A Fast-marching Algorithm for Nonmonotonically Evolving Fronts

Abstract

Abstract. The non-monotonic propagation of fronts is considered. When the speed function F : R n × [0, T ] → R is prescribed, the non-linear advection equation φt + F |∇φ| = 0 is a HamiltonJacobi equation known as the level-set equation. It is argued that a small enough neighbourhood of the zero-level-set M of the solution φ : R n × [0, T ] → R is the graph of ψ : R n → R where ψ solves a Dirichlet problem of the form H(u, ψ(u), ∇ψ(u)) = 0. A fast-marching algorithm is presented where each point is computed using a discretization of such a Dirichlet problem, with no restrictions on the sign of F . The output is a directed graph whose vertices evenly sample M. The convergence, consistency and stability of the scheme are addressed. Bounds on the computational complexity are estimated, and experimentally shown to be on par with the Fast Marching Method. Examples are presented where the algorithm is shown to be globally first-order accurate. The complexities and accuracies observed are independent of the monotonicity of the evolution.

Front Propagation from Radiative Sources

“…This simplification greatly reduces the computational cost. A variant of the fast marching algorithms, which attempts to retain their computational advantages over other level set methods but allow for non-monotonically evolving fronts [7] was also developed to give the flexibility to account for non-uniform interaction around the fronts.…”