arXiv subjects

Boris Thibert

Publications and source records attributed to Boris Thibert.

15 recordsLinked to original sources

arXiv2026

Submanifolds of class $C^{1,\alpha}$ and sets with positive $\mu$-reach

It is well-known since the seminal work of Herbert Federer [Trans. of the AMS, 1959] that submanifolds of class $C^{1,1}$ have positive reach. In this paper, we extend this property to less regular submanifolds by using the notion of $\mu$-reach that was introduced in the 2000's. We first show that every compact $C^1$ submanifold of the Euclidean space $\E^n$ has positive $\mu$-reach for all $\mu<1$. We then show that intermediate regularities $C^{1,\alpha}$ induce more quantitative results on the norm $\|\nabla \d_M\|$ of the generalized gradient of the distance function~$\d_M$ to the submanifold. More precisely, if $M\subset \E^n$ is a submanifold of class $C^{1,\alpha}$, with $\alpha<1$, then there exists a constant $C>0$ such that $$\forall p\in\E^n\setminus M,\quad 1 - \| \nabla \d_M(p) \|^2 \leq C ~ \d_M(p)^{\frac{2 \alpha}{1- \alpha}}.$$ We finally show that the exponent $2\alpha/(1-\alpha)$ in this estimate is sharp.

math.DG↗

arXiv2025

Distributionally Robust Shape and Topology Optimization

This article aims to introduce the paradigm of distributional robustness from the field of convex optimization to tackle optimal design problems under uncertainty. We consider realistic situations where the physical model, and thereby the cost function of the design to be minimized depend on uncertain parameters. The probability distribution of the latter is itself known imperfectly, through a nominal law, reconstructed from a few observed samples. The distributionally robust optimal design problem is an intricate bilevel program which consists in minimizing the worst value of a statistical quantity of the cost function (typically, its expectation) when the law of the uncertain parameters belongs to a certain ``ambiguity set''. We address three classes of such problems: firstly, this ambiguity set is made of the probability laws whose Wasserstein distance to the nominal law is less than a given threshold; secondly, the ambiguity set is based on the first- and second-order moments of the actual and nominal probability laws. Eventually, a statistical quantity of the cost other than its expectation is made robust with respect to the law of the parameters, namely its conditional value at risk. Using techniques from convex duality, we derive tractable, single-level reformulations of these problems, framed over augmented sets of variables. Our methods are essentially agnostic of the optimal design framework; they are described in a unifying abstract framework, before being applied to multiple situations in density-based topology optimization and in geometric shape optimization. Several numerical examples are discussed in two and three space dimensions to appraise the features of the proposed techniques.

math.OC↗

arXiv2023

The Hyperbolic Plane in $\mathbb{E}^3$

We build an explicit $C^1$ isometric embedding $f_{\infty}:\mathbb{H}^2\to\mathbb{E}^3$ of the hyperbolic plane whose image is relatively compact. Its limit set is a closed curve of Hausdorff dimension 1. Given an initial embedding $f_0$, our construction generates iteratively a sequence of maps by adding at each step $k$ a layer of $N_{k}$ corrugations. To understand the behavior of $df_\infty$ we introduce a $formal$ $corrugation$ $process$ leading to a $formal$ $analogue$ $\Phi_{\infty}:\mathbb{H}^2\to \mathcal{L}(\mathbb{R}^2,\mathbb{R}^3)$. We show a self-similarity structure for $\Phi_{\infty}$. We next prove that $df_\infty$ is close to $\Phi_{\infty}$ up to a precision that depends on the sequence $N_*:= (N_{k})_k$. We then introduce the $pattern$ $maps$ $\boldsymbol{\nu}_{\infty}^\Phi$ and $\boldsymbol{\nu}_{\infty}$, of respectively $\Phi_{\infty}$ and $df_\infty$, that together with $df_0$ entirely describe the geometry of the Gauss maps associated to $\Phi_{\infty}$ and $df_\infty$. For well chosen sequences of corrugation numbers, we finally show an asymptotic convergence of $\boldsymbol{\nu}_{\infty}$ towards $\boldsymbol{\nu}_{\infty}^\Phi$ over circles of rational radii.

math.DG↗

arXiv2023

Entropy-regularized Wasserstein distributionally robust shape and topology optimization

