EFFICIENT GENERATION OF CONDITIONAL SIMULATIONS BY CHEBYSHEV MATRIX POLYNOMIAL APPROXIMATIONS TO THE SYMMETRICAL SQUARE-ROOT OF THE COVARIANCE-MATRIX

Consider the problem of generating a realization y(1) of a Gaussian random field on a dense grid of points Omega(1) conditioned on field observations y, collected on a sparse grid of points Omega(2). An approach to this is to generate first an unconditional realization y over the grid Omega = Omega(1) U Omega(2), and then to produce y(1) by conditioning y on the data y(2). As standard methods for generating y, such as the turning bands, spectral or Cholesky approaches can have various limitations, it has been proposed by M. W. Davis to generate realizations from a matrix polynomial approximations to the square root of the covariance matrix. In this paper we describe how to generate a direct approximation to the conditional realization y(1) on Omega(1) using a variant of Davis' approach based on approximation by Chebyshev polynomials. The resulting algorithm is simple to implement, numerically stable, and bounds on the approximation error are readily available. Furthermore we show that the conditional realization y(1) can be generated directly with a lower order polynomial than the unconditional realization y, and that further reductions can be achieved by exploiting a nugget effect if one is present. A pseudocode version of the algorithm is provided that can be implemented using the fast Fourier transform if the field is stationary and the grid Omega 1 is rectangular. Finally, numerical illustrations are given of the algorithm's performance in generating various 2-D realizations of conditional processes on large sampling grids.