Perron's method for quasilinear hyperbolic systems, Part II

In part I (P. Smith, Perron's method for quasilinear hyperbolic systems, part I, J. Math. Anal., in press) of this paper we defined a notion of viscosity solution (sub- (super-)solution) for these systems, proved a comparison principle for viscosity sub- and supersolutions. Here, in part II, we prove existence of viscosity solutions to the Cauchy problem, using a Perron-like method, for long time, and for all time.     

An interval algorithm for sensitivity analysis of coupled vibro-acoustic systems

Sensitivity analysis is a vital part in the optimization design of coupled vibro-acoustic systems. A new interval sensitivity-analysis method for vibro-acoustic systems is proposed in this paper. This method relies on only interval perturbation analysis instead of partial derivatives and difference operations. For strongly nonlinear systems, in particular, this methodology requires parameter variation over narrower ranges in comparison with other methods. To implement sensitivity analysis based on this method, the interval ranges of the responses of the vibro-acoustic system with interval parameters should first be obtained. Therefore, an interval perturbation-analysis method is presented for obtaining the interval bounds of the sound-pressure responses of a coupled vibro-acoustic system with interval parameters. The interval perturbation method is then compared with the Monte Carlo method, which can be taken as the benchmark for comparative accuracy. Two numerical examples involving sensitivity analysis of vibro-acoustic systems illustrate the feasibility and effectiveness of the proposed interval-based method.     

Center-focus problem for analytic systems with nonzero linear part

We suggest a method for solving the center-focus problem for real analytic systems with nonzero linear part. The cases of a linear center (pure imaginary eigenvalues of the linear part) and of a nilpotent center are considered. A unified method for solving the center-focus problem in both cases is indicated.     

On a method of solving dual integral equations : PMM vol. 37, n6, 1973, pp. 1087–1097

We expound a method of reducing a class of dual integral equations which find important practical application to infinite algebraic systems of the first kind. The latter system can be reduced to the systems of the second kind by exact inversion of the principal singular part, and the second kind systems can be solved using the method of consecutive approximations [1–6], The dual integral equations generated by the Kontorovich-Lebedev and Mehler-Fock integral transforms are considered as examples as well as the problems of torsion of a truncated elastic sphere by a punch and that of a circular crack in an elastic space.     

Solution of a system of simultaneous linear equations with a sparse coefficient matrix by elimination methods

A method is described for handling systems of linear equations with a sparse coefficient matrix as economically as possible. In particular, a convenient rule for the order of pivoting is given. Numerical experiments indicating considerable savings have also been performed.This research was supported in part by the National Aeronautics and Space Administration, Washington, D.C., Grant No. NGR-33-015-013.     

线性离散系统的部分稳定性

In this paper, we study the partial stability of linear discrete systems by means of Liapunov's functions of quadratic form . We obtain a necessary and sufficient condition for the system being stable with respect to part of variables and generalize Liapunov's equation to the partial stability of linear discrete systems . A method of constructing Liapunov's function of quadratic form for the stability of the systems is given.     

Stabilization of unknown nonlinear systems using a cognition-based framework

This paper considers an adaptive control method based on a cognition-based framework to stabilize unknown nonlinear systems. In order to fulfill the task of stabilization, neither the information about the systems dynamical structure nor the knowledge about system physical behaviors, but the system states, which are assumed as measurable, are required. The structure of the proposed controller consists of three parts. The first part is based on a recurrent neural network (RNN) to be used for local identification of the unknown nonlinear system in real time. The network can be utilized as system characteristics, which is further used to design the controller within the third part. In the second part, the set of the given input values leading to stable behavior of the closed-loop system will be calculated numerically with a geometrical criterion based on a suitable definition of quadratic stability. In the third part, a suitable control input value is chosen accordingly to a time-relevant criteria from the set of input values generated in the second part of the controller. These three parts and their internal connections are arranged within a so-called cognition framework. The proposed cognitive controller is able to gain useful knowledge (with local validity) and define autonomously a suitable control input with respect to the requirements of the time-relevant criteria. Numerical examples are shown to demonstrate the successful application and performance of the method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized synchronization of chaos based on suitable separation

In this Letter, generalized synchronization with a kind of function relationship between the states of drive and response chaotic systems is achieved. From matrix measure theory, some sufficient conditions for generalized synchronization are derived through suitable separation by decomposing the system as the linear part and the nonlinear one. Simulation results are provided for illustration and verification of the proposed method.     

