Article: McCaslin JO, Courtine E, Desjardins O; (2014) A Fast Marching Approach to Multidimensional Extrapolation, Journal of Computational Physics, 227: 393-412
Abstract: 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 Gamma that separates the subdomain Omega(in) over which the quanity is known from the subdomain
Omega(out) 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 Gamma 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). 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. (C) 2014 Elsevier Inc. All rights reserved.