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 Computer Aided Geometric Design (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.

6.1 New software

7.1 Exact Moment Representation in Polynomial Optimization

Participants: Lorenzo Baldi, Bernard Mourrain.

In 21, we investigate the problem of representing moment sequences by measures in the context of Polynomial Optimization Problems, that consist in finding the infimum of a real polynomial on a real semialgebraic set defined by polynomial inequalities. We analyze the exactness of Moment Matrix (MoM) hierarchies, dual to the Sum of Squares (SoS) hierarchies, which are sequences of convex cones introduced by Lasserre to approximate measures and positive polynomials. We investigate in particular flat truncation properties, which allow testing effectively when MoM exactness holds and recovering the minimizers. We show that the dual of the MoM hierarchy coincides with the SoS hierarchy extended with the real radical of the support of the defining quadratic module $Q$. We deduce that flat truncation happens if and only if the support of the quadratic module associated with the minimizers is of dimension zero. We also bound the order of the hierarchy at which flat truncation holds. As corollaries, we show that flat truncation and MoM exactness hold when regularity conditions, known as Boundary Hessian Conditions, hold (and thus that MoM exactness holds generically); and when the support of the quadratic module $Q$ is zero-dimensional. Effective numerical computations illustrate these flat truncation properties.

7.2 On Łojasiewicz Inequalities and the Effective Putinar’s Positivstellensatz

Participants: Lorenzo Baldi, Bernard Mourrain, Adam Parusinski.

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In 22, we study the effective Putinar's Positivestellensatz on a compact basic semialgebraic set $S$ and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter $\epsilon$ measuring the non-vanishing of the positive function, the constant $c$ and exponent $L$ of a Łojasiewicz inequality for the semi-algebraic distance function associated to the inequalities $g=(g_1,...,g_r)$ defining $S$. They are polynomial in $c$ and ${\epsilon }^{-1}$ with an exponent depending only on $L$. We analyze in details the Łojasiewicz inequality when the defining inequalities $g$ satisfy the Constraint Qualification Condition. We show that, in this case, the Łojasiewicz exponent $L$ is 1 and we relate the Łojasiewicz constant $c$ with the distance of $g$ to the set of singular systems.

7.3 Orbit spaces of Weyl groups acting on compact tori: a unified and explicit polynomial description

Participants: Evelyne Hubert, Tobias Metzlaff, Cordian Riener.

The Weyl group of a crystallographic root system has a nonlinear action on the compact torus. The orbit space of this action is a compact basic semi-algebraic set. In 29, we present a polynomial description of this set for the Weyl groups of type A, B, C, D and G. Our description is given through a polynomial matrix inequality. The novelty lies in an approach via Hermite quadratic forms and a closed formula for the matrix entries. The orbit space of the nonlinear Weyl group action is the orthogonality region of generalized Chebyshev polynomials. In this polynomial basis, we show that the matrices obtained for the five types follow the same, surprisingly simple pattern. This is applied to the optimization of trigonometric polynomials with crystallographic symmetries.

7.4 Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs

Participants: Evelyne Hubert, Tobias Metzlaff, Philippe Moustrou, Cordian Riener.

In 28, we provide a new approach to the optimization of trigonometric polynomials under the hypothesis of a crystallographic symmetry. This approach widens the bridge between trigonometric and polynomial optimization. The trigonometric polynomials considered are supported on weight lattices associated to crystallographic root systems and are assumed invariant under the associated reflection group. On one hand, the invariance allows us to rewrite the objective function in terms of generalized Chebyshev polynomials of the generalized cosines; On the other hand, the generalized cosines parametrize a compact basic semi algebraic set, this latter being given by an explicit polynomial matrix inequality. The initial problem thus boils down to a polynomial optimization problem that is straightforwardly written in terms of generalized Chebyshev polynomials. The minimum is to be computed by a converging sequence of lower bounds as given by a hierarchy of relaxations based on the Hol-Scherer Positivstellensatz and indexed by the weighted degree associated to the root system. This new method for trigonometric optimization was motivated by its application to estimate the spectral bound on the chromatic number of set avoiding graphs. We examine cases of the literature where the avoided set affords crystallographic symmetry. In some cases, we obtain new analytic proofs for sharp bounds on the chromatic number while, in others, we compute new lower bounds numerically.

