ON GRADIENT FLOWS INITIALIZED NEAR MAXIMA

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Abstract

Let (M, g) be a closed Riemannian manifold, and let F : M \rightarrow \BbbR be a smooth function on M. We show the following holds generically for the function F: for each maximum p of F, there exist two minima, denoted by m+(p) and m-(p), so that the gradient flow initialized at a random point close to p converges to either m-(p) or m+(p) with high probability. The statement also holds for F \in C\infty(M) fixed and a generic metric g on M. We conclude by associating to a generic pair (F,g) what we call its max-min graph, which captures the relation between minima and maxima derived in the main result.

Original languageEnglish (US)
Pages (from-to)2526-2548
Number of pages23
JournalSIAM Journal on Control and Optimization
Volume61
Issue number5
DOIs
StatePublished - 2023
Externally publishedYes

Keywords

  • gradient flows
  • qualitative dynamics
  • structural stability

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics

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