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

On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems

The optimal parameter of the Hermitian/skew-Hermitian splitting (HSS) iteration method for a real two-by-two linear system is obtained. The result is used to determine the optimal parameters for linear systems associated with certain two-by-two block matrices and to estimate the optimal parameters of the HSS iteration method for linear systems with n-by-n real coefficient matrices. Numerical examples are given to illustrate the results.

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Optimal Parameter in Hermitian and Skew-Hermitian Splitting Method for Certain Two-by-Two Block Matrices

The convergence properties of a variant of the multisplitting methods for solving the large sparse linear complementarity problems presented by Machida, Fukushima, and Ibaraki [J. Comput. Appl. Math., 62 (1995), pp. 217--227] are further discussed when the system matrices are nonsymmetric and the weighting matrices are nonnegative and diagonal. This directly results in several novel sufficient conditions for guaranteeing the convergence of these multisplitting methods. Moreover, some applicable parallel multisplitting relaxation methods and their corresponding convergence properties are discussed in detail.

On the Convergence of the Multisplitting Methods for the Linear Complementarity Problem

For solving large-scale systems of linear equations by iteration methods, we introduce an effective probability criterion for selecting the working rows from the coefficient matrix and construct a ...

On Greedy Randomized Kaczmarz Method for Solving Large Sparse Linear Systems

We study the eigenvalue bounds of block two-by-two nonsingular and symmetric indefinite matrices whose $(1,1)$ block is symmetric positive definite and Schur complement with respect to its $(2,2)$ block is symmetric indefinite. A constraint preconditioner for this matrix is constructed by simply replacing the $(1,1)$ block by a symmetric and positive definite approximation, and the spectral properties of the preconditioned matrix are discussed. Numerical results show that, for a suitably chosen $(1,1)$ block-matrix, this constraint preconditioner outperforms the block-diagonal and the block-tridiagonal ones in iteration step and computing time when they are used to accelerate the GMRES method for solving these block two-by-two symmetric positive indefinite linear systems. The new results extend the existing ones about block two-by-two matrices of symmetric negative semidefinite $(2,2)$ blocks to those of general symmetric $(2,2)$ blocks.

Zhong-Zhi Bai

Papers

On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems

Optimal Parameter in Hermitian and Skew-Hermitian Splitting Method for Certain Two-by-Two Block Matrices

On the Convergence of the Multisplitting Methods for the Linear Complementarity Problem

On Greedy Randomized Kaczmarz Method for Solving Large Sparse Linear Systems

Constraint Preconditioners for Symmetric Indefinite Matrices