7.5 Isolated singularities, inverse systems and the punctual Hilbert scheme

Participant: Bernard Mourrain.

In 34, we study the punctual Hilbert scheme from an algorithmic point of view. We first present algorithms, which allow to compute the inverse system of an isolated point. We define the punctual Hilbert scheme as a subvariety of a Grassmannian variety and provide explicit equations defining it. Then we localized our study to the algebraic variety ${\mathrm{Hilb}}_B$ of inverse systems, which admits a given dual (monomial) basis B. Building on the algorithm to compute the inverse system and exploiting combinatorial properties of the monomial basis $B$, we derive a minimal set of equations defining ${\mathrm{Hilb}}_B$. Using this effective presentation, we prove new properties of transversality of ${\mathrm{Hilb}}_B$, give new dimension formula for its irreducible components and prove that they are birational.

7.6 Toric Sylvester Forms

Participants: Laurent Busé, Carles Checa.

In 24, we investigate the structure of the saturation of ideals generated by sparse homogeneous polynomials over a projective toric variety X with respect to the irrelevant ideal of X. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on X. In particular, we prove that toric Sylvester forms yield bases of some graded components of Isat/I, where I denotes an ideal generated by n+1 generic forms, n is the dimension of I and Isat the saturation of I with respect to the irrelevant ideal of the Cox ring of X. Then, to illustrate the relevance of toric Sylvester forms, we provide three consequences in elimination theory: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we give a new formula for computing toric residues of the product of two forms.

7.7 Mixed Subdivisions Suitable for the Greedy Canny–Emiris Formula

Participants: Carles Checa, Ioannis Emiris.

The Canny–Emiris formula (Canny and Emiris in International symposium on applied algebra, algebraic algorithms, and error-correcting codes, 1993) gives the sparse resultant as the ratio of the determinant of a Sylvester-type matrix over a minor of it, both obtained via a mixed subdivision algorithm. In Checa and Emiris (International symposium on symbolic and algebraic computation, 2022), the authors gave an explicit class of mixed subdivisions for the greedy approach so that the formula holds, and the dimension of the constructed matrices is smaller than that of the subdivision algorithm, following the approach of Canny and Pedersen (Algorithm for the Newton resultant, 1993). Our method improves upon the dimensions of the matrices when the Newton polytopes are zonotopes and the systems are multihomogeneous. In 25, we provide more such cases, and we conjecture which might be the liftings providing minimal size of the resultant matrices. We also describe two applications of this formula, namely in computer vision and in the implicitization of surfaces, while offering the corresponding JULIA code. We finally introduce a novel tropical approach that leads to an alternative proof of a result in Checa and Emiris (2022).

7.8 From CAD to Representations Suitable for Isogeometric Analysis: a Complete Pipeline

Participants: Michelangelo Marsala, Angelos Mantzaflaris, Bernard Mourrain.

In 31, we present a complete pipeline to convert CAD models into smooth G1 spline representations, which are suitable for isogeometric analysis. Starting from a CAD boundary representation of a mechanical object, we perform an automatic control cage extraction by means of quadrangular faces, such that its limit Catmull-Clark subdivision surface approximates accurately the input model. Then we compute a basis of the G1 spline space over the quad mesh in order to carry out least squares fitting over a point cloud, acquired by sampling the original CAD geometry. The resulting surface is a collection of Bézier patches with G1 regularity, except at the sharp edges. Finally, we use the basis functions to perform isogeometric analysis simulations of realistic PDEs on the reconstructed G1 model. The quality of the construction is demonstrated via several numerical examples performed on a collection of CAD objects presenting various challenging realistic shapes.

