heat flow on alexandrov spaces we prove that on compact alexandrov spaces with curvature bounded below the gradient flow of the dirichlet energy in the space produces the same evolution as the gradient flow of the relative entropy in the wasserstein space this means that the heat flow is well defined by either one of the two gradient flows combining properties of these flows we are able to deduce the lipschitz continuity of the heat kernel as well as bakry emery gradient estimates and the condition our identification is established by purely metric means unlike preceding results relying on pde techniques our approach generalizes to the case of heat flow with drift 2012 wiley periodicals inc

q heat flow and the gradient flow of the renyi entropy in the p wasserstein space abstract based on the idea of a recent paper by ambrosio gigli savare 2014 5 we show that the l 2 gradient flow of the q cheeger energy called q heat flow solves a generalized gradient flow problem of the renyi entropy functional in the p wasserstein for that a further study of the q heat flow is presented including a condition for its mass preservation under a convexity assumption on the upper gradient which holds for all q 2 one gets uniqueness of the gradient flow and the two flows can be identified smooth solutions of the q heat flow are solutions to the parabolic q laplace equation i e t f t q f t

non contraction of heat flow on minkowski spaces we study contractivity properties of gradient flows for functions on normed spaces or more generally on finsler manifolds contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions this is different from the usual convexity along geodesics in non riemannian finsler manifolds as an application we show that the heat flow on minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic wasserstein distance

weak curvature conditions and functional inequalities we give sufficient conditions for a measured length space x d to admit local and global poincare inequalities along with a sobolev inequality we first introduce a condition dm on x d d ef ined in terms of transport of measures we show that dm together with a doubling condition on implies a scaleinvariant local poincare inequality we show that if x d has nonnegative n ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies dm with constant 2 n t he condition dm is preserved by measured gromov hausdorff limits we then prove a sobolev inequality for measured length spaces with n ricci curvature bounded below by k 0 finally we derive a sharp global poincare inequality

heat flow on finsler manifolds this paper studies the heat flow on finsler manifolds a finsler manifold is a smooth manifold m equipped with a minkowski norm f x txm on each tangent space mostly we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x the particular case of a hilbert norm on each tangent space leads to the important subclasses of riemannian manifolds where the heat flow is widely studied and well understood we present two approaches to the heat flow on a finsler manifold as gradient flow on l2 m m for the energy as gradient flow on the reverse l2 wasserstein space 2 m of probability measures on m for the relative entropy both approaches depend on the choice of a measure m on m and then lead to the same nonlinear evolution semigroup we prove 1 regularity for solutions to the nonlinear heat equation on the finsler space m f m typically solutions to the heat equation will not be 2 moreover we derive pointwise comparison results a la cheeger yau and integrated upper gaussian estimates a la davies 2008 wiley periodicals inc

information geometry of q gaussian densities and behaviors of solutions to related diffusion equations this paper presents new geometric aspects of the behaviors of solutions to the porous medium equation pme and its associated equation first we discuss thermostatistical structure with information geometry on a manifold of generalized exponential densities a dualistic relation between the two existing formalisms is elucidated next by equipping the manifold of q gaussian densities with such a structure we derive several physically and geometrically interesting properties of the solutions the manifold is proved invariant and attracting for the evolving solutions which play crucial roles in our analysis we demonstrate that the moment conserving projection of a solution coincides with a geodesic curve on the manifold further the evolutional velocities of the second moments and the convergence rate to the manifold are evaluated in terms of the bregman divergence finally we show that the self similar solution is geometrically special in the sense that it is simultaneously geodesic with respect to the mutually dual two affine connections

irregular perfectly matched layers for 3d elastic wave modeling we have developed a perfectly matched layer pml absorbing boundary condition that can be imposed along an arbitrary geometric boundary in 3d elastic wave modeling the scheme is developed by using the local coordinate system based pml splitting equations and integral approach of the pml equations under a discretization of tetrahedral grids however no explicit coordinate transformations arise the local coordinate system based pml splitting equations make it possible to decay incident waves around the direction normal to the irregular geometric boundaries instead of a coordinate axis direction based on the resulting 3d irregular pml model we can flexibly construct the computational domain with smaller nodes by cutting uninterested zones this results in significant reductions in computational cost and memory requirements for 3d simulations by building a smooth artificial boundary the irregular pml model can avoid the respective treatments to the edges and corners of the artificial boundaries also the irregular pml model may reduce the grazing incidence that makes the pml model less efficient by changing the shapes of the artificial boundaries numerical testing was used to demonstrate the performance of the irregular pml model

gradient flows on wasserstein spaces over compact alexandrov spaces we establish the existence of euclidean tangent cones on wasserstein spaces over compact alexandrov spaces of curvature bounded below by using this riemannian structure we formulate and construct gradient flows of functions on such spaces if the underlying space is a riemannian manifold of nonnegative sectional curvature then our gradient flow of the free energy produces a solution of the linear fokker planck equation

