Hussam Al Daas — Page 2

arXiv2021

Two-level Nyström--Schur preconditioner for sparse symmetric positive definite matrices

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.

math.NA↗

arXiv2020

Low-Rank and Total Variation Regularization and Its Application to Image Recovery

In this paper, we study the problem of image recovery from given partial (corrupted) observations. Recovering an image using a low-rank model has been an active research area in data analysis and machine learning. But often, images are not only of low-rank but they also exhibit sparsity in a transformed space. In this work, we propose a new problem formulation in such a way that we seek to recover an image that is of low-rank and has sparsity in a transformed domain. We further discuss various non-convex non-smooth surrogates of the rank function, leading to a relaxed problem. Then, we present an efficient iterative scheme to solve the relaxed problem that essentially employs the (weighted) singular value thresholding at each iteration. Furthermore, we discuss the convergence properties of the proposed iterative method. We perform extensive experiments, showing that the proposed algorithm outperforms state-of-the-art methodologies in recovering images.

cs.CV↗