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Fast-phase space computation of multiple arrivals.

S Fomel1, J A Sethian

  • 1Department of Mathematics, University of California, Berkeley, CA 94720, USA.

Proceedings of the National Academy of Sciences of the United States of America
|May 29, 2002
PubMed
Summary
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We developed a fast computational method to solve static Hamilton-Jacobi equations by deriving "Escape Equations" from the Liouville formulation. This technique efficiently computes all possible trajectories and arrivals for various boundary conditions.

Area of Science:

  • Computational Mathematics
  • Mathematical Physics

Background:

  • Static Hamilton-Jacobi equations are fundamental in various scientific fields.
  • Solving these equations, especially for all possible boundary conditions, presents significant computational challenges.

Purpose of the Study:

  • To introduce a novel, fast, and general computational technique for solving static Hamilton-Jacobi equations.
  • To enable the computation of all possible trajectories and boundary arrivals efficiently.

Main Methods:

  • Derivation of "Escape Equations" from the Liouville formulation of characteristic equations, resulting in static, time-independent Eulerian PDEs.
  • Numerical solution using a "one-pass" formulation integrating semi-Lagrangian, Dijkstra-like, and Ordered Upwind methods.
  • Computational complexity of O(N log N) for computing all phase-space solutions, with rapid post-processing for specific boundary conditions.

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Main Results:

  • The developed technique provides a fast and general method for computing phase-space solutions of static Hamilton-Jacobi equations.
  • Demonstrated application to the Eikonal equation for computing first, multiple, and most energetic arrivals.
  • The method efficiently handles all possible boundary conditions and starting configurations.

Conclusions:

  • The presented computational technique offers a significant advancement in solving static Hamilton-Jacobi equations.
  • The "Escape Equations" approach provides an efficient framework for analyzing trajectory and arrival problems.
  • Further optimizations are suggested for scenarios with pre-defined source distributions.