3.1 High order geometric modeling

The accurate description of shapes is a long standing problem in mathematics, with an important impact in many domains, inducing strong interactions between geometry and computation. Developing precise geometric modeling techniques is a critical issue in CAD-CAM. Constructing accurate models, that can be exploited in geometric applications, from digital data produced by cameras, laser scanners, observations or simulations is also a major issue in geometry processing. A main challenge is to construct models that can capture the geometry of complex shapes, using few parameters while being precise.

Our first objective is to develop methods, which are able to describe accurately and in an efficient way, objects or phenomena of geometric nature, using algebraic representations.

The approach followed in CAGD, to describe complex geometry is based on parametric representations called NURBS (Non Uniform Rational B-Spline). The models are constructed by trimming and gluing together high order patches of algebraic surfaces. These models are built from the so-called B-Spline functions that encode a piecewise algebraic function with a prescribed regularity at knots. Although these models have many advantages and have become the standard for designing nowadays CAD models, they also have important drawbacks. Among them, the difficulty to locally refine a NURBS surface and also the topological rigidity of NURBS patches that imposes to use many such patches with trims for designing complex models, with the consequence of the appearing of cracks at the seams. To overcome these difficulties, an active area of research is to look for new blending functions for the representation of CAD models. Some examples are the so-called T-Splines, LR-Spline blending functions, or hierarchical splines, that have been recently devised in order to perform efficiently local refinement. An important problem is to analyze spline spaces associated to general subdivisions, which is of particular interest in higher order Finite Element Methods. Another challenge in geometric modeling is the efficient representation and/or reconstruction of complex objects, and the description of computational domains in numerical simulation. To construct models that can represent efficiently the geometry of complex shapes, we are interested in developing modeling methods, based on alternative constructions such as skeleton-based representations. The change of representation, in particular between parametric and implicit representations, is of particular interest in geometric computations and in its applications in CAGD.

We also plan to investigate adaptive hierarchical techniques, which can locally improve the approximation of a shape or a function. They shall be exploited to transform digital data produced by cameras, laser scanners, observations or simulations into accurate and structured algebraic models.

The precise and efficient representation of shapes also leads to the problem of extracting and exploiting characteristic properties of shapes such as symmetry, which is very frequent in geometry. Reflecting the symmetry of the intended shape in the representation appears as a natural requirement for visual quality, but also as a possible source of sparsity of the representation. Recognizing, encoding and exploiting symmetry requires new paradigms of representation and further algebraic developments. Algebraic foundations for the exploitation of symmetry in the context of non linear differential and polynomial equations are addressed. The intent is to bring this expertise with symmetry to the geometric models and computations developed by aromath.

3.2 Robust algebraic-geometric computation

In many problems, digital data are approximated and cannot just be used as if they were exact. In the context of geometric modeling, polynomial equations appear naturally as a way to describe constraints between the unknown variables of a problem. An important challenge is to take into account the input error in order to develop robust methods for solving these algebraic constraints. Robustness means that a small perturbation of the input should produce a controlled variation of the output, that is forward stability, when the input-output map is regular. In non-regular cases, robustness also means that the output is an exact solution, or the most coherent solution, of a problem with input data in a given neighborhood, that is backward stability.

Our second long term objective is to develop methods to robustly and efficiently solve algebraic problems that occur in geometric modeling.

Robustness is a major issue in geometric modeling and algebraic computation. Classical methods in computer algebra, based on the paradigm of exact computation, cannot be applied directly in this context. They are not designed for stability against input perturbations. New investigations are needed to develop methods which integrate this additional dimension of the problem. Several approaches are investigated to tackle these difficulties.

One relies on linearization of algebraic problems based on “elimination of variables” or projection into a space of smaller dimension. Resultant theory provides a strong foundation for these methods, connecting the geometric properties of the solutions with explicit linear algebra on polynomial vector spaces, for families of polynomial systems (e.g., homogeneous, multi-homogeneous, sparse). Important progress has been made in the last two decades to extend this theory to new families of problems with specific geometric properties. Additional advances have been achieved more recently to exploit the syzygies between the input equations. This approach provides matrix based representations, which are particularly powerful for approximate geometric computation on parametrized curves and surfaces. They are tuned to certain classes of problems and an important issue is to detect and analyze degeneracies and to adapt them to these cases.

