A fast marching approach to multidimensional extrapolation

A computationally efficient approach to extrapolating a data field with second order accuracy is presented. This is achieved through the sequential solution of non-homogeneous linear static Hamilton–Jacobi equations, which can be performed rapidly using the fast marching methodology. In particular, the method relies on a fast marching calculation of the distance from the manifold Γ   that separates the subdomain ΩinΩin over which the quanity is known from the subdomain ΩoutΩout over which the quantity is to be extrapolated. A parallel algorithm is included and discussed in the appendices. Results are compared to the multidimensional partial differential equation (PDE) extrapolation approach of Aslam (Aslam (2004) [31]). It is shown that the rate of convergence of the extrapolation within a narrow band near Γ is controlled by both the number of successive extrapolations performed and the order of accuracy of the spatial discretization. For m successive extrapolating steps and a spatial discretization scheme of order N  , the rate of convergence in a narrow band is shown to be min(N+1,m+1)min(N+1,m+1). Results show that for a wide range of error levels, the fast marching extrapolation strategy leads to dramatic improvements in computational cost when compared to the PDE approach.