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Edouard Oudet

Publications and source records attributed to Edouard Oudet.

At least 19 recordsLinked to original sources
arXiv2024

A Lagrangian shape and topology optimization framework based on semi-discrete optimal transport

This article revolves around shape and topology optimization, in the applicative context where the objective and constraint functionals depend on the solution to a physical boundary value problem posed on the optimized domain. We introduce a novel framework based on modern concepts from computational geometry, optimal transport and numerical analysis. Its pivotal feature is a representation of the optimized shape by the cells of an adapted version of a Laguerre diagram. Although such objects are originally described by a collection of seed points and weights, recent results from optimal transport theory suggest a more intuitive parametrization in terms of the seed points and measures of the associated cells. The polygonal mesh of the shape induced by this diagram serves as support for the deployment of the Virtual Element Method for the numerical solution of the physical boundary value problem at play and the calculation of the objective and constraint functionals. The sensitivities of the latter are derived next; at first, we calculate their derivatives with respect to the positions of the vertices of the Laguerre diagram by shape calculus techniques; a suitable adjoint methodology is then developed to express them in terms of the seed points and cell measures of the diagram. The evolution of the shape is realized by first updating the design variables according to these sensitivities and then reconstructing the diagram with efficient algorithms from computational geometry. Our shape optimization strategy is versatile: it can be applied to a wide gammut of physical situations. It is Lagrangian by essence, and it thereby benefits from all the assets of a consistently meshed representation of the shape. Yet, it naturally handles dramatic motions, including topological changes, in a very robust fashion. These features, among others, are illustrated by a series of 2d numerical examples.

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arXiv2023

The Fixed Landscape Inference MethOd (flimo): a versatile alternative to Approximate Bayesian Computation, faster by several orders of magnitude

Modelling in biology must adapt to increasingly complex and massive data. The efficiency of the inference algorithms used to estimate model parameters is therefore questioned. Many of these are based on stochastic optimization processes that require significant computing time. We introduce the Fixed Landscape Inference MethOd (flimo), a new likelihood-free inference method for continuous state-space stochastic models. It applies deterministic gradient-based optimization algorithms to obtain a point estimate of the parameters, minimizing the difference between the data and some simulations according to some prescribed summary statistics. In this sense, it is analogous to Approximate Bayesian Computation (ABC). Like ABC, it can also provide an approximation of the distribution of the parameters. Three applications are proposed: a usual theoretical example, namely the inference of the parameters of g-and-k distributions; a population genetics problem, not so simple as it seems, namely the inference of a selective value from time series in a Wright-Fisher model; and simulations from a Ricker model, representing chaotic population dynamics. In the two first applications, the results show a drastic reduction of the computational time needed for the inference phase compared to the other methods, despite an equivalent accuracy. Even when likelihood-based methods are applicable, the simplicity and efficiency of flimo make it a compelling alternative. Implementations in Julia and in R are available on https://metabarcoding.org/flimo. To run flimo, the user must simply be able to simulate data according to the chosen model.

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arXiv2023

On the numerical approximation of Blaschke-Santaló diagrams using Centroidal Voronoi Tessellations

Identifying Blaschke-Santaló diagrams is an important topic that essentially consists in determining the image $Y=F(X)$ of a map $F:X\to{\mathbb{R}}^d$, where the dimension of the source space $X$ is much larger than the one of the target space. In some cases, that occur for instance in shape optimization problems, $X$ can even be a subset of an infinite-dimensional space. The usual Monte Carlo method, consisting in randomly choosing a number $N$ of points $x_1,\dots,x_N$ in $X$ and plotting them in the target space ${\mathbb{R}}^d$, produces in many cases areas in $Y$ of very high and very low concentration leading to a rather rough numerical identification of the image set. On the contrary, our goal is to choose the points $x_i$ in an appropriate way that produces a uniform distribution in the target space. In this way we may obtain a good representation of the image set $Y$ by a relatively small number $N$ of samples which is very useful when the dimension of the source space $X$ is large (or even infinite) and the evaluation of $F(x_i)$ is costly. Our method consists in a suitable use of {\it Centroidal Voronoi Tessellations} which provides efficient numerical results. Simulations for two and three dimensional examples are shown in the paper.

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arXiv2022

Maximization of Neumann eigenvalues

This paper is motivated by the maximization of the $k$-th eigenvalue of the Laplace operator with Neumann boundary conditions among domains of ${\mathbb R}^N$ with prescribed measure. We relax the problem to the class of (possibly degenerate) densities in ${\mathbb R}^N$ with prescribed mass and prove the existence of an optimal density. For $k=1,2$ the two problems are equivalent and the maximizers are known to be one and two equal balls, respectively. For $k \ge 3$ this question remains open, except in one dimension of the space where we prove that the maximal densities correspond to a union of $k$ equal segments. This result provides sharp upper bounds for Sturm-Liouville eigenvalues and proves the validity of the Pólya conjecture in the class of densities in ${\mathbb R}$. Based on the relaxed formulation, we provide numerical approximations of optimal densities for $k=1, \dots, 8$ in ${\mathbb R}^2$.

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arXiv2021

Longest minimal length partitions

This article provides numerical evidence that under volume constraint the ball is the set which maximizes the perimeter of the least-perimeter partition into cells with prescribed areas. We introduce a numerical maximization algorithm which performs multiple optimizations steps at each iteration to approximate minimal partitions. Using these partitions we compute perturbations of the domain which increase the minimal perimeter. The initialization of the optimal partitioning algorithm uses capacity-constrained Voronoi diagrams. A new algorithm is proposed to identify such diagrams, by computing the gradients of areas and perimeters for the Voronoi cells with respect to the Voronoi points.