An algebraic multigrid method for finite element systems on criss-cross grids

In this paper, we design and analyze an algebraic multigrid method for a condensed finite element system on criss-cross grids and then provide a convergence analysis. Criss-cross grid finite element systems represent a large class of finite element systems that can be reduced to a smaller system by first eliminating certain degrees of freedoms. The algebraic multigrid method that we construct is analogous to many other algebraic multigrid methods for more complicated problems such as unstructured grids, but, because of the specialty of our problem, we are able to provide a rigorous convergence analysis to our algebraic multigrid method. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday The work was supported in part by NSAF(10376031) and National Major Key Project for basic researches and by National High-Tech ICF Committee in China.     

On the Asymptotic Integration of a System of Linear Differential Equations with a Small Parameter in the Coefficients of a Part of Derivatives

We propose an asymptotic method for the integration of one type of systems of linear differential equations with a small parameter in the coefficients of a part of derivatives.     

Output-feedback control robust stabilization of nonlinear pulse systems

A nonlinear pulse systems with a time-variant continuous linear part is considered. A method for synthesizing an output-feedback robust stabilizing control is suggested.     

Numerical simulations and modeling for stochastic biological systems with jumps

This paper gives a numerical method to simulate sample paths for stochastic differential equations (SDEs) driven by Poisson random measures. It provides us a new approach to simulate systems with jumps from a different angle. The driving Poisson random measures are assumed to be generated by stationary Poisson point processes instead of Lévy processes. Methods provided in this paper can be used to simulate SDEs with Lévy noise approximately. The simulation is divided into two parts: the part of jumping integration is based on definition without approximation while the continuous part is based on some classical approaches. Biological explanations for stochastic integrations with jumps are motivated by several numerical simulations. How to model biological systems with jumps is showed in this paper. Moreover, method of choosing integrands and stationary Poisson point processes in jumping integrations for biological models are obtained. In addition, results are illustrated through some examples and numerical simulations. For some examples, earthquake is chose as a jumping source which causes jumps on the size of biological population.     

Scattered data interpolation based upon generalized minimum norm networks

A generalization of G. M. Nielson's method for bivariate scattered data interpolation based upon a minimum norm network is presented. The essential part of the new method is the use of a variational principle for definition of function values as well as cross-boundary derivatives over the edges of a triangulation of the data points. We mainly discuss the case ofC ² interpolants and present some examples including quality control with systems of isophotes.     

Stochastic version of the degenerate kernel method for solving large systems of linear equations with sparse matrix

A stochastic method for solving systems of linear algebraic equations with sparse matrix is proposed. The method is based on the determination of the principal part of a matrix operator in the form of a degenerate kernel. Results of a numerical experiment witnessing the efficiency of the method are presented.     

A Randomized Kaczmarz Algorithm with Exponential Convergence

The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical estimates for its rate of convergence are still scarce. We introduce a randomized version of the Kaczmarz method for consistent, overdetermined linear systems and we prove that it converges with expected exponential rate. Furthermore, this is the first solver whose rate does not depend on the number of equations in the system. The solver does not even need to know the whole system but only a small random part of it. It thus outperforms all previously known methods on general extremely overdetermined systems. Even for moderately overdetermined systems, numerical simulations as well as theoretical analysis reveal that our algorithm can converge faster than the celebrated conjugate gradient algorithm. Furthermore, our theory and numerical simulations confirm a prediction of Feichtinger et al. in the context of reconstructing bandlimited functions from nonuniform sampling. T. Strohmer was supported by NSF DMS grant 0511461. R. Vershynin was supported by the Alfred P. Sloan Foundation and by NSF DMS grant 0401032.     

Stability analysis by comparison technique

In this paper we develop a method of vector Liapunov's functions for finite nonlinear dynamical systems. Our main assumption is the following: every components of vector function may be sign definite with respect to a part of variables. Also, we apply the above results in the stability analysis of large scale systems.     

A NEW ALGORITHM FOR COMPUTING LARGEST REAL PART EIGENVALUE OF MATRICES：COLLATZ ＆ PERRON-FROBERNIUS＇ APPROACH

This paper describes a new method and algorithm for the numerical solution of eigenvalues with the largest real part of positive matrices.The method is based on a numerical implementation of Collatz’s eigenvalue inclusion theorem for non-negative irreducible matrices.Eigenvalues are analyzed for the studies of the stability of linear systems.Finally, a numerical discussion is given to derive the required number of mathematical operations of the new algorithm. Comparisons between the new algorithm and several well known ones, such as Power, and QR methods, are discussed.