arXiv subjects

Katy Craig

Publications and source records attributed to Katy Craig.

23 records · Page 2Linked to original sources

arXiv2015

Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions

Over the past fifteen years, the theory of Wasserstein gradient flows of convex (or, more generally, semiconvex) energies has led to advances in several areas of partial differential equations and analysis. In this work, we extend the well-posedness theory for Wasserstein gradient flows to energies satisfying a more general criterion for uniqueness, motivated by the Osgood criterion for ordinary differential equations. We also prove the first quantitative estimates on convergence of the discrete gradient flow or "JKO scheme" outside of the semiconvex case. We conclude by applying these results to study the well-posedness of constrained nonlocal interaction energies, which have arisen in recent work on biological chemotaxis and congested aggregation.

math.AP↗

arXiv2015

Convergence of regularized nonlocal interaction energies

Inspired by numerical studies of the aggregation equation, we study the effect of regularization on nonlocal interaction energies. We consider energies defined via a repulsive-attractive interaction kernel, regularized by convolution with a mollifier. We prove that, with respect to the 2-Wasserstein metric, the regularized energies $Γ$-converge to the unregularized energy and minimizers converge to minimizers. We then apply our results to prove $Γ$-convergence of the gradient flows, when restricted to the space of measures with bounded density.

math.AP↗

arXiv2014

The Exponential Formula for the Wasserstein Metric

A recurring obstacle in the study of Wasserstein gradient flow is the lack of convexity of the square Wasserstein metric. In this paper, we develop a class of transport metrics that have better convexity properties and use these metrics to prove an Euler-Lagrange equation characterizing Wasserstein discrete gradient flow. We then apply these results to give a new proof of the exponential formula for the Wasserstein metric, mirroring Crandall and Liggett's proof of the corresponding Banach space result. We conclude by using our approach to give simple proofs of properties of the gradient flow, including the contracting semigroup property and energy dissipation inequality.

math.AP↗

arXiv2014

A Blob Method for the Aggregation Equation

Motivated by classical vortex blob methods for the Euler equations, we develop a numerical blob method for the aggregation equation. This provides a counterpoint to existing literature on particle methods. By regularizing the velocity field with a mollifier or "blob function", the blob method has a faster rate of convergence and allows a wider range of admissible kernels. In fact, we prove arbitrarily high polynomial rates of convergence to classical solutions, depending on the choice of mollifier. The blob method conserves mass and the corresponding particle system is both energy decreasing for a regularized free energy functional and preserves the Wasserstein gradient flow structure. We consider numerical examples that validate our predicted rate of convergence and illustrate qualitative properties of the method.

math.NA↗

arXiv2012

Contraction of the proximal map and generalized convexity of the Moreau-Yosida regularization in the 2-Wasserstein metric

We investigate the Moreau-Yosida regularization and the associated proximal map in the context of discrete gradient flow for the 2-Wasserstein metric. Our main results are a stepwise contraction property for the proximal map and an "above the tangent line" inequality for the regularization. Using the latter, we prove a Talagrand inequality and an HWI inequality for the regularization, under appropriate hypotheses. In the final section, the results are applied to study the discrete gradient flow for Rényi entropies. As Otto showed, the gradient flow for these entropies in the 2-Wasserstein metric is a porous medium flow or a fast diffusion flow, depending on the exponent of the entropy. We show that a striking number of the remarkable features of the porous medium and fast diffusion flows are present in the discrete gradient flow and do not simply emerge in the limit as the time-step goes to zero.

math.AP↗