Projected Gradient Method Combined with Homotopy Techniques for Volume-Measure-Preserving Optimal Mass Transportation Problems

Mei Heng Yueh, Tsung Ming Huang, Tiexiang Li*, Wen Wei Lin, Shing Tung Yau

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Optimal mass transportation has been widely applied in various fields, such as data compression, generative adversarial networks, and image processing. In this paper, we adopt the projected gradient method, combined with the homotopy technique, to find a minimal volume-measure-preserving solution for a 3-manifold optimal mass transportation problem. The proposed projected gradient method is shown to be sublinearly convergent at a rate of O(1/k). Several numerical experiments indicate that our algorithms can significantly reduce transportation costs. Some applications of the optimal mass transportation maps—to deformations and canonical normalizations between brains and solid balls—are demonstrated to show the robustness of our proposed algorithms.

Original languageEnglish
Article number64
JournalJournal of Scientific Computing
Volume88
Issue number3
DOIs
Publication statusPublished - 2021 Sep

Keywords

  • Optimal mass transportation
  • Simply connected 3-Manifold
  • Volume-measure-preserving

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

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