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

For a class of nonsymmetric algebraic Riccati equations arising from transport theory, Lu [SIAM J. Matrix Anal. Appl., 26 (2005), pp. 679-685] reformulated them into a couple of fixed-point equations in a vector form and constructed an iterative method for finding their minimal positive solutions. In this paper, based on this fixed-point equation, we propose several fast and effective iterative methods for solving the above-mentioned special type of nonsymmetric algebraic Riccati equations. These iterative methods can be categorized into the class of nonlinear block relaxation iterations and can also be considered as accelerated variants of the known fixed-point iteration according to the principle of using the currently available information as promptly as possible. The monotone convergence theorems about these new iterative methods are established, and the sharper bounds about the solutions of the nonsymmetric algebraic Riccati equations are derived. Numerical implementations show that the new iterative methods are feasible and effective solvers for the nonsymmetric algebraic Riccati equations arising from transport theory.

Fast Iterative Schemes for Nonsymmetric Algebraic Riccati Equations Arising from Transport Theory

The monotone convergence of the parallel matrix multisplitting relaxation method for linear complementarity problems (see Bai, Z. Z. and Evans, D. J. 1997 Int. J. Comput. Math. 63, 309-326) is discussed, and the corresponding comparison theorem about the monotone convergence rate of this method is thoroughly established.

On the monotone convergence of matrix multisplitting relaxation methods for the linear complementarity problem

By further generalizing the skew-symmetric triangular splitting iteration method studied by Krukier, Chikina and Belokon (Applied Numerical Mathematics, 41 (2002), pp. 89–105), in this paper, we present a new iteration scheme, called the modified skew-Hermitian triangular splitting iteration method, for solving the strongly non-Hermitian systems of linear equations with positive definite coefficient matrices. We discuss the convergence property and the optimal parameters of this new method in depth. Moreover, when it is applied to precondition the Krylov subspace methods like GMRES, the preconditioning property of the modified skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that, as both solver and preconditioner, the modified skew-Hermitian triangular splitting iteration method is very effective for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.

Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong Skew-Hermitian parts

We set up a class of parallel nonlinear AOR method in the sense of matrix multi-splitting for solving the large scale system of nonlinear equations Ax + ϕ(x) = b with A ∈ L(Rn) nonsingular, b ∈ Rn and ϕ : Rn → Rn being continuously diagonal. The global as well as the monotone convergence of this method is proved.

Parallel nonlinear AOR method and its convergence

A class of modified block SSOR preconditioners is presented for the symmetric positive definite systems of linear equations, whose coefficient matrices come from the hierarchical-basis finite-element discretizations of the second-order self-adjoint elliptic boundary value problems. These preconditioners include a block SSOR iteration preconditioner, and two inexact block SSOR iteration preconditioners whose diagonal matrices except for the (1,1)-block are approximated by either point symmetric Gauss–Seidel iterations or incomplete Cholesky factorizations, respectively. The optimal relaxation factors involved in these preconditioners and the corresponding optimal condition numbers are estimated in details through two different approaches used by Bank, Dupont and Yserentant (Numer. Math. 52 (1988) 427–458) and Axelsson (Iterative Solution Methods (Cambridge University Press, 1994)). Theoretical analyses show that these modified block SSOR preconditioners are very robust, have nearly optimal convergence rates, and especially, are well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.

Zhong-Zhi Bai

Papers

Fast Iterative Schemes for Nonsymmetric Algebraic Riccati Equations Arising from Transport Theory

On the monotone convergence of matrix multisplitting relaxation methods for the linear complementarity problem

Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong Skew-Hermitian parts

Parallel nonlinear AOR method and its convergence

Modified Block SSOR Preconditioners for Symmetric Positive Definite Linear Systems