A more adaptive approach involves linear algebra computation in a hierarchy of polynomial vector spaces. It produces a description of quotient algebra structures, from which the solutions of polynomial systems can be recovered. This family of methods includes Gröbner Basis , which provides general tools for solving polynomial equations. Border Basis is an alternative approach, offering numerically stable methods for solving polynomial equations with approximate coefficients . An important issue is to understand and control the numerical behavior of these methods as well as their complexity and to exploit the structure of the input system.

In order to compute “only” the (real) solutions of a polynomial system in a given domain, duality techniques can also be employed. They consist in analyzing and adding constraints on the space of linear forms which vanish on the polynomial equations. Combined with semi-definite programming techniques, they provide efficient methods to compute the real solutions of algebraic equations or to solve polynomial optimization problems. The main issues are the completness of the approach, their scalability with the degree and dimension and the certification of bounds.

Singular solutions of polynomial systems can be analyzed by computing differentials, which vanish at these points. This leads to efficient deflation techniques, which transform a singular solution of a given problem into a regular solution of the transformed problem. These local methods need to be combined with more global root localisation methods.

Subdivision methods are another type of methods which are interesting for robust geometric computation. They are based on exclusion tests which certify that no solution exists in a domain and inclusion tests, which certify the uniqueness of a solution in a domain. They have shown their strength in addressing many algebraic problems, such as isolating real roots of polynomial equations or computing the topology of algebraic curves and surfaces. The main issues in these approaches is to deal with singularities and degenerate solutions.

4.1 Geometric modeling for Design and Manufacturing.

The main domain of applications that we consider for the methods we develop is Computer Aided Design and Manufacturing.

Computer-Aided Design (CAD) involves creating digital models defined by mathematical constructions, from geometric, functional or aesthetic considerations. Computer-aided manufacturing (CAM) uses the geometrical design data to control the tools and processes, which lead to the production of real objects from their numerical descriptions.

CAD-CAM systems provide tools for visualizing, understanding, manipulating, and editing virtual shapes. They are extensively used in many applications, including automotive, shipbuilding, aerospace industries, industrial and architectural design, prosthetics, and many more. They are also widely used to produce computer animation for special effects in movies, advertising and technical manuals, or for digital content creation. Their economic importance is enormous. Their importance in education is also growing, as they are more and more used in schools and educational purposes.

CAD-CAM has been a major driving force for research developments in geometric modeling, which leads to very large software, produced and sold by big companies, capable of assisting engineers in all the steps from design to manufacturing.

Nevertheless, many challenges still need to be addressed. Many problems remain open, related to the use of efficient shape representations, of geometric models specific to some application domains, such as in architecture, naval engineering, mechanical constructions, manufacturing ...Important questions on the robustness and the certification of geometric computation are not yet answered. The complexity of the models which are used nowadays also appeals for the development of new approaches. The manufacturing environment is also increasingly complex, with new type of machine tools including: turning, 5-axes machining and wire EDM (Electrical Discharge Machining), 3D printer. It cannot be properly used without computer assistance, which raises methodological and algorithmic questions. There is an increasing need to combine design and simulation, for analyzing the physical behavior of a model and for optimal design.

The field has deeply changed over the last decades, with the emergence of new geometric modeling tools built on dedicated packages, which are mixing different scientific areas to address specific applications. It is providing new opportunities to apply new geometric modeling methods, output from research activities.

4.2 Geometric modeling for Numerical Simulation and Optimization

A major bottleneck in the CAD-CAM developments is the lack of interoperability of modeling systems and simulation systems. This is strongly influenced by their development history, as they have been following different paths.

The geometric tools have evolved from supporting a limited number of tasks at separate stages in product development and manufacturing, to being essential in all phases from initial design through manufacturing.

Current Finite Element Analysis (FEA) technology was already well established 40 years ago, when CAD-systems just started to appear, and its success stems from using approximations of both the geometry and the analysis model with low order finite elements (most often of degree $\le 2$).

