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 local quadratic convergence of inexact simplified Jacobi–Davidson method

On the convergence of parallel chaotic nonlinear multisplitting Newton-type methods

We set up a class of parallel nonlinear multisplitting AOR methods by directly multisplitting the nonlinear mapping involved in the nonlinear complementarity problems. The different choices of the relaxation parameters can yield all the known and a lot of new relaxation methods, as well as a lot of new relaxed parallel nonlinear multisplitting methods for solving the nonlinear complementarity problems. The two-sided approximation properties and the influences on the convergence rates from the relaxation parameters about our new methods are shown, and sufficient conditions guaranteeing the methods to converge globally are discussed. Finally, a lot of numerical results show that our new methods are feasible and efficient.

The monotone convergence of a class of parallel nonlinear relaxation methods for nonlinear complementarity problems

For the system of linear equations arising from discretization of the second-order selfadjoint elliptic Dirichlet-periodic boundary value problems, by making use of the special structure of the coefficient matrix we present a class of combinative preconditioners which are technical combinations of modified incomplete Cholesky factorizations and ShermanMorrison-Woodbury update. Theoretical analyses show that the condition numbers of the preconditioned matrices can be reduced to O(h−1), one order smaller than the condition number O(h−2) of the original matrix. Numerical implementations show that the resulting preconditioned conjugate gradient methods are feasible, robust and efficient for solving this class of linear systems. Mathematics subject classification: 65F10, 65F50.

Combinative preconditioners of modified incomplete cholesky factorization and Sherman-Morrison-Woodbury update for self-adjoint elliptic Dirichlet-periodic boundary value problems

By making use of the nonlinear multisplitting and the nonlinear relaxation techniques, we present, in this paper, a class of parallel nonlinear multisplitting successive overrelaxation methods for solving the large sparse nonlinear complementarity problems on the modern high-speed multiprocessors. These new methods particularly include the so-called nonlinear multisplitting SOR-Newton method. Under suitable conditions, we establish the local convergence theories of the new methods, and investigate their asymptotic convergence rates. A lot of numerical results show that our new methods are feasible and efficient for parallel solving the nonlinear complementarity problems.

Zhong-Zhi Bai

Papers

On local quadratic convergence of inexact simplified Jacobi–Davidson method

On the convergence of parallel chaotic nonlinear multisplitting Newton-type methods

The monotone convergence of a class of parallel nonlinear relaxation methods for nonlinear complementarity problems

Combinative preconditioners of modified incomplete cholesky factorization and Sherman-Morrison-Woodbury update for self-adjoint elliptic Dirichlet-periodic boundary value problems

A class of parallel nonlinear multisplitting relaxation methods for the large sparse nonlinear complementarity problems