This is a joint work with Sam Whyman and Mark Gammon (ITI CADFIX, Cambridge, UK).

7.9 A spline-based regularized method for the reconstruction of complex geological models

Participants: Ayoub Belhachmi, Bernard Mourrain.

The study and exploration of the subsurface requires the construction of geological models. This task can be difficult, especially in complex geological settings, with various unconformities. These models are constructed from seismic or well data, which can be sparse and noisy. In 23, we propose a new method to compute a stratigraphic function that represents geological layers in arbitrary settings. This function interpolates the data using piecewise quadratic C 1 Powell-Sabin splines, and is regularized via a selfadaptive diffusion scheme. For the discretization, we use Powell-Sabin splines on triangular meshes. Compared to classical interpolation methods, the use of piecewise quadratic splines has two major advantages. First, their ability to produce surfaces of higher smoothness and regularity. Second, it is straightforward to discretize high order smoothness energies like the squared Hessian energy (Stein et al. 2018). The regularization is considered as the most challenging part of any implicit modeling approach. Often, existing regularization methods produce inconsistent geological models, in particular for data with high thickness variations. To handle this kind of data, we propose a new scheme in which a diffusion term is introduced and iteratively adapted to the shapes and variations in the data, while minimizing the interpolation error.

This is a joint work with Azeddine Benabbou (SLB, Montpellier).

7.10 Goal-Adaptive Meshing of Isogeometric Kirchhoff-Love Shells

Participants: Angelos Mantzaflaris.

Mesh adaptivity is a technique to provide detail in numerical solutions without the need to refine the mesh over the whole domain. Mesh adaptivity in isogeometric analysis can be driven by Truncated Hierarchical B-splines (THB-splines) which add degrees of freedom locally based on finer B-spline bases. Labeling of elements for refinement is typically done using residual-based error estimators. In 32, an adaptive meshing workflow for isogeometric Kirchhoff-Love shell analysis is developed. This framework includes THB-splines, mesh admissibility for combined refinement and coarsening and the Dual-Weighted Residual (DWR) method for computing element-wise error contributions. The DWR can be used in several structural analysis problems, allowing the user to specify a goal quantity of interest which is used to mark elements and refine the mesh. This goal functional can involve, for example, displacements, stresses, eigenfrequencies etc. The proposed framework is evaluated through a set of different benchmark problems, including modal analysis, buckling analysis and non-linear snap-through and bifurcation problems, showing high accuracy of the DWR estimator and efficient allocation of degrees of freedom for advanced shell computations.

This is a joint work with Hugo M. Verhelst, Matthias M. Möller, J. den Besten (Univ. Delft).

7.11 Adaptive isogeometric gear contact analysis: geometry generation, truncated hierarchical B-Spline refinement and validation

Participants: Christos Karampatzakis, Angelos Mantzaflaris.

Gears are one of the most widely used transmission components. Their operation relies on the contact between mating gear teeth flanks for the transmission of power. Accurate prediction of the contact stresses at these regions, is crucial for the design and dimensioning of these systems. Gear design is centered around highly smooth involute curves that greatly influence their contact behavior. In 30, a fully adaptive isogeometric contact modeling scheme, based on hierarchical splines, is presented and applied to the simulation of gear contact problems. In particular, isogeometric simulation is performed for the modeling of mating pair of gear teeth, regarded as linearly elastic bodies. A boundary fitted B-Spline representation of the teeth is automatically generated from engineering design parameters and is used to define the initial discretisation basis. The numerical integration over the contact region is addressed using the so called, Gauss-Point to Surface formulation and a closest point projection procedure. Truncated hierarchical B-Splines are used to capture the highly localized nature of contact, while effectively reducing the number of degrees of freedom. The adaptivity is driven by the strain energy density gradient, which allows to automatically localize the mesh without a priori knowledge of the contact region between the teeth flanks. In our experiments, we justify the choices made in different steps of our algorithm and we assess the performance of our adaptive solver with respect to classical tensor product B-Splines.

