Optimal Transport and Seismic Rays

Fabrizio Magrini () and Malcolm Sambridge
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Fabrizio Magrini: Research School of Earth Sciences, The Australian National University, Canberra, ACT 2601, Australia
Malcolm Sambridge: Research School of Earth Sciences, The Australian National University, Canberra, ACT 2601, Australia

Mathematics, 2023, vol. 11, issue 22, 1-16

Abstract: We present a theoretical framework that links Fermat’s principle of least time to optimal transport theory via a cost function that enforces local transport. The proposed cost function captures the physical constraints inherent in wave propagation; when paired with specific mass distributions, it yields shortest paths in the considered media through the optimal transport plans. In the discrete setting, our formulation results in physically significant optimal couplings, whose off-diagonal entries identify shortest paths in both directed and undirected graphs. For undirected graphs with positive edge weights, commonly used to parameterize seismic media, our method provides solutions to the Eikonal equation consistent with those from the Dijkstra algorithm. For directed negative-weight graphs, corresponding to transportation cost matrices with negative entries, our approach aligns with the Bellman–Ford algorithm but offers considerable computational advantages. We also highlight potential research directions. These include the use of sparse cost matrices to reduce the number of unknowns and constraints in the considered transportation problem, and solving specific classes of optimal transport problems through the Dijkstra algorithm to enhance computational efficiency.

Keywords: transportation theory; ray theory; shortest path problem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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