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Constrained deepest descent in the 2-Wasserstein metric

Constrained deepest descent in the 2-Wasserstein metric,10.4007/annals.2003.157.807,Annals of Mathematics,E. A. Carlen,Wilfrid Gangbo

Constrained deepest descent in the 2-Wasserstein metric   (Citations: 25)
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We study several constrained variational problem in the 2-Wasserstein metric for which the set of probability densities satisfying the constraint is not closed. For example, given a probability density $F_0$ on $\R^d$ and a time-step $h>0$, we seek to minimize $I(F) = hS(F) + W_2^2(F_0,F)$ over all of the probability densities $F$ that have the same mean and variance as $F_0$, where $S(F)$ is the entropy of $F$. We prove 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 a criterion 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.
Journal: Annals of Mathematics - ANN MATH , vol. 157, no. 3, pp. 807-846, 2003
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    • ...While [17] studied purely dissipative equations, it was the idea of Carlen and Gangbo [5] to apply and extend the JKO scheme to a more general set of equations: equations that not only dissipate, but also satisfy some conservation laws...

    A. Tudorascuet al. On a Nonlinear, Nonlocal Parabolic Problem with Conservation of Mass, ...

    • ...Ledoux [21], Evans and Gangbo [14], Carlen and Gangbo [12]...

    Yinfang Shenet al. On Monge–Kantorovich problem in the plane

    • ...We finally note that the projection on the manifold I = const according to the gradient notion used here, has been considered by Carlen and Gangbo in [4] for a different class of equations...

    E. Cagliotiet al. On a Constrained 2-D Navier-Stokes Equation

    • ...The issue appeared in the study of constrained optimization in the 2-Wasserstein and was left open by Carlen and Gangbo in [4]...
    • ...Carlen and Gangbo [4] perform a comprehensive study of constrained optimization in the space of probability densities with nite second-order moments over RN. An application is provided by the same...
    • ...Discussed in [5] and resolved in the companion paper [4] is a modication of a specic part of the variational scheme (the one accounting for collisions) which imposes conservation of energy even at the discrete level...
    • ...Furthermore, the analysis subsequent to this change in the discrete scheme requires [4] estimates on the fourth-order moments of the minimizers which now need to be proved for the constrained variational problem...
    • ...The issue is left open in [4] and we address it successfully towards the end of this paper...
    • ...To solve the constrained variational problem, the authors of [4] construct an argument based on the dual variational characterization of the Wasserstein distance in a functional setting...
    • ...More precisely, denote by (s) the exponential es 1, which is the Legendre transform of (s) = s logs for s 0 and (s) = +1 if s < 0. Then, it is proved in [4] that the constrained minimization problem discussed above is equivalent to the maximization of...
    • ...The rst open problem left in [4] is, basically, to nd a way to circumvent much of the dicult y incurred by this quite involved maximization problem by building on the unconstrained case analyzed in the seminal paper [9]...
    • ...To explain, the duality argument used in [4], although natural and enlightening, appears complicated and could readily be replaced, as the authors of [4] observe, by an easier one based on Lagrange multipliers if one knew that the unconstrained minimizer 1 2 M of I[ 0; ] satised Z...
    • ...To explain, the duality argument used in [4], although natural and enlightening, appears complicated and could readily be replaced, as the authors of [4] observe, by an easier one based on Lagrange multipliers if one knew that the unconstrained minimizer 1 2 M of I[ 0; ] satised Z...
    • ...The inequality (1.4) is only conjectured in [4]...
    • ...Finally, in Section 5 we obtain an estimate on the fourth-order moments of the minimizers arising from the constrained variational problem, estimate which was conjectured in [4]...
    • ...In this section, we follow the course of action outlined by Carlen and Gangbo in [4]...
    • ...The other question raised in [4] was whether an inequality of the form...
    • ...could be proved for the constrained minimization problem (whereC may depend on some higher moments of ). We thank Carlen and Gangbo for clarifying this, since there seems to be an ambiguity at end of the paper [4]...
    • ...Note that I[ ; ] replaces I[ ; ] in the statement above. We do this to be consistent with [4]...
    • ...Thus, the equivalent of (1.8) for the constrained problem is also true (with, as noted above, instead of ). Proof of Lemma 5: Let be the convex function such that r # 1 = . According to Theorem 4.1 (with the only exception that, here, is only a probability in P2; however, nothing needs to be changed in the proof in [4] to infer that everything still works in this general case), the Euler equation for the constrained problem leads to, after ...

    Adrian Tudorascu. On the Jordan–Kinderlehrer–Otto variational scheme and constrained opt...

    • ...The geometric and analytic features of this structure have been intensively studied, cfr. e.g. [4], [11], [12], [32], [37]...

    Wilfrid Gangboet al. Differential forms on Wasserstein space and infinite-dimensional Hamil...

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