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.

/pdf/attack-detection-and-identification-in-cyber-physical-3fy6vyw4zt.pdf

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

https://www.cerfacs.fr/algor/reports/2008/TR_PA_08_24.pdf

Numerical study on incomplete orthogonal factorization preconditioners

Quasi‐HSS iteration methods for non‐Hermitian positive definite linear systems of strong skew‐Hermitian parts

On convergence conditions of waveform relaxation methods for linear differential-algebraic equations

We present a modified damped Newton method for solving large sparse linear complementarity problems, which adopts a new strategy for determining the stepsize at each Newton iteration The global convergence of the new method is proved when the system matrix is a nondegenerate matrix We then apply the matrix splitting technique to this new method, deriving an inexact splitting method for the linear complementarity problems The global convergence of the resulting inexact splitting method is proved, too Numerical results show that the new methods are feasible and effective for solving the large sparse linear complementarity problems

Zhong-Zhi Bai

Papers

Numerical study on incomplete orthogonal factorization preconditioners

Quasi‐HSS iteration methods for non‐Hermitian positive definite linear systems of strong skew‐Hermitian parts

On convergence conditions of waveform relaxation methods for linear differential-algebraic equations

A modified damped Newton method for linear complementarity problems

Fast matrix splitting preconditioners for higher dimensional spatial fractional diffusion equations