This brief note aims to introduce the recent paradigm of distributional robustness in the field of shape and topology optimization. Acknowledging that the probability law of uncertain physical data is rarely known beyond a rough approximation constructed from observed samples, we optimize the worst-case value of the expected cost of a design when the probability law of the uncertainty is ``close'' to the estimated one up to a prescribed threshold. The ``proximity'' between probability laws is quantified by the Wasserstein distance, a notion pertaining to optimal transport theory. The combination of the classical entropic regularization technique in this field with recent results from convex duality theory allows to reformulate the distributionally robust optimization problem in a way which is tractable for computations. Two numerical examples are presented, in the different settings of density-based topology optimization and geometric shape optimization. They exemplify the relevance and applicability of the proposed formulation regardless of the selected optimal design framework.

math.OC↗

arXiv2022

Strong c-concavity and stability in optimal transport

The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and justifies many of the applications of optimal transport. In this article, we introduce the notion of strong c-concavity, and we show that it plays an important role for proving stability results in optimal transport for general cost functions c. We then introduce a differential criterion for proving that a function is strongly c-concave, under an hypothesis on the cost introduced originally by Ma-Trudinger-Wang for establishing regularity of optimal transport maps. Finally, we provide two examples where this stability result can be applied, for cost functions taking value +$\infty$ on the sphere: the reflector problem and the Gaussian curvature measure prescription problem.

math.NA↗

arXiv2021

Metasurfaces and Optimal transport

This paper provides a theoretical and numerical approach to show existence, uniqueness, and the numerical determination of metalenses refracting radiation with energy patterns. The theoretical part uses ideas from optimal transport and for the numerical solution we study and implement a damped Newton algorithm to solve the semi discrete problem. A detailed analysis is carried out to solve the near field one source refraction problem and extensions to the far field are also mentioned.

math.AP↗

arXiv2021

A Damped Newton Algorithm for Generated Jacobian Equations

Generated Jacobian Equations have been introduced by Trudinger [Disc. cont. dyn. sys (2014), pp. 1663-1681] as a generalization of Monge-Amp{è}re equations arising in optimal transport. In this paper, we introduce and study a damped Newton algorithm for solving these equations in the semi-discrete setting, meaning that one of the two measures involved in the problem is finitely supported and the other one is absolutely continuous. We also present a numerical application of this algorithm to the near-field parallel refractor problem arising in non-imaging problems.

cs.CG↗

arXiv2020

Optimal transport: discretization and algorithms

This chapter describes techniques for the numerical resolution of optimal transport problems. We will consider several discretizations of these problems, and we will put a strong focus on the mathematical analysis of the algorithms to solve the discretized problems. We will describe in detail the following discretizations and corresponding algorithms: the assignment problem and Bertsekas auction's algorithm; the entropic regularization and Sinkhorn-Knopp's algorithm; semi-discrete optimal transport and Oliker-Prussner or damped Newton's algorithm, and finally semi-discrete entropic regularization. Our presentation highlights the similarity between these algorithms and their connection with the theory of Kantorovich duality.

math.NA↗

arXiv2019

Light in Power: A General and Parameter-free Algorithm for Caustic Design

We present in this paper a generic and parameter-free algorithm to efficiently build a wide variety of optical components, such as mirrors or lenses, that satisfy some light energy constraints. In all of our problems, one is given a collimated or point light source and a desired illumination after reflection or refraction and the goal is to design the geometry of a mirror or lens which transports exactly the light emitted by the source onto the target. We first propose a general framework and show that eight different optical component design problems amount to solving a light energy conservation equation that involves the computation of visibility diagrams. We then show that these diagrams all have the same structure and can be obtained by intersecting a 3D Power diagram with a planar or spherical domain. This allows us to propose an efficient and fully generic algorithm capable to solve these eight optical component design problems. The support of the prescribed target illumination can be a set of directions or a set of points located at a finite distance. Our solutions satisfy design constraints such as convexity or concavity. We show the effectiveness of our algorithm on simulated and fabricated examples.

cs.GR↗

arXiv2018

Distributable Consistent Multi-Object Matching

In this paper we propose an optimization-based framework to multiple object matching. The framework takes maps computed between pairs of objects as input, and outputs maps that are consistent among all pairs of objects. The central idea of our approach is to divide the input object collection into overlapping sub-collections and enforce map consistency among each sub-collection. This leads to a distributed formulation, which is scalable to large-scale datasets. We also present an equivalence condition between this decoupled scheme and the original scheme. Experiments on both synthetic and real-world datasets show that our framework is competitive against state-of-the-art multi-object matching techniques.

