Convergence of the vertical gradient flow for the Gaussian Monge problem

IF 1 Q3 Engineering

Journal of Computational Dynamics Pub Date : 2023-01-01 DOI:10.3934/jcd.2023008

Erik Jansson, Klas Modin

引用次数: 0

Abstract

We investigate a matrix dynamical system related to optimal mass transport in the linear category, namely, the problem of finding an optimal invertible matrix by which two covariance matrices are congruent. We first review the differential geometric structure of the problem in terms of a principal fiber bundle. The dynamical system is a gradient flow restricted to the fibers of the bundle. We prove global existence of solutions to the flow, with convergence to the polar decomposition of the matrix given as initial data. The convergence is illustrated in a numerical example.

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高斯蒙日问题中垂直梯度流的收敛性

研究了线性范畴中与最优质量输运相关的矩阵动力系统，即寻找两个协方差矩阵全等的最优可逆矩阵的问题。我们首先从一个主纤维束的角度回顾了这个问题的微分几何结构。动力系统是一种限于束纤维的梯度流动。我们证明了该流解的全局存在性，并收敛于作为初始数据的矩阵的极分解。通过数值算例说明了该方法的收敛性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Computational Dynamics Engineering-Computational Mechanics

CiteScore

2.30

自引率

10.00%

发文量

期刊介绍： JCD is focused on the intersection of computation with deterministic and stochastic dynamics. The mission of the journal is to publish papers that explore new computational methods for analyzing dynamic problems or use novel dynamical methods to improve computation. The subject matter of JCD includes both fundamental mathematical contributions and applications to problems from science and engineering. A non-exhaustive list of topics includes * Computation of phase-space structures and bifurcations * Multi-time-scale methods * Structure-preserving integration * Nonlinear and stochastic model reduction * Set-valued numerical techniques * Network and distributed dynamics JCD includes both original research and survey papers that give a detailed and illuminating treatment of an important area of current interest. The editorial board of JCD consists of world-leading researchers from mathematics, engineering, and science, all of whom are experts in both computational methods and the theory of dynamical systems.