Many applications in spatial statistics, geostatistics and image analysis require efficient techniques for sampling from large Gaussian Markov random ﬁelds (GMRFs). A suite of methods, based on the Cholesky decomposition, for sampling from GMRFs, sampling conditioned on a set of linear constraints, and computing the likelihood were presented by Rue \ud (2001). In this paper, we present an alternate set of methods based on Krylov subspace approaches. These methods have the advantage of requiring far less storage than the Cholesky decomposition and may be useful in problems where computing a Cholesky decomposition is infeasible
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