There has been no requirement between CAD and numerical simulation, based on Finite Element Analysis, leading to incompatible mathematical representations in CAD and FEA. This incompatibility makes interoperability of CAD/CAM and FEA very challenging. In the general case today this challenge is addressed by expensive and time-consuming human intervention and software developments.

Improving this interaction by using adequate geometric and functional descriptions should boost the interaction between numerical analysis and geometric modeling, with important implications in shape optimization. In particular, it could provide a better feedback of numerical simulations on the geometric model in a design optimization loop, which incorporates iterative analysis steps.

The situation is evolving. In the past decade, a new paradigm has emerged to replace the traditional Finite Elements by B-Spline basis element of any polynomial degree, thus in principle enabling exact representation of all shapes that can be modeled in CAD. It has been demonstrated that the so-called isogeometric analysis approach can be far more accurate than traditional FEA.

It opens new perspectives for the interoperability between geometric modeling and numerical simulation. The development of numerical methods of high order using a precise description of the shapes raises questions on piecewise polynomial elements, on the description of computational domains and of their interfaces, on the construction of good function spaces to approximate physical solutions. All these problems involve geometric considerations and are closely related to the theory of splines and to the geometric methods we are investigating. We plan to apply our work to the development of new interactions between geometric modeling and numerical solvers.

5.1 New software

6.1 On the Effective Putinar’s Positivstellensatz and Moment Approximation

Participants: Lorenzo Baldi, Bernard Mourrain.

6.2 Curve Valuations and Mixed Volumes in the Implicitization of Rational Varieties

Participants: Alicia Dickenstein, Maria Isabel Herrero, Bernard Mourrain.

In 21, we address the description of the tropicalization of families of rational varieties under parametrizations with prescribed support, via curve valuations. We recover and extend results by Sturmfels, Tevelev and Yu for generic coefficients, considering rational parametrizations with non-trivial denominator. The advantage of our point of view is that it can be generalized to deal with non-generic parametrizations. We provide a detailed analysis of the degree of the closed image, based on combinatorial conditions on the relative positions of the supports of the polynomials defining the parametrization. We obtain a new formula and finer bounds on the degree, when the supports of the polynomials are different. We also present a new formula and bounds for the order at the origin in case the closed image is a hypersurface.

6.3 Univariate Rational Sums of Squares

Participants: Teresa Krick, Bernard Mourrain, Agnes Szanto.

6.4 A certified iterative method for isolated singular roots

Participants: Angelos Mantzaflaris, Bernard Mourrain, Agnes Szanto.

6.5 Newton-Type Methods For Simultaneous Matrix Diagonalization

Participants: Rima Khouja, Bernard Mourrain, Jean-Claude Yakoubsohn.

6.6 Multigraded Sylvester forms, Duality and Elimination Matrices

Participants: Laurent Busé, Marc Chardin, Navid Nemati.

In 16 we study the equations of the elimination ideal associated with n+1 generic multihomogeneous polynomials defined over a product of projective spaces of dimension n. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods.

6.7 Formulas for the eigendiscriminants of ternary and quaternary forms

Participants: Laurent Busé.

A $d$-dimensional tensor $A$ of format $n\times n\times \cdots \times n$ defines naturally a rational map $\Psi$ from the projective space ${\mathbb{P}}^{n-1}$ to itself and its eigenscheme is then the subscheme of ${\mathbb{P}}^{n-1}$ of fixed points of $\Psi$. The eigendiscriminant is an irreducible polynomial in the coefficients of $A$ that vanishes for a given tensor if and only if its eigenscheme is singular. In 17 we contribute to formulas for the computation of eigendiscriminants in the cases $n=3$ and $n=4$. In particular, by restriction to symmetric tensors, we obtain closed formulas for the eigendiscriminants of plane curves and surfaces in ${\mathbb{P}}^3$ as the ratio of some determinants of resultant matrices.

6.8 Tri-linear birational maps in dimension three

Participants: Laurent Busé, Pablo Gonzalez Mazon, Josef Schicho.

