Solution and linear estimation of 2-D nearest-neighbor models

The solution and linear estimation of 2-D nearest-neighbor models (NNMs) are considered. The class of problems that can be described by NNMs is quite large, as models of this type arise whenever partial differential equations are discretized with finite-difference methods. A general solution technique that relies on converting the system to an equivalent 1-D two-point boundary-value descriptor system (TPBVDS) of large dimension, for which a recursive and stable solution technique is developed, is proposed. Under slightly restrictive assumptions, an even faster procedure can be obtained by using the fast Fourier transform (FFT) with respect to one of the space dimensions to convert the 1-D TPBVDS into a set of decoupled TPBVDS of low order, which can be solved in parallel. The smoothing problem for 2-D random fields described by stochastic NNMs is also examined. The smoother is expressed as a Hamiltonian system of twice the dimension of the original system and is also in NNM form. NNM solution techniques are therefore directly applicable to this solution. The results are illustrated by two examples corresponding to the discretized Poisson and heat equations, respectively.