7.12 Parameterization learning with convolutional neural networks for gridded data fitting

Participants: Angelos Mantzaflaris.

The approximation of point clouds in terms of parametric representations is a fundamental task for geometric modeling and processing applications to properly analyze the (re-)constructed model in its full complexity. A necessary and key step in this process requires the identification of a suitable data parameterization. If the data connectivity is determined by a rectilinear grid, standard closed-form methods are available for general multivariate configurations. Existing data-driven parameterization methods instead are limited to the univariate case and require pre/post-processing steps of the input data to handle point sequences without fixed length. In 35, we propose a dimension independent method based on convolutional neural networks to assign parameter values to gridded point clouds of arbitrary size, without the need for additional data processing steps. We train the proposed networks by considering polynomial least squares approximations and demonstrate, both in the univariate and bivariate settings, that the accuracy of the final model properly scales when uniform and adaptive spline refinement is considered. A selection of numerical experiments on point clouds of different sizes highlights the performance of our parameterization scheme. Noisy data sets which simulate measurement errors are also considered.

This is joint work with Michele de Vita, Carlotta Giannelli, Sofia Imperatore (Univ. Firenze).

7.13 BIDGCN: Boundary informed dynamic graph convolutional network for adaptive spline fitting of scattered data

Participants: Angelos Mantzaflaris.

In 26, we propose a Boundary Informed Dynamic Graph Convolutional Network (BIDGCN) characterized by a novel boundary informed input layer, with special focus on applications related to adaptive spline approximation of scattered data. The newly introduced layer propagates given boundary information to the interior of the point cloud, in order to let the input data be suitably processed by successive graph convolutional network layers. We apply our BIDGCN model to the problem of parameterizing three-dimensional unstructured data sets over a planar domain. The parameterization problem is a key step in the solution of different geometric modeling tasks and in particular for the design of surface reconstruction schemes with smooth spline surfaces. A selection of numerical examples shows the effectiveness of the proposed approach for adaptive spline fitting with (truncated) hierarchical B-spline constructions.

This is a joint work with Carlotta Giannelli, Sofia Imperatore (Univ. Firenze), Felix Scholz (MTU, Munich).

7.14 Improved markerless gait kinematics measurement using a biomechanically-aware algorithm with subject-specific geometric modeling

Participants: Laurent Busé, Mehran Hatamzadeh.

Despite the advancements in developing markerless gait analysis systems, they still demonstrate lower accuracy compared to gold-standard systems. Hence, in 27, a novel approach is presented to improve the lower limb kinematics accuracy in markerless gait analysis. This approach refines the 3D lower-limb skeletons obtained by AI-based pose estimation algorithms in a subject-specific geometric manner, preserves skeleton links’ length, benefits from gait phases information that adds biomechanical awareness to the algorithm, and utilizes an embedded trajectory smoothing. Validation of the proposed method shows that it reduces 12.6%-43.5% of root mean square error (RMSE) and significantly improves kinematic curves’ similarity to the gold-standard ones. Results also prove the feasibility of more accurate lower limb kinematics calculation using a single (2.02°-7.57° RMSE) or dual RGB-D camera (1.66°-7.25° RMSE). Development of such algorithms could result in requirement of fewer cameras that deliver comparable or even superior measurement accuracy compared to multi-camera approaches.

7.15 HydraProt: A New Deep Learning Tool for Fast and Accurate Prediction of Water Molecule Positions for Protein Structures

Participants: Ioannis Emiris.