A tri-linear rational map in dimension three is a rational map $\varphi :{\left({\mathbb{P}}_{{ℂ}^1}\right)}^3⤏{\mathbb{P}}_{ℂ}^3$ defined by four tri-linear polynomials without a common factor. If $\varphi$ admits an inverse rational map ${\varphi }^{-1}$, it is a tri-linear birational map. In 18 we address computational and geometric aspects about these transformations. We prove a characterization of birationality based on the first syzygies of the entries. More generally, we describe all the possible minimal free resolutions of the ideal generated by these entries. With respect to geometry, we show that the set ${\mathrm{Bir}}_{(1,1,1)}$ of tri-linear birational maps, up to automorphism of ${\mathbb{P}}_{ℂ}^3$, is a locally closed algebraic subset of the Grassmannian of 4-dimensional subspaces in the vector space of tri-linear polynomials, and has eight irreducible components. Moreover, the group action on ${\mathrm{Bir}}_{(1,1,1)}$ given by composition with automorphisms of ${\left({\mathbb{P}}_{{ℂ}^1}\right)}^3$ defines 19 orbits, and each of these orbits determines an isomorphism class of the base loci of these transformations.

6.9 Efficient sampling in spectrahedra and volume approximation

Participants: Apostolos Chalkis, Ioannis Emiris, Vissarion Fisikopoulos, Panagiotis Repouskos, Elias Tsigaridas.

We present algorithmic, complexity, and implementation results on the problem of sampling points from a spectrahedron, that is, the feasible region of a semidefinite program. Our main tool is geometric random walks. We analyze the arithmetic and bit complexity of certain primitive geometric operations that are based on the algebraic properties of spectrahedra and the polynomial eigenvalue problem. This study leads to the implementation of a broad collection of random walks for sampling from spectrahedra that experimentally show faster mixing times than methods currently employed either in theoretical studies or in applications, including the popular family of Hit-and-Run walks. The different random walks offer a variety of advantages, thus allowing us to efficiently sample from general probability distributions, for example the family of log-concave distributions which arise in numerous applications. We focus on two major applications of independent interest: (i) approximate the volume of a spectrahedron, and (ii) compute the expectation of functions coming from robust optimal control. We exploit efficient linear algebra algorithms and implementations to address the aforementioned computations in very high dimension. In particular, we provide a C++ open source implementation of our methods that scales efficiently, for the first time, up to dimension 200. We illustrate its efficiency on various data sets 19.

6.10 New upper bounds for the number of embeddings of minimally rigid graphs

Participants: Evangelos Bartzos, Ioannis Emiris, Ilias Kotsireas, Charalambos Tzamos.

Determining the number of solutions of a multi-homogeneous polynomial system is a fundamental problem in algebraic geometry. The multi-homogeneous Bézout (m-Bézout) number bounds from above the number of non-singular solutions of a multi-homogeneous system, but its computation is a $\#P$-hard problem. Recent work related the m-Bézout number of certain multi-homogeneous systems derived from rigidity theory with graph orientations, cf Bartzos et al. (2020). A first generalization applied graph orientations for bounding the root count of a multi-homogeneous system that can be modeled by simple undirected graphs, as shown by three of the authors (Bartzos et al., 2021). Here, we prove that every multi-homogeneous system can be modeled by hypergraphs and the computation of its m-Bézout bound is related to constrained hypergraph orientations. Thus, we convert the algebraic problem of bounding the number of roots of a polynomial system to a purely combinatorial problem of analyzing the structure of a hypergraph. We also provide a formulation of the orientation problem as a constraint satisfaction problem (CSP), hence leading to an algorithm that computes the multi-homogeneous bound by finding constrained hypergraph orientations 32.

6.11 New upper bounds for the number of embeddings of minimally rigid graphs

Participants: Evangelos Bartzos, Ioannis Emiris, Raimundas Vidunas.

6.12 A Greedy Approach to the Canny-Emiris Formula

Participants: Carles Checa, Ioannis Emiris.

6.13 Modeling complex root motion of real random polynomials under differentiation

Participants: André Galligo.

6.14 Symmetry in Multivariate Ideal Interpolation

Participants: Evelyne Hubert, Erick Rodriguez-Bazan.

