Fractional Differential Equations

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

Cyber-physical systems are ubiquitous in power systems, transportation networks, industrial control processes, and critical infrastructures. These systems need to operate reliably in the face of unforeseen failures and external malicious attacks. In this paper: (i) we propose a mathematical framework for cyber-physical systems, attacks, and monitors; (ii) we characterize fundamental monitoring limitations from system-theoretic and graph-theoretic perspectives; and (ii) we design centralized and distributed attack detection and identification monitors. Finally, we validate our findings through compelling examples.

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Attack Detection and Identification in Cyber-Physical Systems

Cyber-physical systems integrate computation, communication, and physical capabilities to interact with the physical world and humans. Besides failures of components, cyber-physical systems are prone to malignant attacks, and specific analysis tools as well as monitoring mechanisms need to be developed to enforce system security and reliability. This paper proposes a unified framework to analyze the resilience of cyber-physical systems against attacks cast by an omniscient adversary. We model cyber-physical systems as linear descriptor systems, and attacks as exogenous unknown inputs. Despite its simplicity, our model captures various real-world cyber-physical systems, and it includes and generalizes many prototypical attacks, including stealth, (dynamic) false-data injection and replay attacks. First, we characterize fundamental limitations of static, dynamic, and active monitors for attack detection and identification. Second, we provide constructive algebraic conditions to cast undetectable and unidentifiable attacks. Third, by using the system interconnection structure, we describe graph-theoretic conditions for the existence of undetectable and unidentifiable attacks. Finally, we validate our findings through some illustrative examples with different cyber-physical systems, such as a municipal water supply network and two electrical power grids.

Attack Detection and Identification in Cyber-Physical Systems -- Part I: Models and Fundamental Limitations

This paper reveals the inner links between two known frameworks of multisplitting relaxation methods as completely as possible. By meticulously investigating the specific structures of these two frameworks, the asymptotic convergence rates as well as the monotone convergence rates of them are compared theoretically. This then definitely answers the question that which converges faster between these two frameworks of parallel matrix multisplitting relaxation methods from the standpoint of pure mathematics. At last, an example further confirms the correctness of the theoretical results.

On the comparisons of the multisplitting unsymmetric aor methods for M-matrices

For computing the smallest eigenvalue and the corresponding eigenvector of a Hermitian matrix, by introducing a concept of perfect Krylov subspace, we propose a class of perfect Krylov subspace methods. For these methods, we prove their local, semilocal, and global convergence properties, and discuss their inexact implementations and preconditioning strategies. In addition, we use numerical experiments to demonstrate the convergence properties and exhibit the competitiveness of these methods with a few state-of-the art iteration methods such as Lanczos, rational Krylov sequence, and Jacobi-Davidson, when they are employed to solve large and sparse Hermitian eigenvalue problems.

Computing eigenpairs of Hermitian matrices in perfect Krylov subspaces

A new comparison theorem is given for the nonlinear multisplitting relaxation method [1], and an important modification is proposed for it, too

Improved comparison theorem for the nonlinear multisplitting relaxation method

On preconditioned and relaxed AVMM methods for quadratic programming problems with equality constraints

When the artificial compressibility method in conjunction with high-order upwind compact finite difference schemes is employed to discretize the steady-state incompressible Navier-Stokes equations, in each pseudo-time step we need to solve a structured system of linear equations approximately by, for example, a Krylov subspace method such as the preconditioned GMRES. In this paper, based on the special structure and concrete property of the linear system we construct a structured preconditioner for its coefficient matrix and estimate eigenvalue bounds of the correspondingly preconditioned matrix. Numerical examples are given to illustrate the effectiveness of the proposed preconditioning methods.

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Zhong-Zhi Bai

Papers

On the comparisons of the multisplitting unsymmetric aor methods for M-matrices

Computing eigenpairs of Hermitian matrices in perfect Krylov subspaces

Improved comparison theorem for the nonlinear multisplitting relaxation method

On preconditioned and relaxed AVMM methods for quadratic programming problems with equality constraints

On approximated ILU and UGS preconditioning methods for linearized discretized steady incompressible Navier-Stokes equations