cs.DS↗

arXiv2017

An algorithm for optimal transport between a simplex soup and a point cloud

We propose a numerical method to find the optimal transport map between a measure supported on a lower-dimensional subset of R^d and a finitely supported measure. More precisely, the source measure is assumed to be supported on a simplex soup, i.e. on a union of simplices of arbitrary dimension between 2 and d. As in [Aurenhammer, Hoffman, Aronov, Algorithmica 20 (1), 1998, 61--76] we recast this optimal transport problem as the resolution of a non-linear system where one wants to prescribe the quantity of mass in each cell of the so-called Laguerre diagram. We prove the convergence with linear speed of a damped Newton's algorithm to solve this non-linear system. The convergence relies on two conditions: (i) a genericity condition on the point cloud with respect to the simplex soup and (ii) a (strong) connectedness condition on the support of the source measure defined on the simplex soup. Finally, we apply our algorithm in R^3 to compute optimal transport plans between a measure supported on a triangulation and a discrete measure. We also detail some applications such as optimal quantization of a probability density over a surface, remeshing or rigid point set registration on a mesh.

cs.CG↗

arXiv2016

Convergence of a Newton algorithm for semi-discrete optimal transport

Many problems in geometric optics or convex geometry can be recast as optimal transport problems: this includes the far-field reflector problem, Alexandrov's curvature prescription problem, etc. A popular way to solve these problems numerically is to assume that the source probability measure is absolutely continuous while the target measure is finitely supported. We refer to this setting as semi-discrete optimal transport. Among the several algorithms proposed to solve semi-discrete optimal transport problems, one currently needs to choose between algorithms that are slow but come with a convergence speed analysis (e.g. Oliker-Prussner) or algorithms that are much faster in practice but which come with no convergence guarantees Algorithms of the first kind rely on coordinate-wise increments and the number of iterations required to reach the solution up to an error of $\epsilon$ is of order $N^3/\epsilon$, where $N$ is the number of Dirac masses in the target measure. On the other hand, algorithms of the second kind typically rely on the formulation of the semi-discrete optimal transport problem as an unconstrained convex optimization problem which is solved using a Newton or quasi-Newton method. The purpose of this article is to bridge this gap between theory and practice by introducing a damped Newton's algorithm which is experimentally efficient and by proving the global convergence of this algorithm with optimal rates. The main assumptions is that the cost function satisfies a condition that appears in the regularity theory for optimal transport (the Ma-Trudinger-Wang condition) and that the support of the source density is connected in a quantitative way (it must satisfy a weighted Poincar\'e-Wirtinger inequality).

math.NA↗

arXiv2015

Robust Geometry Estimation using the Generalized Voronoi Covariance Measure

The Voronoi Covariance Measure of a compact set K of R^d is a tensor-valued measure that encodes geometric information on K and which is known to be resilient to Hausdorff noise but sensitive to outliers. In this article, we generalize this notion to any distance-like function delta and define the delta-VCM. We show that the delta-VCM is resilient to Hausdorff noise and to outliers, thus providing a tool to estimate robustly normals from a point cloud approximation. We present experiments showing the robustness of our approach for normal and curvature estimation and sharp feature detection.

cs.CG↗

arXiv2014

Intersection of paraboloids and application to Minkowski-type problems

In this article, we study the intersection (or union) of the convex hull of N confocal paraboloids (or ellipsoids) of revolution. This study is motivated by a Minkowski-type problem arising in geometric optics. We show that in each of the four cases, the combinatorics is given by the intersection of a power diagram with the unit sphere. We prove the complexity is O(N) for the intersection of paraboloids and Omega(N^2) for the intersection and the union of ellipsoids. We provide an algorithm to compute these intersections using the exact geometric computation paradigm. This algorithm is optimal in the case of the intersection of ellipsoids and is used to solve numerically the far-field reflector problem.

cs.CG↗

arXiv2008

Stability of Curvature Measures

We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive $μ$-reach can be estimated by the same curvature measures of the offset of a compact set K' close to K in the Hausdorff sense. We show how these curvature measures can be computed for finite unions of balls. The curvature measures of the offset of a compact set with positive $μ$-reach can thus be approximated by the curvature measures of the offset of a point-cloud sample. These results can also be interpreted as a framework for an effective and robust notion of curvature.

cs.CG↗