62 research outputs found
On the reconstruction of convex sets from random normal measurements
We study the problem of reconstructing a convex body using only a finite
number of measurements of outer normal vectors. More precisely, we suppose that
the normal vectors are measured at independent random locations uniformly
distributed along the boundary of our convex set. Given a desired Hausdorff
error eta, we provide an upper bounds on the number of probes that one has to
perform in order to obtain an eta-approximation of this convex set with high
probability. Our result rely on the stability theory related to Minkowski's
theorem
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
Lower bounds for k-distance approximation
Consider a set P of N random points on the unit sphere of dimension ,
and the symmetrized set S = P union (-P). The halving polyhedron of S is
defined as the convex hull of the set of centroids of N distinct points in S.
We prove that after appropriate rescaling this halving polyhedron is Hausdorff
close to the unit ball with high probability, as soon as the number of points
grows like . From this result, we deduce probabilistic lower
bounds on the complexity of approximations of the distance to the empirical
measure on the point set by distance-like functions
Minimal geodesics along volume preserving maps, through semi-discrete optimal transport
We introduce a numerical method for extracting minimal geodesics along the
group of volume preserving maps, equipped with the L2 metric, which as observed
by Arnold solve Euler's equations of inviscid incompressible fluids. The method
relies on the generalized polar decomposition of Brenier, numerically
implemented through semi-discrete optimal transport. It is robust enough to
extract non-classical, multi-valued solutions of Euler's equations, for which
the flow dimension is higher than the domain dimension, a striking and
unavoidable consequence of this model. Our convergence results encompass this
generalized model, and our numerical experiments illustrate it for the first
time in two space dimensions.Comment: 21 pages, 9 figure
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
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
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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