Bruno Eric Levy

• PhD
• Research Director at National Institute for Research in Computer Science and Control

We study Monge-Ampère gravity (MAG) as an effective theory of cosmological structure formation through optimal transport theory. MAG is based on the Monge-Ampère equation, a nonlinear version of the Poisson equation, that relates the Hessian determinant of the potential to the density field. We explain how MAG emerges from a conditioned system of i...

In this article, I propose a numerical method to solve semi-discrete optimal transport problems for gigantic pointsets ($10^8$ points and more). By pushing the limits by several orders of magnitude, it opens the path to new applications in cosmology, fluid simulation and data science to name but a few. The method is based on a new algorithm that co...

We demonstrate the effectiveness of one of the many multitracer analyses enabled by optimal transport (OT) reconstruction. Leveraging a semidiscrete OT algorithm, we determine the displacements between initial and observed positions of biased tracers and the remaining matter field. With only redshift-space distorted final positions of biased tracer...

This article introduces a general mesh intersection algorithm that exactly computes the so-called Weiler model and that uses it to implement boolean operations with arbitrary multi-operand expressions, CSG (constructive solid geometry) and some mesh repair operations. From an input polygon soup, the algorithm first computes the co-refinement, with...

Recent research has emphasized the benefits of accurately reconstructing the initial Lagrangian positions of biased tracers from their positions at a later time, to gain cosmological information. A weighted semidiscrete optimal transport algorithm can achieve the required accuracy, provided the late-time positions are known, with minimal informatio...

Recent research has emphasized the benefits of accurately reconstructing the initial Lagrangian positions of biased tracers from their positions at a later time, to gain cosmological information. A weighted semi-discrete optimal transport algorithm can achieve the required accuracy, provided the late-time positions are known, with minimal informati...

A weighted, semidiscrete, fast optimal transport (OT) algorithm for reconstructing the Lagrangian positions of protohalos from their evolved Eulerian positions is presented. The algorithm makes use of a mass estimate of the biased tracers and of the distribution of the remaining mass (the “dust”) but is robust to errors in the mass estimates. Tests...

Interpolating between measures supported by polygonal or polyhedral domains is a problem that has been recently addressed by the semi-discrete optimal transport framework. Within this framework, one of the domains is discretized with a set of samples, while the other one remains continuous. In this paper we present a method to introduce some symmet...

Optimal transport theory has recently re-emerged as a vastly resourceful field of mathematics with elegant applications across physics and computer science. Harnessing methods from geometry processing, we report on the efficient implementation for a specific problem in cosmology-the reconstruction of the linear density field from low redshifts, in...

A model-independent, weighted semi-discrete, fast optimal transport algorithm to reconstruct the Lagrangian positions of proto-halos from their evolved Eulerian positions is presented. Tests with state-of-art cosmological simulations show that the positions of proto-halos are reconstructed accurately, without having to assume a background cosmology...

This article introduces a representation of dynamic meshes, adapted to some numerical simulations that require controlling the volume of objects with free boundaries, such as incompressible fluid simulation, some astrophysical simulations at cosmological scale, and shape/topology optimization. The algorithm decomposes the simulated object into a se...

Optimal transport theory has recently reemerged as a vastly resourceful field of mathematics with elegant applications across physics and computer science. Harnessing methods from geometry processing, we report on the efficient implementation for a specific problem in cosmology -- the reconstruction of the linear density field from low redshifts, i...

We leverage powerful mathematical tools stemming from optimal transport theory and transform them into an efficient algorithm to reconstruct the fluctuations of the primordial density field, built on solving the Monge-Ampère-Kantorovich equation. Our algorithm computes the optimal transport between an initial uniform continuous density field, parti...

We propose a method to simultaneously decompose a 3D object into power diagram cells and to integrate given functions in each of the obtained simple regions. We offer a novel, highly parallel algorithm that lends itself to an efficient GPU implementation. It is optimized for algorithms that need to compute many decompositions, for instance, centroi...

This article introduces a representation of dynamic meshes, adapted to some numerical simulations that require controlling the volume of objects with free boundaries, such as incompressible fluid simulation, some astrophysical simulations at cosmological scale, and shape/topology optimization. The algorithm decomposes the simulated object into a se...

We leverage powerful mathematical tools stemming from optimal transport theory and transform them into an efficient algorithm to reconstruct the fluctuations of the primordial density field, built on solving the Monge-Amp\`ere-Kantorovich equation. Our algorithm computes the optimal transport between an initial uniform continuous density field, par...

