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

Eric A. Carlen, Katy Craig

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)33-65
Number of pages33
JournalMathematics and Mechanics of Complex Systems
Volume1
Issue number1
DOIs
StatePublished - 2013

All Science Journal Classification (ASJC) codes

  • Civil and Structural Engineering
  • Numerical Analysis
  • Computational Mathematics

Keywords

  • Gradient flow
  • Moreau-Yosida regularization
  • Wasserstein metric

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