Two-Level Nyström--Schur Preconditioner for Sparse Symmetric Positive Definite Matrices

Abstract

Randomized methods are becoming increasingly popular in numerical linear algebra. However, few attempts have been made to use them in developing preconditioners. Our interest lies in solving large-scale sparse symmetric positive definite linear systems of equations where the system matrix is preordered to doubly bordered block diagonal form (for example, using a nested dissection ordering). We investigate the use of randomized methods to construct high quality preconditioners.In particular, we propose a new and efficient approach that employs Nyström's method for computing low rank approximations to develop robust algebraic two-level preconditioners. Construction of the new preconditioners involves iteratively solving a smaller but denser symmetric positive definite Schur complement system with multiple right-hand sides. Numerical experiments on problems coming from a range of application areas demonstrate that this inner system can be solved cheaply using block conjugate gradients and that using a large convergence tolerance to limit the cost does not adversely affect the quality of the resulting Nyström-Schur two-level preconditioner.

Physical human locomotion prediction using manifold regularization

“…Another algorithm called Nyström-PCG has been suggested by Daas, Rees & Scott (2021) , which consists of two steps. First, Nyström uniform subsampling is done using the previous matrix from LapRLS and secondly, preconditioning has been introduced to accelerate the solution reducing the time to .…”

A low-rank update for relaxed Schur complementpreconditioners in fluid flow problems

“…An overview of the numerical solution of saddle-point problems arising in fluid flow applications and suitable preconditioners are given in [4] or [10]. A typical choice for preconditioners of 2 × 2-block systems as given in (1) is based on (approximations of) their block LU factorizations. Such block preconditioners typically require approximations to the inverse of the upper left block A of the system matrix in (1) as well as the (negative) pressure Schur complement S := BA −1 B T .…”

“…A typical choice for preconditioners of 2 × 2-block systems as given in (1) is based on (approximations of) their block LU factorizations. Such block preconditioners typically require approximations to the inverse of the upper left block A of the system matrix in (1) as well as the (negative) pressure Schur complement S := BA −1 B T . Well-known preconditioning techniques for the Schur complement are SIMPLE-type preconditioners [24,13], the least-squares commutator (LSC) [9,10], the pressure-convection-diffusion commutator [17], grad-div preconditioners [7,19,11], augmented Lagrangian preconditioners [8,13,14] as well as hierarchical matrix preconditioners [18].…”

“…In [25], a low-rank update is applied to the velocity-related block A of the system matrix but not to the Schur complement. In [27,1,26], low-rank techniques are applied to precondition the Schur complement that arises in domain decomposition methods.…”

Single-pass Nyström approximation in mixed precision

“…The Nyström method, a randomised approach, arises in two forms based on column-sampling and general random projections. The columnsampling approach is often analysed and used in machine learning setting [32,9] and the general projection version has been explored for, e.g., approximating matrices in a streaming model [28,29] and preconditioning linear systems of equations [3,6,10].…”