Computationally Efficient Models of Urban and Natural Terrain by Non-iterative Domain Decomposition for L1 Smoothing Splines

In this paper, we propose and validate a computationally efficient non-iterative domain decomposition procedure for calculating bivariate cubic L1 smoothing splines. This domain decomposition procedure involves calculating local L1 smoothing splines individually on overlapping “extended subdomains” that cover the global domain and then creating the global L1 smoothing spline by patching together the local L1 smoothing splines. Using this procedure, we calculate the global L1 smoothing splines of one urban terrain data set (Baltimore) and one natural terrain data set (Killeen, Texas). The local L1 smoothing splines generally match well at subdomain boundaries but do not always do so. The current hypothesis is that the cases in which the local L1 smoothing splines do not match well at the boundaries of the subdomains are due to limitations in the compressed primal-dual algorithm that is used to calculate the local L1 smoothing splines. The non-iterative nature of this new domain decomposition procedure is in strong contrast to and is a large improvement over the iterative nature of all previously known domain decomposition procedures. With sequential and especially with parallel computation, the non-iterative L1 smoothing spline domain decomposition procedure will be a large factor in reducing computing time so that complex terrain models can be calculated and manipulated in real time.