The Linear Smoothing (LS) scheme \cite{francisa.ortiz-bernardin2017} ameliorates linear and quadratic approximations over convex polytopes by employing a three-point integration scheme. In this work, we propose a linearly consistent one point integration scheme which possesses the properties of the LS scheme with three integration points but requir...

In this paper, the recently proposed linearly consistent one point integration rule for the meshfree methods is extended to arbitrary polytopes. The salient feature of the proposed technique is that it requires only one integration point within each n-sided polytope as opposed to 3n in Francis et al. (2017) and 13n integration points in the convent...

Incorporating prior geological knowledge in geophysical process models often meets practical meshing challenges and raises the question of how much detail is to be included in the geometric model. We introduce a strategy to automatically repair and simplify geological maps, geological cross-sections and the associated meshes while preserving elemen...

We propose a GPU algorithm that computes a 3D Voronoi diagram. Our algorithm is tailored for applications that solely make use of the geometry of the Voronoi cells, such as Lloyd's relaxation used in meshing, or some numerical schemes used in fluid simulations and astrophysics. Since these applications only require the geometry of the Voronoi cells...

In this article, I present recent methods for the numerical simulation of fluid dynamics and the associated computational algorithms. The goal of this article is to explain how to model an incompressible fluid, and how to write a computer program that simulates it. I will start from Newton laws "$F = ma$" applied to a bunch of particles, then show...

This article presents a new method to compute a self-intersection free high-dimensional Euclidean embedding (SIFHDE²) for surfaces and volumes equipped with an arbitrary Riemannian metric. It is already known that given a high-dimensional (high-d) embedding, one can easily compute an anisotropic Voronoi diagram by back-mapping it to 3D space. We sh...

We propose a robust pipeline that can generate hex-dominant meshes from any global parameterization of a tetrahedral mesh. We focus on robustness in order to be able to benchmark different parameterizations on a large database. Our main contribution is a new method that integrates the hexahedra (extracted from the parameterization) into the origina...

This article gives an introduction to optimal transport, a mathematical theory that makes it possible to measure distances between functions (or distances between more general objects), to interpolate between objects or to enforce mass/volume conservation in certain computational physics simulations. Optimal transport is a rich scientific domain, w...

This article introduces a new method to efficiently compute the distance (i.e., Lp norm of the difference) between two functions supported by two different meshes of the same 3D domain. The functions that we consider are typically finite element solutions discretized in different function spaces supported by meshes that are potentially completely u...

This article gives an introduction to optimal transport, a mathematical theory that makes it possible to measure distances between functions (or distances between more general objects), to interpolate between objects or to enforce mass/volume conservation in certain computational physics simulations. Optimal transport is a rich scientific domain, w...

The structural modeling of a macromolecular machine is like a "Lego" approach that is challenged when blocks, like proteins imported from the Protein Data Bank, are to be assembled with an element adopting a serpentine shape, such as DNA templates. DNA must then be built ex nihilo, but modeling approaches are either not user-friendly or very long a...

The structural modeling of a macromolecular machine is like a “Lego” approach that is challenged when blocks, like proteins imported from the Protein Data Bank, are to be assembled with an element adopting a serpentine shape, such as DNA templates. DNA must then be built ex nihilo, but modeling approaches are either not user-friendly or very long a...

We present a method for reconstructing a 3D surface triangulation from an input point set. The main component of the method is an algorithm that computes the restricted Voronoi diagram. In our specific case, it corresponds to the intersection between the 3D Voronoi diagram of the input points and a set of disks centered at the points and orthogonal...

RINGMesh is a C++ open-source programming library for manipulating discretized geological models. It is designed to ease the development of applications and workflows that use discretized 3D models. It is neither a geomodeler, nor a meshing software. RINGMesh implements functionalities to read discretized surface-based or volumetric structural mode...

Given a tetrahedral mesh, the algorithm described in this article produces a smooth 3D frame field, i.e. a set of three orthogonal directions associated with each vertex of the input mesh. The field varies smoothly inside the volume, and matches the normals of the volume boundary. Such a 3D frame field is a key component for some hexahedral meshing...

This article presents a new method to optimally partition a geometric domain with capacity constraints on the partitioned regions. It is an important problem in many fields, ranging from engineering to economics. It is known that a capacity-constrained partition can be obtained as a power diagram with the squared L2 metric. We present a method with...

This article introduces a method that generates a hexahedral-dominant mesh from an input tetrahedral mesh. It follows a three-step pipeline similar to the one proposed by Carrier Baudoin et al.: (1) generate a frame field, (2) generate a pointset P that is mostly organized on a regular grid locally aligned with the frame field, and (3) generate the...

