontaine.

The work in describes a new generic parametric modeler integrated into an automated optimization loop for shape optimization. The modeler enables the generation of shapes by selecting a set of design parameters that controls a twofold parameterization: geometrical - based on a skeleton approach - and architectural - based on the experience of practitioners - to impact the system performance. The resulting forms are relevant and effective, thanks to a smoothing procedure that ensures the consistency of the shapes produced. As an application, we propose to perform a multi-objective shape optimization of an AC45 foil. The modeler is linked to the fluid solver AVANTI, coupled with Xfoil, and to the optimization toolbox FAMOSA.

This is a joint work with Régis Duvigneau, Matthieu Sacher, Yann Roux.