Alexandra Tcheng

Alexandra Tcheng, Jean-Christophe Nave

The non-monotonic propagation of fronts is considered. When the speed function $F:\mathbb{R}^{n} \times [0,T]\rightarrow \mathbb{R}$ is prescribed, the non-linear advection equation $ϕ_{t}+F|\nabla ϕ|=0$ is a Hamilton-Jacobi equation known as the level-set equation. It is argued that a small enough neighbourhood of the zero-level-set $\mathcal{M}$ of the solution $ϕ: \mathbb{R}^{n} \times [0,T] \rightarrow \mathbb{R}$ is the graph of $ψ:\mathbb{R}^{n} \rightarrow \mathbb{R}$ where $ψ$ solves a Dirichlet problem of the form $H(\vec{u},ψ(\vec{u}),\nabla ψ(\vec{u}))=0$. 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 $\mathcal{M}$. 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.

Alexandra Tcheng, Jean-Christophe Nave

A new algorithm is proposed to describe the propagation of fronts advected in the normal direction with prescribed speed function F. The assumptions on F are that it does not depend on the front itself, but can depend on space and time. Moreover, it can vanish and change sign. To solve this problem the Level-Set Method [Osher, Sethian; 1988] is widely used, and the Generalized Fast Marching Method [Carlini et al.; 2008] has recently been introduced. The novelty of our method is that its overall computational complexity is predicted to be comparable to that of the Fast Marching Method [Sethian; 1996], [Vladimirsky; 2006] in most instances. This latter algorithm is O(N^n log N^n) if the computational domain comprises N^n points. Our strategy is to use it in regions where the speed is bounded away from zero -- and switch to a different formalism when F is approximately 0. To this end, a collection of so-called sideways partial differential equations is introduced. Their solutions locally describe the evolving front and depend on both space and time. The well-posedness of those equations, as well as their geometric properties are addressed. We then propose a convergent and stable discretization of those PDEs. Those alternative representations are used to augment the standard Fast Marching Method. The resulting algorithm is presented together with a thorough discussion of its features. The accuracy of the scheme is tested when F depends on both space and time. Each example yields an O(1/N) global truncation error. We conclude with a discussion of the advantages and limitations of our method.