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

A new comparison theorem on the monotone convergence rates of the parallel nonlinear multisplitting accelerated overrelaxation (AOR) method for solving the large scale nonlinear complementarity problem is established. Thus, the monotone convergence theory of this class of method is completed.

New comparison theorem for the nonlinear multisplitting relaxation method for the nonlinear complementarity problems

A stationary convection-diffusion problem with a small parameter multiplying the highest derivative is considered. The problem is discretized on a uniform rectangular grid by the central-difference scheme. A new class of two-step iterative methods for solving this problem is proposed and investigated. The convergence of the methods is proved, optimal iterative methods are chosen, and the rate of convergence is estimated. Numerical results are presented that show the high efficiency of the methods.

Two-step iterative methods for solving the stationary convection-diffusion equation with a small parameter at the highest derivative on a uniform grid

Directional secant method for nonlinear equations

To solve the linear complementarity problems efficiently on the high-speed multiprocessor systems, we set up a class of asynchronous parallel matrix multisplitting accelerated over-relaxation (AOR) method by technical combination of the matrix multisplitting and the accelerated overrelaxation techniques. The convergence theory of this new method is thoroughly established under the condition that the system matrix of the linear complementarity problem is an H-matrix with positive diagonal elements. At last, we also make multi-parameter extension for this new asynchronous multisplitting AOR method, and investigate the convergence property of the resulted asynchronous multisplitting unsymmetric AOR method. Thereby, an extensive sequence of asynchronous parallel relaxed iteration methods in the sense of multisplitting is presented for solving the large scale linear complementarity problems in the asynchronous parallel computing environments. This not only affords various choices, but also presents systematic ...

Asynchronous multisplitting relaxation methods for linear complementarity problems

By applying the reduced-order sinc discretization to the two-point boundary value problem of a linear third-order ordinary differential equation, we can obtain a block two-by-two system of linear equations, with each block of its coefficient matrix being a combination of Toeplitz and diagonal matrices. This class of linear systems can be effectively solved by Krylov subspace iteration methods such as GMRES and BiCGSTAB. We construct block-triangular preconditioning matrices to accelerate the convergence rates of the Krylov subspace iteration methods, and demonstrate that the eigenvalues of certain approximations to the preconditioned matrices are uniformly bounded within a rectangle, being independent of the size of the discrete linear system, on the complex plane. In addition, we use numerical examples to show the effectiveness of the proposed preconditioning methods.

Zhong-Zhi Bai

Papers

New comparison theorem for the nonlinear multisplitting relaxation method for the nonlinear complementarity problems

Two-step iterative methods for solving the stationary convection-diffusion equation with a small parameter at the highest derivative on a uniform grid

Directional secant method for nonlinear equations

Asynchronous multisplitting relaxation methods for linear complementarity problems

Block-triangular preconditioning methods for linear third-order ordinary differential equations based on reduced-order sinc discretizations