An interpolation problem is defined by a set of linear forms on the (multivariate) polynomial ring and values to be achieved by an interpolant. For Lagrange interpolation the linear forms consist of evaluations at some nodes, while Hermite interpolation also considers the values of successive derivatives. Both are examples of ideal interpolation in that the kernels of the linear forms intersect into an ideal. For an ideal interpolation problem with symmetry, we address in 30 the simultaneous computation of a symmetry adapted basis of the least interpolation space and the symmetry adapted H-basis of the ideal. Beside its manifest presence in the output, symmetry is exploited computationally at all stages of the algorithm. For an ideal invariant, under a group action, defined by a Groebner basis, the algorithm allows to obtain a symmetry adapted basis of the quotient and of the generators. We shall also note how it applies surprisingly but straightforwardly to compute fundamental invariants and equivariants of a reflection group.

6.15 Algorithms for Fundamental Invariants and Equivariants

Participants: Evelyne Hubert, Erick Rodriguez- Bazan.

For a finite group, we presented in 24 three algorithms to compute a generating set of invariants simultaneously to generating sets of basic equivariants, i.e., equivariants for the irreducible representations of the group. The main novelty resides in the exploitation of the orthogonal complement of the ideal generated by invariants; Its symmetry adapted basis delivers the fundamental equivariants. Fundamental equivariants allow to assemble symmetry adapted bases of polynomial spaces of higher degrees, and these are essential ingredients in exploiting and preserving symmetry in computations. They appear within algebraic computation and beyond, in physics, chemistry and engineering. Our first construction applies solely to reflection groups and consists in applying symmetry preserving interpolation, as developed by the same authors, along an orbit in general position. The fundamental invariants can be read off the H-basis of the ideal of the orbit while the fundamental equivariants are obtained from a symmetry adapted basis of an invariant direct complement to this ideal in the polynomial ring. The second algorithm takes as input primary invariants and the output provides not only the secondary invariants but also free bases for the modules of basic equivariants. These are constructed as the components of a symmetry adapted basis of the orthogonal complement, in the polynomial ring, to the ideal generated by primary invariants. The third algorithm proceeds degree by degree, determining the fundamental invariants as forming a H-basis of the Hilbert ideal, i.e., the polynomial ideal generated by the invariants of positive degree. The fundamental equivariants are simultaneously computed degree by degree as the components of a symmetry adapted basis of the orthogonal complement of the Hilbert ideal.

6.16 G1-smooth Biquintic Approximation of Catmull-Clark Subdivision Surfaces

Participants: Michelangelo Marsala, Angelos Mantzaflaris, Bernard Mourrain.

6.17 Near-neighbor preserving dimension reduction via coverings for doubling subsets of l1

Participants: Ioannis Emiris, Vasilis Margonis, Ioannis Psarros.

Randomized dimensionality reduction has been recognized as one of the cornerstones in handling high-dimensional data, originating in various foundational works such as the celebrated Johnson-Lindenstrauss Lemma. More specifically, nearest neighbor-preserving embeddings exist for L2 (Euclidean) and L1 (Manhattan) metrics, as well as doubling subsets of L2, where doubling dimension is today the most effective way of capturing intrinsic dimensionality, as well as input structure in various applications. These randomized embeddings bound the distortion only for distances between the query point and a point set. Motivated by the foundational character of fast Approximate Nearest Neighbor search in L1, this paper settles an important missing case, namely that of doubling subsets of L1. In particular, we introduce a randomized dimensionality reduction by means of a near neighbor-preserving embedding, which is related to the decision-with-witness problem. The input set gets represented with a carefully chosen covering point set; in a second step, the algorithm randomly projects the latter. In order to obtain the covering point sets, we leverage either approximate r-nets or randomly shifted grids, with different tradeoffs between preprocessing time and target dimension. We exploit Cauchy random variables, and derive a concentration bound of independent interest. Our algorithms are rather simple and should therefore be useful in practice 22.

6.18 Flux-aligned quad mesh generation in magnetohydrodynamic simulation

Participants: Meng Wu, Xuhui Wang, Bernard Mourrain, Gang Xu, Qian Ni, Yuan Liu, Boniface Nkonga.

6.19 A kinematic-geometric model based on ankles’ depth trajectory in frontal plane for gait analysis using a single RGB-D camera

Participants: Mehran Hatamzadeh, Laurent Busé, Frédéric Chorin, Pierre Alliez, Jean-Dominique Favreau, Raphaël Zory.