This article introduces a method that generates a hexahedral-dominant mesh from an input tetrahedral mesh. It follows a three-step pipeline similar to the one proposed by Carrier Baudoin et al.: (1) generate a frame field, (2) generate a pointset P that is mostly organized on a regular grid locally aligned with the frame field, and (3) generate the...

Corner-point gridding is widely used in reservoir and basin modeling but generally yields approximations in the representation of geological interfaces. This paper introduces an indirect method to generate a hex-dominant mesh conformal to 3D geological surfaces and well paths suitable for finite-element and control-volume finite-element simulations...

This article deals with solving partial differential equations with the finite element method on hybrid non-conforming hexahedral-tetrahedral meshes. By non-conforming, we mean that a quadrangular face of a hexahedron can be connected to two triangular faces of tetrahedra. We introduce a set of low-order continuous (C0) finite element spaces define...

We present a practical reconstruction algorithm that generates a surface triangulation from an input point set. In the result, the input points appear as vertices of the generated triangulation. The algorithm has several interesting properties: it is very simple to implement, it is time and memory efficient, and it is trivially parallelized. On a s...

In this article, I focus on the robustness of geometric programs (e.g., De-launay triangulation, intersection between surfacic or volumetric meshes, Voronoi-based meshing. . .) w.r.t. numerical degeneracies. Some of these geometric programs require " exotic " predicates, not available in standard libraries (e.g., J.-R. Shewchuk's implementation and...

We present a geometric representation of a tetrahedral mesh that is solely based on dihedral angles. We first show that the shape of a tetrahedral mesh is completely defined by its dihedral angles. This proof leads to a set of angular constraints that must be satisfied for an immersion to exist in R-3. This formulation lets us easily specify condit...

This work describes an automatic method to anisotropically remesh an input bad quality mesh while preserving sharp features. We extend the method of Lévy and Bonneel (2012), based on the lifting of the input mesh in a 6D space (position and normal), and the optimization of a restricted Voronoï diagram in that space. The main advantage of this metho...

In this paper we propose a method to generate mixed-element meshes (tetrahedra, triangular prisms, square pyramids) for B-Rep models. The vertices, edges, facets, and cells of the final volumetric mesh are determined from the combinatorial analysis of the intersections between the model components and the Voronoi diagram of sites distributed to sam...

Corner-point gridding of reservoir and basin models is widely used but generally yields approximations in the geological interfaces representation in flow simulation. This paper introduces an indirect method to generate a hexdominant mesh conformal to 3D geological surfaces suitable for Finite-Element and Control-Volume Finite-Element simulations....

This paper introduces a numerical algorithm to compute the $L_2$ optimal transport map between two measures $\mu$ and $\nu$, where $\mu$ derives from a density $\rho$ defined as a piecewise linear function (supported by a tetrahedral mesh), and where $\nu$ is a sum of Dirac masses. I first give an elementary presentation of some known results on op...

Flow simulation in a reservoir can be highly impacted by upscaling errors. These errors can be reduced by using simulation grids with cells as homogeneous as possible, hence conformable to horizons and faults. In this paper, the coordinates of 3D Voronoi seeds are optimized so that Voronoi cell facets honor the structural features. These features a...

We propose a method to remesh the surfaces of 3D sealed geological structural models for subsequent volumetric meshing. The input of the method is a set of triangulated surfaces that are in contact along given lines and at given points. The output is a set of surfaces meshed with triangles as equilateral as possible. The method relies on a global C...

This paper proposes a new algorithm to generate a graded three-dimensional tetrahedral mesh. It revisits the class of methods based on optimal Delaunay triangulation (ODT) and proposes a proper way of injecting a background density function into the objective function minimized by ODT. This continuous/analytic point of view leads to an objective fu...

Triangle meshes have been nearly ubiquitous in computer graphics, and a large body of data structures and geometry processing algorithms based on them has been developed in the literature. At the same time, quadrilateral meshes, especially semi‐regular ones, have advantages for many applications, and significant progress was made in quadrilateral m...

We propose a method that computes a piecewise constant approximation of a function defined on a mesh. The approximation is associated with the cells of a restricted Voronoï diagram. Our method optimizes an objective function measuring the quality of the approximation. This objective function depends on the placement of the samples that define the r...

We propose a method for mapping polynomial volumes. Given a closed surface and an initial template volume grid, our method deforms the template grid by fitting its boundary to the input surface while minimizing a volume distortion criterion. The result is a point‐to‐point map distorting linear cells into curved ones. Our method is based on several...

