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. Contrary to the common practice in the optimal design literature, 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 that show up in different situations regarding the available knowledge about the parameter uncertainty and the nature of the optimal design problem. Firstly, the 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 a priori intricate problems, framed over augmented sets of variables. Our methods are independent of the optimal design framework. We thus describe these reformulations in a unifying abstract framework, before specializing them to multiple situations in density-based topology optimization and in geometric shape optimization. Multiple numerical examples are presented in both academic benchmark situations and relatively realistic settings to illustrate the efficiency and broad range of application of the proposed techniques.

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 μ-reach that was introduced in the 2000's. We first show that every compact C^1 submanifold of the Euclidean space ^n has positive μ-reach for all μ<1. We then show that intermediate regularities C^1,α induce more quantitative results on the norm ∇_̣M of the generalized gradient of the distance function _̣M to the submanifold. More precisely, if M⊂^n is a submanifold of class C^1,α, with α<1, then there exists a constant C>0 such that ∀ p∈^n∖ M, 1 - ∇_̣M(p) ^2 ≤ C _̣M(p)^2 α/1- α. We finally show that the exponent 2α/(1-α) in this estimate is sharp.

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.

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.

We build an explicit C^1 isometric embedding f_∞:ℍ^2→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_∞ we introduce a formal corrugation process leading to a formal analogue Φ_∞:ℍ^2→ℒ(ℝ^2,ℝ^3). We show a self-similarity structure for Φ_∞. We next prove that df_∞ is close to Φ_∞ up to a precision that depends on the sequence N_*:= (N_k)_k. We then introduce the pattern maps ν_∞^Φ and ν_∞, of respectively Φ_∞ and df_∞, that together with df_0 entirely describe the geometry of the Gauss maps associated to Φ_∞ and df_∞. For well chosen sequences of corrugation numbers, we finally show an asymptotic convergence of ν_∞ towards ν_∞^Φ over circles of rational radii.

The estimation of differential quantities on oriented point cloud is a classical step for many geometry processing tasks in computer graphics and vision. Even if many solutions exist to estimate such quantities, they usually fail at satisfying both a stable estimation with theoretical guarantee, and the efficiency of the associated algorithm. Relying on the notion of corrected curvature measures [LRT22, LRTC20] designed for surfaces, the method introduced in this paper meets both requirements. Given a point of interest and a few nearest neighbours, our method estimates the whole curvature tensor information by generating random triangles within these neighbours and normalising the corrected curvature measures by the corrected area measure. We provide a stability theorem showing that our pointwise curvatures are accurate and convergent, provided the noise in position and normal information has a variance smaller than the radius of neighbourhood. Experiments and comparisons with the state-of-the-art confirm that our approach is more accurate and much faster than alternatives. The method is fully parallelizable, requires only one nearest neighbour request per point of computation, and is trivial to implement.

Generated Jacobian Equations have been introduced by Trudinger (Discrete Contin Dyn Syst A 34(4):1663–1681, 2014) as a generalization of Monge–Ampère 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 reflector problem arising in non-imaging problems.

This paper proposes a new mathematical and computational tool for inferring the geometry of shapes known only through approximations such as triangulated or digital surfaces. The main idea is to decouple the position of the shape boundary from its normal vector field. To do so, we extend a classical tool of geometric measure theory, the normal cycle, so that it takes as input not only a surface but also a normal vector field. We formalize it as a current in the oriented Grassmann bundle $$\mathbb {R}^3 \times \mathbb {S}^2$$ . By choosing adequate differential forms, we define geometric measures like area, mean and Gaussian curvatures. We then show the stability of these measures when both position and normal input data are approximations of the underlying continuous shape. As a byproduct, our tool is able to correctly estimate curvatures over polyhedral approximations of shapes with explicit bounds, even when their natural normal are not correct, as long as an external convergent normal vector field is provided. Finally, the accuracy, convergence and stability under noise perturbation is evaluated experimentally onto digital surfaces.

Is it possible to shape a piece of glass so that it refracts and concentrates sunlight in order to produce a given image? The modelling of this kind of problem leads to nonlinear second-order partial differential equations, which belong to the family of Monge–Ampère equations. We will see how semi-discrete methods, that can be traced back to Minkowski’s works, allow us to numerically solve such equations.

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.

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.

A consistent and yet practically accurate definition of curvature onto polyhedral meshes remains an open problem. We propose a new framework to define curvature measures, based on the Corrected Normal Current, which generalizes the normal cycle: it uncouples the positional information of the polyhedral mesh from its geometric normal vector field, and the user can freely choose the corrected normal vector field at vertices for curvature computations. A smooth surface is then built in the Grassmannian ℝ3 × 𝕊2 by simply interpolating the given normal vector field. Curvature measures are then computed using the usual Lipschitz–Killing forms, and we provide closed‐form formulas per triangle. We prove a stability result with respect to perturbations of positions and normals. Our approach provides a natural scale‐space for all curvature estimations, where the scale is given by the radius of the measuring ball. We show on experiments how this method outperforms state‐of‐the‐art methods on clean and noisy data, and even achieves pointwise convergence on difficult polyhedral meshes like digital surfaces. The framework is also well suited to curvature computations using normal map information.

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).

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.

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.

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.

Many real-world tasks for autonomous agents benefit from understanding dynamic inter-object interactions. Detecting, analyzing and differentiating between the various ways that an object can be interacted with provides implicit information about its function. This can help train autonomous agents to handle objects and understand unknown scenes. We describe a general mathematical framework to analyze and classify interactions, defined as dynamic motions performed by an active object onto a passive one. We factorize interactions via motion features computed in the spatio-temporal domain, and encoded into a global, object-centric signature. Equipped with a similarity measure to compare such signatures, we showcase classification of interactions with a single object. We also propose a novel acquisition setup combining RGBD sensing with a virtual reality (VR) display, to capture interactions with purely virtual objects.

Geometric inference deals with the problem of recovering the geometry and topology of a compact subset K of R-d from an approximation by a finite set P. This problem has seen several important developments in the previous decade. Many of the proposed constructions share a common feature: they estimate the geometry of the underlying compact set K using offsets of P, that is r-sublevel set of the distance function to P. These offset correspond to what is called tubular neighborhoods in differential geometry. First and second-order geometric quantities are encoded in the tube K-r around a manifold. For instance, the classical tube formula asserts that it is possible to estimate the curvature of a compact smooth submanifold K from the volume of its offsets. One can hope that if the finite set P is close to K in the Hausdorff sense, some of this geometric information remains in the offsets of P. In this chapter, we will see how this idea can be used to infer generalized notions of curvature such as Federer's curvature measures.