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

a family of functional inequalities łojasiewicz inequalities and displacement convex functions abstract for displacement convex functionals in the probability space equipped with the monge kantorovich metric we prove the equivalence between the gradient and functional type ojasiewicz inequalities we also discuss the more general case of convex functions and we provide a general convergence theorem for the corresponding gradient dynamics specialising our results to the boltzmann entropy we recover otto villani s theorem asserting the equivalence between logarithmic sobolev and talagrand s inequalities the choice of power type entropies shows a new equivalence between gagliardo nirenberg inequality and a nonlinear talagrand inequality some nonconvex results and other types of equivalences are discussed

calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below this paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces x mathsf d mathfrak m our main results are a general study of the relations between the hopf lax semigroup and hamilton jacobi equation in metric spaces x d the equivalence of the heat flow in l 2 x mathfrak m generated by a suitable dirichlet energy and the wasserstein gradient flow of the relative entropy functional mathrm ent mathfrak m in the space of probability measures open image in new window the proof of density in energy of lipschitz functions in the sobolev space w 1 2 x mathsf d mathfrak m a fine and very general analysis of the differentiability properties of a large class of kantorovich potentials in connection with the optimal transport problem is the fourth achievement of the paper our results apply in particular to spaces satisfying ricci curvature bounds in the sense of lott and villani ann math 169 903 991 2009 and sturm acta math 196 65 131 2006 and acta math 196 133 177 2006 and require neither the doubling property nor the validity of the local poincare inequality

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media we present and analyze a perfectly matched absorbing layer model for the velocity stress formulation of elastodynamics the principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components a component orthogonal to the boundary and a component parallel to it a system of equations governing these new unknowns then is constructed a damping term finally is introduced for the component orthogonal to the boundary this layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium in practice both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers as we will show with several experiments this model gives very satisfactory results namely the reflection coefficient even in the case of heterogeneous anisotropic media is about 1 for a layer thickness of five space discretization steps

displacement convexity of generalized relative entropies abstract we investigate the m relative entropy which stems from the bregman divergence on weighted riemannian and finsler manifolds we prove that the displacement k convexity of the m relative entropy is equivalent to the combination of the nonnegativity of the weighted ricci curvature and the k convexity of the weight function we use this to show appropriate variants of the talagrand hwi and the logarithmic sobolev inequalities as well as the concentration of measures we also prove that the gradient flow of the m relative entropy produces a solution to the porous medium equation or the fast diffusion equation

a finite edge gtd analysis of the h plane horn radiation pattern the earlier geometrical theory of diffraction gtd approach to the h plane horn radiation problem is reconsidered with spherical source excitation corner diffraction terms are included to provide a gtd model for the finiteness of the horn edges a new heuristic corner slope diffraction csd coefficient for a finite edge in a conducting plane is presented the h plane horn pattern obtained with the addition of the corner diffraction and the new csd terms to the earlier infinite edge gtd approach is found to be in better agreement with measured results compared to earlier gtd formulations

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

information geometry of u boost and bregman divergence we aim at an extension of adaboost to u boost in the paradigm to build a stronger classification machine from a set of weak learning machines a geometric understanding of the bregman divergence defined by a generic convex function u leads to the u boost method in the framework of information geometry extended to the space of the finite measures over a label set we propose two versions of u boost learning algorithms by taking account of whether the domain is restricted to the space of probability functions in the sequential step we observe that the two adjacent and the initial classifiers are associated with a right triangle in the scale via the bregman divergence called the pythagorean relation this leads to a mild convergence property of the u boost algorithm as seen in the expectation maximization algorithm statistical discussions for consistency and robustness elucidate the properties of the u boost methods based on a stochastic assumption for training data

isoperimetric inequality for radial probability measures on euclidean spaces abstract we generalize the poincare limit which asserts that the n dimensional gaussian measure is approximated by the projections of the uniform probability measure on the euclidean sphere of appropriate radius to the first n coordinates as the dimension diverges to infinity the generalization is done by replacing the projections with certain maps using this generalization we derive a gaussian isoperimetric inequality for an absolutely continuous probability measure on euclidean spaces with respect to the lebesgue measure whose density is a radial function

gradient flows of the entropy for finite markov chains let k be an irreducible and reversible markov kernel on a finite set x we construct a metric w on the set of probability measures on x and show that with respect to this metric the law of the continuous time markov chain evolves as the gradient flow of the entropy this result is a discrete counterpart of the wasserstein gradient flow interpretation of the heat flow in rn by jordan kinderlehrer and otto 1998 the metric w is similar to but different from the l2 wasserstein metric and is defined via a discrete variant of the benamou brenier formula

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

wasserstein space over the wiener space the goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the wiener space based on and extending fundamental results of feyel ustunel carrying out the program of ambrosio gigli savare we present a complete characterization of the derivative processes for certain class of absolutely continuous curves we prove existence of the gradient flow curves for the relative entropy w r t the wiener measure and identify these gradient flow curves with solutions of the ornstein uhlenbeck evolution equation

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

localization and tensorization properties of the curvature dimension condition for metric measure spaces abstract this is an addendum to the paper k bacher k t sturm localization and tensorization properties of the curvature dimension condition for metric measure spaces j funct anal 259 2010 28 56 we prove the tensorization property for the curvature dimension condition add some detailed calculations including explicit dependence of constants and comment on assumptions and conjectures concerning the local to global statement in bacher and sturm 2010 1 and villani 2009 6 respectively