Water molecules are integral to the structural stability of proteins and vital for facilitating molecular interactions. However, accurately predicting their precise position around protein structures remains a significant challenge, making it a vibrant research area. In 33, we introduce HydraProt (deep Hydration of Proteins), a novel methodology for predicting precise positions of water molecule oxygen atoms around protein structures, leveraging two interconnected deep learning architectures: a 3D U-net and a Multi-Layer Perceptron (MLP). Our approach starts by introducing a coarse voxel-based representation of the protein, which allows for rapid sampling of candidate water positions via the 3D U-net. These water positions are then assessed by embedding the water–protein relationship in the Euclidean space by means of an MLP. Finally, a postprocessing step is applied to further refine the MLP predictions. HydraProt surpasses existing state-of-the-art approaches in terms of precision & recall and has been validated on large data sets of protein structures. Notably, our method offers rapid inference runtime and should constitute the method of choice for protein structure studies and drug discovery applications. Our pretrained models, data, and the source code required to reproduce these results are accessible at github.com/azamanos/HydraProt.

This is a joint work with Andreas Zamanos and George Ioannakis from Athena Research and Innovation Centre (Greece).

8.1 Bilateral grants with industry

Participants: Ioannis Emiris, Panagiotis Repouskos, Ioannis Psarros.

$•$ Geometric computing.

Ioannis Emiris coordinates a research contract with the industrial partner ANSYS Inc. (Greece), in collaboration with Athena Research Center. MSc students P. Repouskos, M. Dioletis, and T. Pappas, postdoc fellow I. Psarros, and Athena researcher George Ioannakis 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 2024, 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. The implementation of prototypes has led into the development of a software library including implementations of parallel algorithms (Cuda).

It has continued during the entire year 2024 along with a tripartite grant from the Greek ministry of Development (Athena RC, ANSYS, and University of Patras, Greece).

Participants: Ayoub Belhachmi, Bernard Mourrain.

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

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.

9.1 European initiatives

9.2 National initiatives

Participants: Angelos Mantzaflaris, Bernard Mourrain, Régis Duvigneau [ACUMES].

• Title:
  ANR PRCI: “RFF-Splines: High-order Isogeometric simulation with geometrically continuous functions”
• Duration:
  2025–2028, administrative start October 2025
• Coordinator:
  Angelos Mantzaflaris
• Partners:
  University of Linz, Austria
• Summary:

  Our project aims at the development of a novel framework for high-order discretization of partial differential equations on general domains. Smoothness requirements and superior approximation power are paramount for efficient simulations. We focus on the paradigm of isogeometric analysis that uses spline functions for design and analysis on real-world (both man-made and occurring in nature) geometries. General domains pose challenges due to their topology, in particular in the vicinity of so-called extraordinary vertices, which are essentially artifacts created by the discretization (i.e., meshing) procedures. We propose a framework of geometrically continuous splines, called Refinable FreeForm (RFF-) Splines, to enable numerical schemes for topologically unrestricted design and analysis.

  The novelty of the construction is based on three main points: First, we will establish theoretical guarantees for approximation power. These will be derived based on the property of local polynomial reproduction, using an adapted construction for spline projectors on domain manifolds. Second, we will focus on the efficient construction of the basis functions, notably concerning the evaluation and the matrix assembly for simulation via numerical integration. Third, we will emphasize adaptivity and approximate evaluation through local refinement. We will employ the truncation mechanism to reduce the support of the basis functions (thereby increasing sparsity) and to preserve the partition of unity property, again with theoretical guarantees regarding the approximation power.

  The starting point of our work is provided by a now classic work of Prautzsch in 1997, which focused on the fundamental property of polynomial reproduction. Obviously, this property is essential for the derivation of theoretical guarantees for the approximation performance. The project goes all the way from the design of the construction and the theory development to the algorithmic aspects and the efficient implementation in C++, as well as experimental evaluation in demanding applications that involve high order partial differential equations.