Centroidal Voronoi tessellation (CVT) and its extensions have a wide spectrum of applications including computational geometry, image processing, cellular biology and scientific visualization etc. In this paper, we propose the concept of the complete streamline and the CVT of streamlines, and then formulate the computation of CVT of complete stream...

This paper introduces a particle-based approach for anisotropic surface meshing. Given an input polygonal mesh endowed with a Riemannian metric and a specified number of vertices, the method generates a metric-adapted mesh. The main idea consists of mapping the anisotropic space into a higher dimensional isotropic one, called "embedding space". The...

The GraphiteLifeExplorer tool enables biologists to reconstruct 3D cellular complexes built from proteins and DNA molecules. Models of DNA molecules can be drawn in an intuitive way and assembled to proteins or others globular structures. Real time navigation and immersion offer a unique view to the reconstructed biological machinery.

This paper introduces Voronoi squared distance minimization (VSDM), an algorithm that fits a surface to an input mesh. VSDM minimizes an objective function that corresponds to a Voronoi-based approximation of the overall squared distance function between the surface and the input mesh (SDM). This objective function is a generalization of the one mi...

This paper introduces a new method for anisotropic surface meshing. From an input polygonal mesh and a specified number of vertices, the method generates a curvature-adapted mesh. The main idea consists in transforming the 3d anisotropic space into a higher dimensional isotropic space (typically 6d or larger). In this high dimensional space, the me...

This paper introduces a new method for anisotropic surface meshing. From an input polygonal mesh and a specified number of vertices, the method generates a curvature-adapted mesh. The main idea consists in transforming the 3d anisotropic space into a higher dimensional isotropic space (typically 6d or larger). In this high dimensional space, the me...

We propose a new free surface fluid front tracking algorithm. Based on Centroidal Voronoi Diagram optimizations, our method creates a serie of either isotropic or anisotropic meshes that confroms with an evolving surface. Fig. 1. Enright's test at steps 0, 50, 75, 100, 150, 200, 225, 250 and 300 (top) and details of the isotropic and anisotropic 15...

Centroidal Voronoi Tessellation (CVT) of points has many applications in geometry processing, including re-meshing and segmentation, to name but a few. In this paper, we generalize the CVT concept to graphs via a variational characterization. Given a graph and a 3D polygonal surface, our method optimizes the placement of the vertices of the graph i...

This paper focuses on fault-related uncertainties in the subsurface, which can significantly affect the numerical simulation of physical processes. Our goal is to use dynamic data and process-based simulation to update structural uncertainty in a Bayesian inverse approach. We propose a stochastic fault model where the number and features of faults...

Geometry processing is a fast-growing area of research that designs efficient algorithms for the acquisition, reconstruction, analysis, manipulation, simulation and transmission of 3D models. This course covers different aspects of Geometry Processing, related with the reconstruction of high-level information from raw data. The first part of the co...

Anisotropic triangle meshes are used for efficient approximation of surfaces and flow data in finite element analysis, and in these applications it is desirable to have as few obtuse triangles as possible to reduce the discretization error. We present a variational approach to suppressing obtuse triangles in anisotropic meshes. Specifically, we int...

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Reservoir production curves provide significant information about reservoir compartmentalization and connectivity, but are most often used at late stages of reservoir modeling workflows, leading to possible inconsistencies between static and dynamic reservoir models. In this paper, we propose to use this information during structural modeling to re...

Ray casting on graphics processing units (GPUs) opens new possibilities for molecular visualization. We describe the implementation and calculation of diverse molecular representations such as licorice, ball-and-stick, space-filling van der Waals spheres, and approximated solvent-accessible surfaces using GPUs. We introduce HyperBalls, an improved...

The Voronoi diagram is a fundamental geometric structure widely used in various fields, especially in computer graphics and geometry computing. For a set of points in a compact domain (i.e. a bounded and closed 2D region or a 3D volume), some Voronoi cells of their Voronoi diagram are infinite or partially outside of the domain, but in practice onl...

For numerical reservoir flow simulation, grids that are conformal to the geological features are needed in order to reduce the homogenization error (in particular between horizons) and to retrieve the major flow features (such as faults). In this paper, Voronoi Tessellations are obtained by an optimization method where the minimized function is mod...

In this paper we propose a method to remesh non-manifold surfaces with triangles as equilateral as possible. We adapt an existing Voronoi based remeshing framework to recover the topology of non-manifold surfaces and their boundaries. The input of the procedure is a non-manifold triangulated surface constituted of several connected components repre...

In this paper, we propose an efficient algorithm to compute the centroidal Voronoi tessellation in 2D periodic space. We first present a simple algorithm for constructing the periodic Voronoi diagram (PVD) from a Euclidean Voronoi diagram. The presented PVD algorithm considers only a small set of periodic copies of the input sites, which is more ef...