The emergence of RGB-D cameras and the development of pose estimation algorithms offer opportunities in biomechanics. However, some challenges still remain when using them for gait analysis, including noise which leads to misidentification of gait events and inaccuracy. Therefore, in 23 we present a novel kinematic-geometric model for spatio-temporal gait analysis, based on ankles’ trajectory in the frontal plane and distance-to-camera data (depth). Our approach consists of three main steps: identification of the gait pattern and modeling via parameterized curves, development of a fitting algorithm, and computation of locomotive indices. The proposed fitting algorithm applies on both ankles’ depth data simultaneously, by minimizing through numerical optimization some geometric and biomechanical error functions. For validation, 15 subjects were asked to walk inside the walkway of the OptoGait, while the OptoGait and an RGB-D camera (Microsoft Azure Kinect) were both recording. Then, the spatiotemporal parameters of both feet were computed using the OptoGait and the proposed model. Validation results show that the proposed model yields good to excellent absolute statistical agreement ($0.86\le Rc\le 0.99$). Our kinematic-geometric model offers several benefits: (1) It relies only on the ankles’ depth trajectory both for gait events extraction and spatio-temporal parameters’ calculation; (2) it is usable with any kind of RGB-D camera or even with 3D marker-based motion analysis systems in absence of toes’ and heels’ markers; and (3) it enables improving the results by denoising and smoothing the ankles’ depth trajectory. Hence, the proposed kinematic-geometric model facilitates the development of portable markerless systems for accurate gait analysis.

6.20 Investigating the Bioactive Conformation of Angiotensin II Using Markov State Modeling Revisited with Web-Scale Clustering

Participants: Ioannis Emiris, Emmanouil Christoforou, Hari Leontiadou, Frank Noé, Jannis Samios, Zoe Cournia.

Molecular dynamics simulation is a powerful technique for studying the structure and dynamics of biomolecules in atomic-level detail by sampling their various conformations in real time. Because of the long timescales that need to be sampled to study biomolecular processes and the big and complex nature of the corresponding data, relevant analyses of important biophysical phenomena are challenging. Clustering and Markov state models (MSMs) are efficient computational techniques that can be used to extract dominant conformational states and to connect those with kinetic information. In this work, we perform Molecular Dynamics simulations to investigate the free energy landscape of Angiotensin II (AngII) in order to unravel its bioactive conformations using different clustering techniques and Markov state modeling. AngII is an octapeptide hormone, which binds to the AT1 transmembrane receptor, and plays a vital role in the regulation of blood pressure, conservation of total blood volume, and salt homeostasis. To mimic the water–membrane interface as AngII approaches the AT1 receptor and to compare our findings with available experimental results, the simulations were performed in water as well as in water–ethanol mixtures. Our results show that in the water–ethanol environment, AngII adopts more compact U-shaped (folded) conformations than in water, which resembles its structure when bound to the AT1 receptor. For clustering of the conformations, we validate the efficiency of an inverted-quantized k-means algorithm, as a fast approximate clustering technique for web-scale data (millions of points into thousands or millions of clusters) compared to k-means, on data from trajectories of molecular dynamics simulations with reasonable trade-offs between time and accuracy. Finally, we extract MSMs using various clustering techniques for the generation of microstates and macrostates, and for the selection of the macrostate representatives 20.

7.1 Bilateral grants with industry

$•$ Geometric computing.

Participants: Ioannis Emiris, Apostolos Chalkis, Panagiotis Repouskos, Ioannis Psarros.

Ioannis Emiris coordinates a research contract with the industrial partner ANSYS (Greece), in collaboration with Athena Research Center. MSc students P. Repouskos and T. Pappas, PhD candidate A. Chalkis and postdoc fellow I. Psarros are partially funded.

Electronic design automation (EDA) and simulating Integrated Circuits requires robust geometric operations on thousands of electronic elements (capacitors, resistors, coils etc) represented by polyhedral objects in 2.5 dimensions, not necessarily convex. A special case may concern axis-aligned objects but the real challenge is the general case. The project, extended into 2022, focuses on 3 axes: (1) efficient data structures and prototype implementations for storing the aforementioned polyhedral objects so that nearest neighbor queries are fast in the L-max metric, which is the primary focus of the contract, (2) random sampling of the free space among objects, (3) data-driven algorithmic design for problems concerning data-structures and their construction and initialization.