10.1 Promoting scientific activities

10.2 Teaching - Supervision - Juries

11.1 Major publications

• 1 articleL.Lorenzo Baldi and B.Bernard Mourrain. On the Effective Putinar’s Positivstellensatz and Moment Approximation.Mathematical Programming, Series ASeptember 2022HALDOI
• 2 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, 325-357HAL
• 3 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, 796HALDOI
• 4 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, 743-787HALDOI
• 5 articleL.Laurent Busé and A.Anna Karasoulou. Resultant of an equivariant polynomial system with respect to the symmetric group.Journal of Symbolic Computation762016, 142-157HALDOI
• 6 articleL.Ludovic Calès, A.Apostolos Chalkis, I. Z.Ioannis Z. Emiris and V.Vissarion Fisikopoulos. Practical volume approximation of high-dimensional convex bodies, applied to modeling portfolio dependencies and financial crises.Computational Geometry109February 2023, 101916HALDOI
• 7 articleA.Apostolos Chalkis, I. Z.Ioannis Z. Emiris and V.Vissarion Fisikopoulos. A Practical Algorithm for Volume Estimation based on Billiard Trajectories and Simulated Annealing.ACM Journal of Experimental Algorithmics28May 2023, 1-34HALDOI
• 8 articleA.Apostolos Chalkis, I. Z.Ioannis Z. Emiris, V.Vissarion Fisikopoulos, E.Elias Tsigaridas and H.Haris Zafeiropoulos. Geometric algorithms for sampling the flux space of metabolic networks.Journal of Computational Geometry1412023HALDOI
• 9 articleE.Emmanouil Christoforou, H.Hari Leontiadou, F.Frank Noé, J.Jannis Samios, I. Z.Ioannis Z. Emiris and Z.Zoe Cournia. Investigating the Bioactive Conformation of Angiotensin II Using Markov State Modeling Revisited with Web-Scale Clustering.Journal of Chemical Theory and Computation189September 2022, 5636-5648HALDOI
• 10 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, 1-20HALDOI
• 11 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 - 93HALDOI
• 12 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, 101865HALDOI
• 13 articleP.Paul Görlach, E.Evelyne Hubert and T.Théo Papadopoulo. Rational invariants of even ternary forms under the orthogonal group.Foundations of Computational Mathematics192019, 1315-1361HALDOI
• 14 articleE.Evelyne Hubert and E.Erick Rodriguez Bazan. Algorithms for fundamental invariants and equivariants: (of finite groups).Mathematics of Computation913372022, 2459-2488HALDOI
• 15 articleZ.Zbigniew Jelonek and A.André Galligo. Elimination ideals and Bezout relations.Journal of Algebra5622020, 621-626HALDOI
• 16 inproceedingsL.Loukas Kavouras, K.Konstantinos Tsopelas, G.Giorgos Giannopoulos, D.Dimitris Sacharidis, E.Eleni Psaroudaki, N.Nikolaos Theologitis, D.Dimitrios Rontogiannis, D.Dimitris Fotakis and I. Z.Ioannis Z. Emiris. Fairness Aware Counterfactuals for Subgroups.NeurIPS 2023 - 37th Conference on Neural Information Processing SystemsProceedings Thirty-seventh Conference on Neural Information Processing SystemsNew-Orleans, Lousiane, United StatesJune 2023HAL
• 17 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, 1062-1085HALDOI
• 18 articleB.Bernard Mourrain. Polynomial-Exponential Decomposition from Moments.Foundations of Computational Mathematics186December 2018, 1435--1492HALDOI
• 19 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--1447HALDOI
• 20 inproceedingsK.Konstantinos Tertikas, D.Despoina Paschalidou, B.Boxiao Pan, J. J.Jeong Joon Park, M. A.Mikaela Angelina Uy, I. Z.Ioannis Z. Emiris, Y.Yannis Avrithis and L.Leonidas Guibas. PartNeRF: Generating Part-Aware Editable 3D Shapes without 3D Supervision.CVPR 2023 - IEEE/CVF Conference on Computer Vision and Pattern RecognitionProceedings of the 2023 IEEE/CVF Conference on Computer Vision and Pattern RecognitionVancouver, CanadaIEEEMarch 2023, 4466-4478HALDOI

11.2 Publications of the year