We present several variations on Centroidal Voronoi Tesselations. First we review the classical definition, as a stable critical point of an objective function (quantization noise power), then we propose some modifications of the objective function (anisotropy, Lp norm). The so-modified Centroidal Voronoi Tesselations are useful for applications in...

3D geological models commonly built to manage natural resources are much affected by uncertainty because most of the subsurface is inaccessible to direct observation. Appropriate ways to intuitively visualize uncertainties are therefore critical to draw appropriate decisions. However, empirical assessments of uncertainty visualization for decision...

Recent advances in experimental structure determination provide a wealth of structural data on huge macromolecular assemblies such as the ribosome or viral capsids, available in public databases. Further structural models arise from reconstructions using symmetry orders or fitting crystal structures into low-resolution maps obtained by electron-mic...

3D structural modeling is a major instrument in geosciences, e.g. for the assessment of groundwater and energy resources or nuclear waste underground storage. Fault network modeling is a particularly crucial step during this task, for faults compartmentalize rock units and plays a key role in subsurface flow, whether faults are sealing barriers or...

This paper introduces Lp-Centroidal Voronoi Tessellation (Lp-CVT), a generalization of CVT that minimizes a higher-order moment of the coordinates on the Voronoi cells. This generalization allows for aligning the axes of the Voronoi cells with a predefined background tensor field (anisotropy). Lp-CVT is computed by a quasi-Newton optimization frame...

We introduce Lp-Centroidal Voronoi Tessellation (Lp-CVT), a generalization of CVT that minimizes a higher-order moment of the coordinates on the Voronoi cells. This generalization allows for aligning the axes of the Voronoi cells with a predefined background tensor field (anisotropy). Lp-CVT is computed by a quasi-Newton optimization framework, bas...

The Voronoi diagram is a fundamental geometry structure widely used in various fields, especially in computer graphics and geometry computing. For a set of points in a compact 3D domain (i.e. a finite 3D volume), some Voronoi cells of their Voronoi diagram are infinite, but in practice only the parts of the cells inside the domain are needed, as wh...

Surface materials are commonly described by attributes stored in textures (for instance, color, normal, or displacement). Interpolation during texture lookup provides a continuous value field everywhere on the surface, except at the chart boundaries where visible discontinuities appear. We propose a solution to make these seams invisible, while sti...

The generation of reservoir grids has to take into account numerous flow parameters, static and dynamic, from the fine-scale geological models to minimize discretization errors. These parameters are generally encoded separately as constraints on cell size, orientation and aspect ratio. In this paper, we propose to encode them all at a time in a Rie...

The structural interpretation of 2D seismic lines and their correlation to deduce a 3D structure, e.g. a fault network, is a hard task for geoscientists. Interpretations may introduce a prior geological knowledge bias and/or human bias (Bond et al., 2007). Consequently, decisions based on one single deterministic model of the subsurface may be high...

Spectral mesh processing is an idea that was proposed at the beginning of the 1990s to port the "signal processing toolbox" to the setting of 3D mesh models. Recent advances in both computing power and numerical software make it possible to fully implement this vision. In the classical context of sound and image processing, Fourier analysis was a c...

This reference is for an abstract only. A full paper was not submitted for this conference. Abstract 1 Introduction 3D visualization of flow simulation results is valuable in addressing field development issues such as appropriate placement of infill wells. However, integration of static and dynamic uncertainty in reservoir management decisions o...

Many algorithms in texture synthesis, nonphotorealistic rendering (hatching), or remeshing require to define the orientation of some features (texture, hatches, or edges) at each point of a surface. In early works, tangent vector (or tensor) fields were used to define the orientation of these features. Extrapolating and smoothing such fields is usu...

3D visualization of flow simulation results is valuable in addressing field development issues such as appropriate placement of infill wells. However, integration of static and dynamic uncertainty in reservoir management decisions often lacks proper visualization methods - typically, alternative reservoir models are considered one after another, or...

Many objects have patterns that vary in appearance at different surface locations. We say that these are differences in materials, and we present a material-space approach for interactively designing such textures. At the heart of our approach is a new method to pre-calculate and use a 3D texture tile that is periodic in the spatial dimensions (s;t...

Centroidal Voronoi tessellation (CVT) is a particular type of Voronoi tessellation that has many applications in computational sciences and engineering, including computer graphics. The prevailing method for computing CVT is Lloyd's method, which has linear convergence and is inefficient in practice. We develop new efficient methods for CVT computa...