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

The finite difference or the finite element discretizations of many differential or integral equations often result in a class of systems of weakly nonlinear equations. In this paper, by reasonably applying both the multisplitting and the two-stage iteration techniques, and in accordance with the special properties of this system of weakly nonlinear equations, we first propose a general multisplitting two-stage iteration method through the two-stage multiple splittings of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear iterative methods, we present a multisplitting two-stage AOR method, which particularly uses the AOR-like iteration as inner iteration and is substantially a relaxed variant of the afore-presented method. These two methods have a forceful parallel computing function and are much more suitable to the high-speed multiprocessor systems. For these two classes of methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the involved nonlinear mapping is only directionally differentiable. When the system matrix is either an H-matrix or a monotone matrix, and the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover, under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence properties of the new multisplitting two-stage iteration methods, and investigate the influence of the multiple splittings as well as the relaxation parameters upon the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and efficient for parallel solving of the system of weakly nonlinear equations.

Parallel multisplitting two-stage iterative methods for large sparse systems of weakly nonlinear equations

On inexact Newton methods based on doubling iteration scheme for non‐symmetric algebraic Riccati equations

Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix ☆

We construct, analyze and implement SSOR-like preconditioners for non-Hermitian positive definite system of linear equations when its coefficient matrix possesses either a dominant Hermitian part or a dominant skew-Hermitian part. We derive tight bounds for eigenvalues of the preconditioned matrices and obtain convergence rates of the corresponding SSOR-like iteration methods as well as the corresponding preconditioned GMRES iteration methods. Numerical implementations show that Krylov subspace iteration methods such as GMRES, when accelerated by the SSOR-like preconditioners, are efficient solvers for these classes of non-Hermitian positive definite linear systems.

Zhong-Zhi Bai

Papers

Parallel multisplitting two-stage iterative methods for large sparse systems of weakly nonlinear equations

On inexact Newton methods based on doubling iteration scheme for non‐symmetric algebraic Riccati equations

Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix ☆

On SSOR‐like preconditioners for non‐Hermitian positive definite matrices

Product-type skew-Hermitian triangular splitting iteration methods for strongly non-Hermitian positive definite linear systems