It is expected to continue into 2023 along with a grant from the Greek ministry of Development.

$•$ Interactive construction of 3D models - Application to the modeling of complex geological structures.

Participants: Ayoub Belhachmi, Bernard Mourrain.

CIFRE collaboration between Schlumberger Montpellier (A. Azzedine) and Inria Sophia Antipolis (B. Mourrain). The PhD candidate is A. Belhachmi. The objective of the work is the development of a new spline based high quality geomodeler for reconstructing the stratigraphy of geological layers from the adaptive and efficient processing of large terrain information.

8.1 International initiatives

8.2 International research visitors

8.3 European initiatives

8.4 National initiatives

GdR EFI and GDM: Evelyne Hubert is part of the Scientific Committee of the GdR Equations Fonctionnelles et Interactions and participates to the GdR Géometrie Differentielle et Mécanique (gdr-gdm.univ-lr.fr).

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

9.3 Popularization

10.1 Major publications

• 1 articleE.Evangelos Bartzos, I. Z.Ioannis Z. Emiris and J.Josef Schicho. On the multihomogeneous Bézout bound on the number of embeddings of minimally rigid graphs.Applicable Algebra in Engineering, Communication and Computing315-62020, 325-357
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• 2 articleE.Evangelos Bartzos, I. Z.Ioannis Z. Emiris and R.Raimundas Vidunas. New upper bounds for the number of embeddings of minimally rigid graphs.Discrete and Computational Geometry6832022, 796
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• 3 articleL.Laurent Busé, Y.Yairon Cid-Ruiz and C.Carlos D'Andrea. Degree and birationality of multi-graded rational maps.Proceedings of the London Mathematical Society12142020, 743-787
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• 4 articleL.Laurent Busé and A.Anna Karasoulou. Resultant of an equivariant polynomial system with respect to the symmetric group.Journal of Symbolic Computation762016, 142-157
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• 5 articleI. Z.Ioannis Z. Emiris and I.Ioannis Psarros. Products of Euclidean Metrics, Applied to Proximity Problems among Curves.ACM Transactions on Spatial Algorithms and Systems64August 2020, 1-20
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• 6 articleA. J.Alvaro Javier Fuentes Suárez and E.Evelyne Hubert. Scaffolding skeletons using spherical Voronoi diagrams: feasibility, regularity and symmetry.Computer-Aided Design102May 2018, 83 - 93
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• 7 articleA.Alessandro Giust, B.Bert Jüttler and A.Angelos Mantzaflaris. Local (T)HB-spline projectors via restricted hierarchical spline fitting.Computer Aided Geometric Design80June 2020, 101865
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• 8 articleP.Paul Görlach, E.Evelyne Hubert and T.Théo Papadopoulo. Rational invariants of even ternary forms under the orthogonal group.Foundations of Computational Mathematics192019, 1315-1361
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• 9 articleE.Evelyne Hubert and E.Erick Rodriguez Bazan. Algorithms for fundamental invariants and equivariants: (of finite groups).Mathematics of Computation913372022, 2459-2488
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• 10 articleZ.Zbigniew Jelonek and A.André Galligo. Elimination ideals and Bezout relations.Journal of Algebra5622020, 621-626
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• 11 articleA.Angelos Mantzaflaris, B.Bert Jüttler, B.Boris Khoromskij and U.Ulrich Langer. Low Rank Tensor Methods in Galerkin-based Isogeometric Analysis.Computer Methods in Applied Mechanics and Engineering316April 2017, 1062-1085
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• 12 articleB.Bernard Mourrain. Polynomial-Exponential Decomposition from Moments.Foundations of Computational Mathematics186December 2018, 1435--1492
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• 13 articleS.Simon Telen, B.Bernard Mourrain and M.Marc Van Barel. Solving Polynomial Systems via a Stabilized Representation of Quotient Algebras.SIAM Journal on Matrix Analysis and Applications393October 2018, 1421--1447
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10.2 Publications of the year