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