Euler Sprays and Wasserstein Geometry of the Space of Shapes

نویسندگان

  • JIAN-GUO LIU
  • ROBERT L. PEGO
چکیده

We study a distance between shapes defined by minimizing the integral of kinetic energy along transport paths constrained to measures with characteristic-function densities. The formal geodesic equations for this shape distance are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The minimization problem exhibits an instability associated with microdroplet formation, with the following outcomes: Shape distance is equal to Wasserstein distance. Furthermore, any two shapes of equal volume can be approximately connected by an Euler spray—a countable superposition of ellipsoidal droplet solutions of incompressible Euler equations. Every Wasserstein geodesic between shape densities is a weak limit of Euler sprays. Each Wasserstein geodesic is also the unique minimizer of a relaxed least-action principle for a fluid-vacuum mixture.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Least Action Principles for Incompressible Flows and Optimal Transport between Shapes

As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to be shape densities (characteristic functions). The formal geo...

متن کامل

Natural gradient via optimal transport I

We study a natural Wasserstein gradient flow on manifolds of probability distributions with discrete sample spaces. We derive the Riemannian structure for the probability simplex from the dynamical formulation of the Wasserstein distance on a weighted graph. We pull back the geometric structure to the parameter space of any given probability model, which allows us to define a natural gradient f...

متن کامل

Vibration Analysis of Multi-Step Bernoulli-Euler and Timoshenko Beams Carrying Concentrated Masses

In this paper, vibration analysis of multiple-stepped Bernoulli-Euler and Timoshenko beams carrying point masses is presented analytically for various boundary conditions. Each attached element is considered to have both translational and rotational inertias. The method of solution is “transfer matrix method” which is based on the changes in the vibration modes at the vicinity of any discontinu...

متن کامل

To Vladimir Igorevich Arnold on the occasion of his 70th birthday SHOCK WAVES FOR THE BURGERS EQUATION AND CURVATURES OF DIFFEOMORPHISM GROUPS

We establish a simple relation between curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently, to a ...

متن کامل

Shock Waves for the Burgers Equation and Curvatures of Diffeomorphism Groups

We establish a simple relation between certain curvatures of the group of volumepreserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2016