Effective computation for nonlinear systems

Nonlinear dynamical and control systems are an important source of applications for theories of computation over the the real numbers, since these systems are usually to complicated to study analytically, but may be extremely sensitive to numerical error. Further, computerassisted proofs and verification problems require a rigorous treatment of numerical errors. In this paper we will describe how to provide a semantics for effective computations on sets and maps and show how these operations have been implemented in the tool Ariadne for the analysis, design and verification of nonlinear and hybrid systems

Nonlinear Markov Processes in Big Networks

Big networks express various large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big networks, and applies the mean-field theory and the nonlinear Markov processes to set up a broad class of nonlinear continuous-time block-structured Markov processes, which can be applied to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of interacting big networks with symmetric interactions, each of which is described as a continuous-time block-structured Markov process. Secondly, some effective algorithms are given for computing the fixed points of the nonlinear Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff center, the Lyapunov functions and the relative entropy are used to analyze stability or metastability of the big network, and several interesting open problems are proposed with detailed interpretation. We believe that the results given in this paper can be useful and effective in the study of big networks.Comment: 28 pages in Special Matrices; 201

Nonlinear response and fluctuation dissipation relations

A unified derivation of the off equilibrium fluctuation dissipation relations (FDR) is given for Ising and continous spins to arbitrary order, within the framework of Markovian stochastic dynamics. Knowledge of the FDR allows to develop zero field algorithms for the efficient numerical computation of the response functions. Two applications are presented. In the first one, the problem of probing for the existence of a growing cooperative length scale is considered in those cases, like in glassy systems, where the linear FDR is of no use. The effectiveness of an appropriate second order FDR is illustrated in the test case of the Edwards-Anderson spin glass in one and two dimensions. In the second one, the important problem of the definition of an off equilibrium effective temperature through the nonlinear FDR is considered. It is shown that, in the case of coarsening systems, the effective temperature derived from the second order FDR is consistent with the one obtained from the linear FDR.Comment: 24 pages, 6 figure

New family of iterative methods with high order of convergence for solving nonlinear systems

In this paper we present and analyze a set of predictor-corrector iterative methods with increasing order of convergence, for solving systems of nonlinear equations. Our aim is to achieve high order of convergence with few Jacobian and/or functional evaluations. On the other hand, by applying the pseudocomposition technique on each proposed scheme we get to increase their order of convergence, obtaining new high-order and efficient methods. We use the classical efficiency index in order to compare the obtained schemes and make some numerical test.This research was supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02 and by FONDOCYT 2011-1-B1-33, República Dominicana.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Penkova Vassileva, M. (2013). New family of iterative methods with high order of convergence for solving nonlinear systems. En Numerical Analysis and Its Applications. Springer Verlag. 222-230. https://doi.org/10.1007/978-3-642-41515-9_23S222230Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A modified Newton-Jarratt’s composition. Numer. Algor. 55, 87–99 (2010)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: Efficient high-order methods based on golden ratio for nonlinear systems. Applied Mathematics and Computation 217(9), 4548–4556 (2011)Cordero, A., Torregrosa, J.R.: Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation 190, 686–698 (2007)Cordero, A., Torregrosa, J.R.: On interpolation variants of Newton’s method for functions of several variables. Journal of Computational and Applied Mathematics 234, 34–43 (2010)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Pseudocomposition: a technique to design predictor-corrector methods for systms of nonlinear equtaions. Applied Mathematics and Computation 218(23), 11496–11504 (2012)Nikkhah-Bahrami, M., Oftadeh, R.: An effective iterative method for computing real and complex roots of systems of nonlinear equations. Applied Mathematics and Computation 215, 1813–1820 (2009)Ostrowski, A.M.: Solutions of equations and systems of equations. Academic Press, New York (1966)Shin, B.-C., Darvishi, M.T., Kim, C.-H.: A comparison of the Newton-Krylov method with high order Newton-like methods to solve nonlinear systems. Applied Mathematics and Computation 217, 3190–3198 (2010

On CSCS-based iteration methods for Toeplitz system of weakly nonlinear equations

AbstractFor Toeplitz system of weakly nonlinear equations, by using the separability and strong dominance between the linear and the nonlinear terms and using the circulant and skew-circulant splitting (CSCS) iteration technique, we establish two nonlinear composite iteration schemes, called Picard-CSCS and nonlinear CSCS-like iteration methods, respectively. The advantage of these methods is that they do not require accurate computation and storage of Jacobian matrix, and only need to solve linear sub-systems of constant coefficient matrices. Therefore, computational workloads and computer storage may be saved in actual implementations. Theoretical analysis shows that these new iteration methods are local convergent under suitable conditions. Numerical results show that both Picard-CSCS and nonlinear CSCS-like iteration methods are feasible and effective for some cases

Nonlinear Control and Estimation with General Performance Criteria

This dissertation is concerned with nonlinear systems control and estimation with general performance criteria. The purpose of this work is to propose general design methods to provide systematic and effective design frameworks for nonlinear system control and estimation problems. First, novel State Dependent Linear Matrix Inequality control approach is proposed, which is optimally robust for model uncertainties and resilient against control feedback gain perturbations in achieving general performance criteria to secure quadratic optimality with inherent asymptotic stability property together with quadratic dissipative type of disturbance reduction. By solving a state dependent linear matrix inequality at each time step, the sufficient condition for the control solution can be found which satisfies the general performance criteria. The results of this dissertation unify existing results on nonlinear quadratic regulator, Hinfinity and positive real control. Secondly, an H2-Hinfinity State Dependent Riccati Equation controller is proposed in this dissertation. By solving the generalized State Dependent Riccati Equation, the optimal control solution not only achieves the optimal quadratic regulation performance, but also has the capability of external disturbance reduction. Numerically efficient algorithms are developed to facilitate effective computation. Thirdly, a robust multi-criteria optimal fuzzy control of nonlinear systems is proposed. To improve the optimality and robustness, optimal fuzzy control is proposed for nonlinear systems with general performance criteria. The Takagi-Sugeno fuzzy model is used as an effective tool to control nonlinear systems through fuzzy rule models. General performance criteria have been used to design the controller and the relative weighting matrices of these criteria can be achieved by choosing different coefficient matrices. The optimal control can be achieved by solving the LMI at each time step. Lastly, since any type of controller and observer is subject to actuator failures and sensors failures respectively, novel robust and resilient controllers and estimators are also proposed for nonlinear stochastic systems to address these failure problems. The effectiveness of the proposed control and estimation techniques are demonstrated by simulations of nonlinear systems: the inverted pendulum on a cart and the Lorenz chaotic system, respectively