Marta Abril Bucero

GALAAD 2 Team
INRIA/Université de Nice Sophia Antipolis

Bio

Español | Hola! Soy Doctora en matematicas aplicadas por el instituto de Inria Sophia Antipolis y la Université de Nice Sophia Antipolis. Actualmente trabajo como profesora en la Université de Nice Sophia Antipolis y continuo colaborando como postdoc en el equipo Galaad 2 de Inria, en el cual hice tambien mi doctorado.

Francais | Salut! J'ai fait ma thèse en mathematiques appliquées dans l'equipe Galaad 2 a l'Inria/Université de Nice Sophia Antipolis. Je suis maintenant ATER a l'Université de Nice et je fais aussi une collaboration postdoc dans le meme equipe où j'ai fait ma thèse.

English | Hi! I have a Ph.D. in applied mathematics from Inria/Université de Nice Sophia Antipolis. I'm currently assistant teacher at Université de Nice Sophia Antipolis and I also collaborate as a postdoc in the team Galaad 2 at Inria where I did my Ph.D.

Academic Positions and Non academic Positions

Present 01/2015

PostDoc - collaborator

Galaad 2 Team , Inria Sophia Antipolis
Present 09/2014

Teaching Assistant

Université de Nice Sophia Antipolis
12/2014 12/2011

Ph.D. student in Mathematics

Inria Sophia Antipolis / Université de Nice Sophia Antipolis , Advisor: Bernard Mourrain
08/2010 04/2007

Analist

Bankinter / BBVA

Education & Training

Ph.D. in Mathematics 2014

Ph.D. in Mathematics (ANR GEOLMI project)

Inria Sophia Antipolis /Université de Nice Sophia Antipolis
M. Sc. 2011

Master in Mathematic Research

Universidad Complutense de Madrid
B. Sc.2007

B. Sc. of Mathematics

Universidad Complutense de Madrid

Research Summary

Interests

Real Algebric Geometry
Polynomial optimization
Sum of squares
Polynomial System solving
Applications to cryptography

Selected Publications

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The borderbasix package

Philippe Trebuchet, Bernard Mourrain, Marta Abril Bucero

Conference Papers MEGA 2015

Abstract

We describe the software package borderbasix dedicated to the computation of border bases and the solution of polynomial equations. We present the main ingredients of the algorithm as well as the other main available tools: numerical solution from multiplication matrices, real radical computation, polynomial optimization. The implementation parameterized by the coefficient type and thechoice function provide a versatile family of tools for computation with modular arithmetic, floating point arithmetic or rational arithmetic. It relies on linear algebra solvers for dense and sparse matrices for these various types of coefficients. A connection with SDP solvers has been integrated for the combination of re- laxation approaches with border basis computation. Extensive benchmarks on typical polynomial systems are reported, which show the very good performance of the tool.

On the construction of general cubature formula by flat extension

Marta Abril Bucero,Chandrajit Bajaj , Bernard Mourrain

Journal Papers Linear Algebra and its Applications. (Accepted 2015)

Abstract

This paper describes a new method to compute general cubature formula. The problem is transformed into the computation of truncated Hankel operators with flat extension. We analyse the algebraic properties associated to flat extensions and show how to recover the cubature points and weights from the truncated Hankel operator. A new algorithm to test the flat extension property and to compute the decomposition is given. To compute cubature formula with a minimal number of points, we propose a new relaxation hierarchy of convex optimization problems, which minimizes the nuclear norm of the Hankel operators. We show that, when the order of relaxation is high enough, a minimizer of this optimization problem cor- responds to a cubature formula and the cubature formulae with a minimal number of points are associated to a face of these convex sets. This new method is illustrated on some examples, for which we obtain new minimal cubature formulae.

Moments Matrices, Real algebraic geometry and polynomial optimization

Marta Abril Bucero

Thesis Ph.D. Thesis, 2014.

Abstract

The objective of this thesis is to compute the optimum of a polynomial on a closed basic semialgebraic set and the points where this optimum is reached. To achieve this goal we combine border basis method with Lasserre's hierarchy in order to reduce the size of the moment matrices in the SemiDefinite Programming (SDP) problems. In order to verify if the minimum is reached we describe a new criterion to verify the flat extension condition using border basis. Combining these new results we provide a new algorithm which computes the optimum and the minimizers points. We show several experimentations and some applications in different domains which prove the perfomance of the algorithm. Theorethically we also prove the finite convergence of a SDP hierarchie contructed from a Karush-Kuhn-Tucker ideal and its consequences in particular cases. We also solve the particular case where the minimizers are not KKT points using Fritz-John Variety.

New polynomial optimization algorithm using border basis

Marta Abril Bucero, Bernard Mourrain

Conference Papers Proceedings Encuentro de \'Algebra Computacional y Aplicaciones (EACA), Barcelona, June 2014: 37-39.

Abstract

A relaxation method based on border basis reduction which improves the efficiency of Lasserre’s approach is proposed to compute the optimum of a polynomial function on a basic closed semi algebraic set. A new stopping criteria is given to detect when the relaxation sequence reaches the minimum, using a sparse flat extension criteria. We also provide a new algorithm to reconstruct a finite sum of weighted Dirac measures from a truncated sequence of moments, which can be applied to other sparse reconstruction problems. As an application, we obtain a new algorithm to compute zero-dimensional minimizer ideals and the minimizer points.

Border Basis relaxation for polynomial optimization

Marta Abril Bucero, Bernard Mourrain

Journal Papers Journal of symbolic computation. (Submitted 2014)

Abstract

A relaxation method based on border basis reduction which improves the efficiency of Lasserre's approach is proposed to compute the optimum of a polynomial function on a basic closed semi algebraic set. A new stopping criterion is given to detect when the relaxation sequence reaches the minimum, using a sparse flat extension criterion. We also provide a new algorithm to reconstruct a finite sum of weighted Dirac measures from a truncated sequence of moments, which can be applied to other sparse reconstruction problems. As an application, we obtain a new algorithm to compute zero-dimensional minimizer ideals and the minimizer points or zero-dimensional G-radical ideals. Experimentations show the impact of this new method on significant benchmarks.

Normal Bundle in Grassmannian G(k,n)

Marta Abril Bucero

Thesis Master Thesis, 2011.

Abstract

The aim of my research is to give a resolution for the conormal bundle of the general Grassmannian G(k, n). We first recall some concepts about Grassmannians, like Plücker embedding, Plücker coordenates, vector bundles, universal bundles,....The main tools we use are several exact sequences, among them the universal exact sequence, the Eagon- Northcott complex and the Euler sequence in order to build a conmutative diagram. Finally the snake lemma applied to the conmutative diagram produces the resolution sought. .

Teaching

Grad School Teaching

Present 01/2015

Analysis

Degree of Economic and Social Administration, Université de Nice Sophia Antipolis
12/2015 09/2015

Analysis 2

Degree of Phisics and Chemistry Sciences, Université de Nice Sophia Antipolis
12/2015 09/2015

Probability and Statistics

Degree of Computer Sciences, Université de Nice Sophia Antipolis