Distributed fast marching methods

Fast Marching represents a very efficient technique for solving the front propagation problems which can be formulated as boundary value partial differential equations |∇T(x, y)| = 1/F(x, y) on Ω, with Dirichlet boundary condition T(x, y) = 0 on ∂Ω. We show that the problem of computing the distance map across a smooth sampling domain can be posed in the form of such a partial differential equation, called Eikonal Equation and outline the Fast Marching approach towards approximating its solution. To prevent the Eikonal Equation from becoming non-differentiable, the Fast Marching Methods consider only numerical schemes which guarantee a correct upwind direction and a correct approximation of the viscosity solution.

Fast Marching Methods are a necessary step in Level Set Methods, which are widely used today in scientific computing. The classical Fast Marching Methods, based on finite differences, proposed by Osher and Sethian, are theoretically optimal in terms of operation count: the algorithm's complexity is closer to the optimum of O(NlogN), where N represents the number of grid points. Since Fast Marching Methods are typically sequential, they are an obstacle to the deployment of Level Set Methods on parallel computers. By distributing the entire domain on multiple processors, we do not have to consider huge amount of grid points per processor. Hence, it is not necessary to use a special data structure, i.e. heap sort tree, like in the classical Fast Marching Methods to keep track of exact causal relationships between grid points. We maintain a looser relationship and update all visited points by using a double chained list.

In this poster, we present several parallel implementations of the Fast Marching Methods for the Eikonal equation combining Fast Sweeping with Fast Marching, in such a way that allows fast convergence. We will illustrate the power of these approaches on a series of benchmarks which include the study of the convergence and the error estimates, and show the monotonicity and stability properties of the algorithms. For obtaining all the numerical results, the simulations are done using the LONI resources.