Fast symmetric factorization of hierarchical matrices with applications

We present a fast direct algorithm for computing symmetric factorizations, i.e. A = WWT , of symmetric positive-definite hierarchical matrices with weak-admissibility conditions. The computational cost for the symmetric factorization scales as O(n log n) for hierarchically off-diagonal low-rank matrices. Once this factorization is obtained, the cost for inversion, application, and determinant computation scales as O(n logn). In particular, this allows for the near optimal generation of correlated random variates in the case where A is a covariance matrix. This symmetric factorization algorithm depends on two key ingredients. First, we present a novel symmetric factorization formula for low-rank updates to the identity of the form I + UKUT . This factorization can be computed in O(n) time if the rank of the perturbation is sufficiently small. Second, combining this formula with a recursive divide-and-conquer strategy, near linear complexity symmetric factorizations for hierarchically structured matrices can be obtained. We present numerical results for matrices relevant to problems in probability & statistics (Gaussian processes), interpolation (Radial basis functions), and Brownian dynamics calculations in fluid mechanics (the Rotne-Prager-Yamakawa tensor).