A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

The aim of this survey is to review some recent developments in devising efficient preconditioners for sequences of symmetric positive definite (SPD) linear systems A k x k = b k , k = 1 , … arising in many scientific applications, such as discretization of transient Partial Differential Equations (PDEs), solution of eigenvalue problems, (Inexact) Newton methods applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. In this paper, we will analyze a number of techniques of updating a given initial preconditioner by a low-rank matrix with the aim of improving the clustering of eigenvalues around 1, in order to speed-up the convergence of the Preconditioned Conjugate Gradient (PCG) method. We will also review some techniques to efficiently approximate the linearly independent vectors which constitute the low-rank corrections and whose choice is crucial for the effectiveness of the approach. Numerical results on real-life applications show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates.

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Low-rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

10.20944/preprints202003.0094.v1 ◽

2020 ◽

Author(s):

Luca Bergamaschi

Keyword(s):

Linear Systems ◽

Eigenvalue Problems ◽

Iterative Solution ◽

Low Rank ◽

Nonlinear Pdes ◽

Inexact Newton Method ◽

Krylov Methods ◽

Rank Matrix ◽

Approximate Inverses ◽

Low Rank Matrix

The aim of this survey is to review some recent developements in devising efficient preconditioners for sequences of linear systems A x = b. Such a problem arise in many scientific applications, such as discretization of transient PDEs, solution of eigenvalue problems, (Inexact) Newton method applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. Full purpose preconditioners such as the Incomplete Cholesky (IC) factorization or approximate inverses are aimed at clustering eigenvalues of the preconditioned matrices around one. In this paper we will analyze a number of techniques of updating a given IC preconditioner (which we denote as P0 in the sequel) by a low-rank matrix with the aim of further improving this clustering. The most popular low-rank strategies are aimed at removing the smallest eigenvalues (deflation) or at shifting them towards the middle of the spectrum. The low-rank correction is based on a (small) number of linearly independent vectors whose choice is crucial for the effectiveness of the approach. In many cases these vectors are approximations of eigenvectors corresponding to the smallest eigenvalues of the preconditioned matrix P0 A. We will also review some techniques to efficiently approximate these vectors when incorporated within a sequence of linear systems all possibly having constant (or slightly changing) coefficient matrices. Numerical results concerning sequences arising from discretization of linear/nonlinear PDEs and iterative solution of eigenvalue problems show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates.

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Multi-View Human Activity Recognition in Distributed Camera Sensor Networks

10.32920/14637105.v1 ◽

2021 ◽

Author(s):

Ehsan Adeli Mosabbeb ◽

Kaamran Raahemifar ◽

Mahmood Fathy

Keyword(s):

Human Activity ◽

Matrix Completion ◽

Real Life ◽

Low Rank ◽

Activity Classification ◽

Activity Class ◽

Classification Framework ◽

Rank Matrix ◽

Increasing Demand ◽

Low Rank Matrix

With the increasing demand on the usage of smart and networked cameras in intelligent and ambient technology environments, development of algorithms for such resource-distributed networks are of great interest. Multi-view action recognition addresses many challenges dealing with view-invariance and occlusion, and due to the huge amount of processing and communicating data in real life applications, it is not easy to adapt these methods for use in smart camera networks. In this paper, we propose a distributed activity classification framework, in which we assume that several camera sensors are observing the scene. Each camera processes its own observations, and while communicating with other cameras, they come to an agreement about the activity class. Our method is based on recovering a low-rank matrix over consensus to perform a distributed matrix completion via convex optimization. Then, it is applied to the problem of human activity classification. We test our approach on IXMAS and MuHAVi datasets to show the performance and the feasibility of the method.

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