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Ordered upwind methods for static Hamilton-Jacobi equations.

J A Sethian1, A Vladimirsky

  • 1Department of Mathematics, University of California, Berkeley, CA 94720, USA. sethian@math.berkeley.edu

Proceedings of the National Academy of Sciences of the United States of America
|September 27, 2001
PubMed
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We present fast, iterative methods for solving static Hamilton-Jacobi equations. These techniques achieve O(M log M) complexity, improving computational efficiency for complex problems.

Area of Science:

  • Numerical analysis
  • Partial differential equations
  • Computational mathematics

Background:

  • Static Hamilton-Jacobi equations are crucial in various scientific fields.
  • Traditional iterative methods for solving these equations can be computationally expensive.
  • Discretized partial differential equations often require iterative convergence.

Purpose of the Study:

  • To introduce a family of fast, non-iterative methods for solving static Hamilton-Jacobi equations.
  • To improve the computational efficiency of solving these equations.
  • To provide a robust method for problems with Dirichlet boundary conditions.

Main Methods:

  • Development of fast ordered upwind methods.
  • Utilizing characteristic directions of the partial differential equation to avoid iteration.

Related Experiment Videos

  • Analysis of computational complexity, achieving O(M log M).

    • Main Results:

    • Demonstrated a family of fast methods for static Hamilton-Jacobi equations.
    • Achieved significant reduction in computational time compared to iterative methods.
    • Successfully applied the methods to anisotropic test problems.

    Conclusions:

    • The proposed fast ordered upwind methods offer an efficient alternative to traditional iterative techniques.
    • These methods provide a valuable tool for solving problems in optimal control, seismology, and pathfinding.
    • The O(M log M) complexity makes these methods suitable for large-scale computations.