A Novel Factorized Sparse Approximate Inverse Preconditioner with Supernodes

AbstractKrylov methods preconditioned by Factorized Sparse Approximate Inverses (FSAI) are an efficient approach for the solution of symmetric positive definite linear systems on massively parallel computers. However, FSAI often suffers from a high set-up cost, especially in ill-conditioned problems. In this communication we propose a novel algorithm for the FSAI computation that makes use of the concept of supernode borrowed from sparse LU factorizations and direct methods

Sparse Cholesky factorization by Kullback-Leibler minimization

We propose to compute a sparse approximate inverse Cholesky factor L of a dense covariance matrix Θ by minimizing the Kullback-Leibler divergence between the Gaussian distributions N(0,Θ) and N(0,L−⊤L−1), subject to a sparsity constraint. Surprisingly, this problem has a closed-form solution that can be computed efficiently, recovering the popular Vecchia approximation in spatial statistics. Based on recent results on the approximate sparsity of inverse Cholesky factors of Θ obtained from pairwise evaluation of Green's functions of elliptic boundary-value problems at points {x_i}_(1≤i≤N) ⊂ ℝ^d, we propose an elimination ordering and sparsity pattern that allows us to compute ϵ-approximate inverse Cholesky factors of such Θ in computational complexity O(Nlog(N/ϵ)^d) in space and O(Nlog(N/ϵ)^(2d)) in time. To the best of our knowledge, this is the best asymptotic complexity for this class of problems. Furthermore, our method is embarrassingly parallel, automatically exploits low-dimensional structure in the data, and can perform Gaussian-process regression in linear (in N) space complexity. Motivated by the optimality properties of our methods, we propose methods for applying it to the joint covariance of training and prediction points in Gaussian-process regression, greatly improving stability and computational cost. Finally, we show how to apply our method to the important setting of Gaussian processes with additive noise, sacrificing neither accuracy nor computational complexity