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Methods for inference and simulation of linearly constrained Gaussian MarkovRandom Fields (GMRF) are computationally prohibitive when the number ofconstraints is large. In some cases, such as for intrinsic GMRFs, they may evenbe unfeasible. We propose a new class of methods to overcome these challengesin the common case of sparse constraints, where one has a large number ofconstraints and each only involves a few elements. Our methods rely on a basistransformation into blocks of constrained versus non-constrained subspaces, andwe show that the methods greatly outperform existing alternatives in terms ofcomputational cost. By combining the proposed methods with the stochasticpartial differential equation approach for Gaussian random fields, we also showhow to formulate Gaussian process regression with linear constraints in a GMRFsetting to reduce computational cost. This is illustrated in two applicationswith simulated data.

Efficient methods for Gaussian Markov random fields under sparse linear constraints