Abstract We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. On the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed.

Martin Burger

Matthias Erbar

Franca Hoffmann

Daniel Matthes

André Schlichting

Archive for Rational Mechanics and Analysis

理性力学与分析档案将力学作为经典传统中的一门演绎数学科学滋养起来，并促进分析，特别是在应用方面。其目的是快速、全面地发表关于特殊时刻、深度和持久性的研究。

The Archive for Rational Mechanics and Analysis is a journal dedicated to the study of mechanics, emphasizing the importance of mechanics as a classical mathematical science and promoting the application of analytical methods. The journal aims to publish research findings on specific moments, depth, and permanence quickly and comprehensively.

《Archive for Rational Mechanics and Analysis》是一本致力于力学研究的期刊，强调力学作为一门经典的数学科学的重要性，并推动分析方法的应用。期刊的目标是快速而全面地发表关于特定时刻、深度和持久性的研究成果。

Category	Quartile
数学	1
数学, 应用数学	1
数学, 力学	2

Covariance-Modulated Optimal Transport and Gradient Flows

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