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arXiv2020

Cut locus on compact manifolds and uniform semiconcavity estimates for a variational inequality

We study a family of gradient obstacle problems on a compact Riemannian manifold. We prove that the solutions of these free boundary problems are uniformly semiconcave and, as a consequence, we obtain some fine convergence results for the solutions and their free boundaries. Precisely, we show that the elastic and the $λ$-elastic sets of the solutions Hausdorff converge to the cut locus and the $λ$-cut locus of the manifold.

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arXiv2019

Optimal reinforcing networks for elastic membranes

In this paper we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a network (connected onedimensional structure), that has to be found in a suitable admissible class. We show the existence of an optimal solution that may present multiplicities, that is regions where the optimal structure overlaps. Some numerical simulations are shown to confirm this issue and to illustrate the complexity of the optimal network when their total length becomes large.

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arXiv2019

Variational approximation of functionals defined on $1$-dimensional connected sets in $\mathbb{R}^n$

In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert--Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in $\mathbb{R}^n$. Following the the analysis for the planar case presented in [4], we provide a variational approximation through Ginzburg--Landau type energies proving a $Γ$-convergence result for $n \geq 3$.

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arXiv2018

Variational approximation of functionals defined on 1-dimensional connected sets: the planar case

In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a $Γ$-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to $n$-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [11].

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arXiv2017

The minimal resistance problem in a class of non convex bodies

We characterize the solution to the Newton minimal resistance problem in a class of radial q-concave profiles. We also give the corresponding result for one-dimensional profiles. Moreover, we provide a numerical optimization algorithm for the general nonradial case.

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arXiv2016

Partitions of Minimal Length on Manifolds

We study partitions on three dimensional manifolds which minimize the total geodesic perimeter. We propose a relaxed framework based on a $Γ$-convergence result and we show some numerical results. We compare our results to those already present in the literature in the case of the sphere. For general surfaces we provide an optimization algorithm on meshes which can give a good approximation of the optimal cost, starting from the results obtained using the relaxed formulation.

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arXiv2015

A free boundary problem arising in PDE optimization

A free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form $$\sup_{\int_Dθ\,dx=m}\ \inf_{u\in H^1_0(D)}\int_D\Big(\frac{1+θ}{2}|\nabla u|^2-fu\Big)\,dx.$$ We prove the existence of an optimal reinforcement $θ$ and that it has some higher integrability properties. We also provide some numerical computations for $θ$ and $u$.

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arXiv2014

Numerical methods for matching for teams and Wasserstein barycenters

Equilibrium multi-population matching (matching for teams) is a problem from mathematical economics which is related to multi-marginal optimal transport. A special but important case is the Wasserstein barycenter problem, which has applications in image processing and statistics. Two algorithms are presented: a linear programming algorithm and an efficient nonsmooth optimization algorithm, which applies in the case of the Wasserstein barycenters. The measures are approximated by discrete measures: convergence of the approximation is proved. Numerical results are presented which illustrate the efficiency of the algorithms.

math.NA↗
arXiv2014

Discretization of functionals involving the Monge-Ampère operator

Gradient flows in the Wasserstein space have become a powerful tool in the analysis of diffusion equations, following the seminal work of Jordan, Kinderlehrer and Otto (JKO). The numerical applications of this formulation have been limited by the difficulty to compute the Wasserstein distance in dimension >= 2. One step of the JKO scheme is equivalent to a variational problem on the space of convex functions, which involves the Monge-Ampère operator. Convexity constraints are notably difficult to handle numerically, but in our setting the internal energy plays the role of a barrier for these constraints. This enables us to introduce a consistent discretization, which inherits convexity properties of the continuous variational problem. We show the effectiveness of our approach on nonlinear diffusion and crowd-motion models.

math.NA↗
arXiv2014

Minimal Dirichlet energy partitions for graphs

Motivated by a geometric problem, we introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreasing and converges in a finite number of iterations to a local minimum of the relaxed objective function. Our method is applied to several clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations. The model has a semi-supervised extension and provides a natural representative for the clusters as well.

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arXiv2014

Handling convexity-like constraints in variational problems

We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give estimates of the distance between the approximation space and the admissible set. This framework applies to the approximation of convex functions by piecewise linear functions on a mesh of the domain and by other finite-dimensional spaces such as tensor-product splines. We show how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints. Our implementation relies on proximal algorithms, and can be easily parallelized, thus making it applicable to large scale problems in dimension two and three. We illustrate the versatility and the efficiency of our approach on the numerical solution of three problems in calculus of variation : 3D denoising, the principal agent problem, and optimization within the class of convex bodies.

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arXiv2013

Phase-field approximations of the Willmore functional and flow

We discuss in this paper phase-field approximations of the Willmore functional and the associated L2-flow. After recollecting known results on the approximation of the Willmore energy and its L1-relaxation, we derive the expression of the flows associated with various approximations, and we show their behavior by formal arguments based on matched asymptotic expansions. We introduce an accurate numerical scheme, whose local convergence can be proved, to describe with more details the behavior of two flows, the classical and the flow associated with an approximation model due to Mugnai. We propose a series of numerical simulations in 2D and 3D to illustrate their behavior in both smooth and singular situations.

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