7,407 research outputs found

    Geodesics in Heat

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    We introduce the heat method for computing the shortest geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The method represents a significant breakthrough in the practical computation of distance on a wide variety of geometric domains, since the resulting linear systems can be prefactored once and subsequently solved in near-linear time. In practice, distance can be updated via the heat method an order of magnitude faster than with state-of-the-art methods while maintaining a comparable level of accuracy. We provide numerical evidence that the method converges to the exact geodesic distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where more regularity is required

    Morse homology for the heat flow

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    We use the heat flow on the loop space of a closed Riemannian manifold to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined in the spirit of Floer theory by counting, modulo time shift, heat flow trajectories that converge asymptotically to nondegenerate closed geodesics of Morse index difference one.Comment: 89 pages, 3 figure

    Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group

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    By adapting a technique of Molchanov, we obtain the heat kernel asymptotics at the sub-Riemannian cut locus, when the cut points are reached by an rr-dimensional parametric family of optimal geodesics. We apply these results to the bi-Heisenberg group, that is, a nilpotent left-invariant sub-Rieman\-nian structure on R5\mathbb{R}^{5} depending on two real parameters α1\alpha_{1} and α2\alpha_{2}. We develop some results about its geodesics and heat kernel associated to its sub-Laplacian and we illuminate some interesting geometric and analytic features appearing when one compares the isotropic (α1=α2\alpha_{1}=\alpha_{2}) and the non-isotropic cases (α1≠α2\alpha_{1}\neq \alpha_{2}). In particular, we give the exact structure of the cut locus, and we get the complete small-time asymptotics for its heat kernel.Comment: 17 pages, 1 figur

    Riemannian-like structures on the set of probability measures: a comparison between Euclidean and discrete spaces

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    The purpose of this thesis is to present in detail two theories, not deductible from each other, but which obtain very similar results, giving a Riemannian-like structure to the set of probability measures. Both theories lead to a correspondence between heat equation and gradient flow of entropy. Chapter 1 contains a quick but self-contained treatment of the theory of optimal transport, including Kantorovich’s duality, Brenier’s theorem, the Wasserstein spaces P_p and the characterization of their geodesics. Chapter 2 focuses on P_2(R^n). Using the continuity equation, we describe its geodesics and absolutely continuous curves, and determine a “tangent” velocity to them; this enables the definition of (sub)differential of a functional. We study convexity and differentiability of the “internal energy” and “potential energy” functionals. We give a definition of gradient flows, and exploit its equivalence with the purely metric EVI formulation to show some of its basic properties. We conclude characterizing the gradient flows of the entropy by the condition that the densities solve the heat equation. In Chapter 3, we move to a finite space endowed with an irreducible Markov kernel. Guided by an explicit study of the two-point space, we define a family of distances between probabilities via a discrete analogue of “Benamou-Brenier’s formula” from Chapter 2, and characterize their finiteness. The AC curves are described as in the continuous case. Appropriate subsets of probability measures turn out to be Riemannian manifolds, on which we can reproduce results like the Eulerian description of geodesics, the differentiability of potential energy, the identification of heat flow with gradient flow of the entropy
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