82 research outputs found
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.Comment: 23 page
Large volume minimizers of a non local isoperimetric problem: theoretical and numerical approaches
We consider the volume-constrained minimization of the sum of the perimeter
and the Riesz potential. We add an external potential of the form
that provides the existence of a minimizer for any volume
constraint, and we study the geometry of big minimizers. Then we provide a
numerical method to adress this variational problem.Comment: This improved version contains some minor modifications, new
references, and one extra numerical experimen
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
-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
Numerical Minimization of eigenmodes of a membrane with respect to the domain
International audienceIn this paper we introduce a numerical approach adapted to the minimization of the eigenmodes of a membrane with respect to the domain. This method is based on the combination of the Level Set method of J. Osher and J. Sethian with the relaxed approach. This algorithm enables both changing the topology and working on a fixed regular grid
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
We prove the
existence of an optimal reinforcement and that it has some higher
integrability properties. We also provide some numerical computations for
and .Comment: 29 pages, 42 figure
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 -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 -dimensional Euclidean space or to more general
manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure
Handling convexity-like constraints in variational problems
International audienceWe 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
Discrete optimal transport: complexity, geometry and applications
International audienceIn this article, we introduce a new algorithm for solving discrete optimal transport based on iterative resolutions of local versions of the dual linear program. We show a quantitative link between the complexity of this algorithm and the geometry of the underlying measures in the quadratic Euclidean case. This discrete method is then applied to investigate to wo optimal transport problems with geometric flavor: the regularity of optimal transport plan on oblate ellipsoids, and Alexandrov's problem of reconstructing a convex set from its Gaussian measure
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