Tiến Sĩ Constrained steepest descent in the 2-Wasserstein metric

Thảo luận trong 'Khoa Học Tự Nhiên' bắt đầu bởi Củ Đậu Đậu, 2/4/14.

  1. Củ Đậu Đậu

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    Abstract
    Westudy several constrained variational problems in the 2-Wasserstein
    metric for which the set of probability densities satisfying the constraint is
    not closed. For example, given a probability densityF0onRd
    and a time-step h>0, we seek to minimizeI(F)=hS(F)+W22
    (F0,F)overall of the probability densities Fthat have the same mean and variance asF0, whereS(F)isthe
    entropy ofF.Weprove existence of minimizers. We also analyze the induced
    geometry of the set of densities satisfying the constraint on the variance and
    means, and we determine all of the geodesics on it. From this, we determine
    acriterion for convexity of functionals in the induced geometry. It turns out,
    for example, that the entropy is uniformly strictly convex on the constrained
    manifold, though not uniformly convex without the constraint. The problems
    solved here arose in a study of a variational approach to constructing and
    studying solutions of the nonlinear kinetic Fokker-Planck equation, which is
    briefly described here and fully developed in a companion paper.
    Contents
    1. Introduction
    2. Riemannian geometry of the 2-Wasserstein metric
    3. Geometry of the constraint manifold
    4. The Euler-Lagrange equation
    5. Existence of minimizers
    References
     

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