Incomplete Cholesky decomposition for the kriging of large datasets

Kriging of very large spatial datasets is a challenging problem. The size n of the dataset causes problems in computing the kriging estimate: solving the kriging equations directly involves inverting an n×n covariance matrix. This operation requires O(n3) computations and a storage of O(n2). Under these circumstances, straightforward kriging of massive datasets is not possible. Several approaches have been proposed in the literature among which two main families exist: sparse approximation of the covariance function and low rank approaches. We propose here an approach that is built upon a low rank approximation of the covariance matrix obtained by incomplete Cholesky decomposition. This algorithm requires O(nk) storage and takes O(nk2) arithmetic operations, where k is the rank of the approximation, whose accuracy is controlled by a parameter. We detail the main properties of this method and explore its links with existing methods. Its benefits are illustrated on simple examples and compared to those of existing approaches. Finally, we show that this low rank representation is also suited for inverse conditioning of Gaussian random fields.