Discrete-time ZD, GD and NI for solving nonlinear time-varying equations

A special class of neural dynamics called Zhang dynamics (ZD), which is different from gradient dynamics (GD), has recently been proposed, generalized, and investigated for solving time-varying problems by following Zhang et al.’s design method. In view of potential digital hardware implemetation, discrete-time ZD (DTZD) models are proposed and investigated in this paper for solving nonlinear time-varying equations in the form of $f(x,t)=0$ . For comparative purposes, the discrete-time GD (DTGD) model and Newton iteration (NI) are also presented for solving such nonlinear time-varying equations. Numerical examples and results demonstrate the efficacy and superiority of the proposed DTZD models for solving nonlinear time-varying equations, as compared with the DTGD model and NI.     

Different Zhang functions leading to different Zhang-dynamics models illustrated via time-varying reciprocal solving

Along with neural dynamics (based on analog solvers) widely arising in scientific computation and optimization fields in recent decades which attracts extensive interest and investigation of researchers, a novel type of neural dynamics, called Zhang dynamics (ZD), has been formally proposed by Zhang et al. for the online solution of time-varying problems. By following Zhang et al.’s neural-dynamics design method, the ZD model, which is based on an indefinite Zhang function (ZF), can guarantee the exponential convergence performance for the online time-varying problems solving. In this paper, different indefinite Zhang functions, which can lead to different ZD models, are proposed and developed as the error-monitoring functions for the time-varying reciprocal problem solving. Additionally, for the goal of developing the floating-point processors or coprocessors for the future generation of computers, the MATLAB Simulink modeling and simulative verifications of such different ZD models are further presented for online time-varying reciprocal solving. The modeling results substantiate the efficacy of such different ZD models for time-varying reciprocal solving.     

Design and analysis of two discrete-time ZD algorithms for time-varying nonlinear minimization

Nonlinear minimization, as a subcase of nonlinear optimization, is an important issue in the research of various intelligent systems. Recently, Zhang et al. developed the continuous-time and discrete-time forms of Zhang dynamics (ZD) for time-varying nonlinear minimization. Based on this previous work, another two discrete-time ZD (DTZD) algorithms are proposed and investigated in this paper. Specifically, the resultant DTZD algorithms are developed for time-varying nonlinear minimization by utilizing two different types of Taylor-type difference rules. Theoretically, each steady-state residual error in the DTZD algorithm changes in an O(τ ³) manner with τ being the sampling gap. Comparative numerical results are presented to further substantiate the efficacy and superiority of the proposed DTZD algorithms for time-varying nonlinear minimization.     

Continuous and discrete Zhang dynamics for real-time varying nonlinear optimization

Online solution of time-varying nonlinear optimization problems is considered an important issue in the fields of scientific and engineering research. In this study, the continuous-time derivative (CTD) model and two gradient dynamics (GD) models are developed for real-time varying nonlinear optimization (RTVNO). A continuous-time Zhang dynamics (CTZD) model is then generalized and investigated for RTVNO to remedy the weaknesses of CTD and GD models. For possible digital hardware realization, a discrete-time Zhang dynamics (DTZD) model, which can be further reduced to Newton-Raphson iteration (NRI), is also proposed and developed. Theoretical analyses indicate that the residual error of the CTZD model has an exponential convergence, and that the maximum steady-state residual error (MSSRE) of the DTZD model has an O(τ²) pattern with τ denoting the sampling gap. Simulation and numerical results further illustrate the efficacy and advantages of the proposed CTZD and DTZD models for RTVNO.     

A new smoothing and regularization Newton method for <Emphasis Type="Italic">P</Emphasis><Subscript>0</Subscript>-NCP

The nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the solution of a nonsmooth system of equations. In this paper, we propose a new smoothing and regularization Newton method for solving nonlinear complementarity problem with P ₀-function (P ₀-NCP). Without requiring strict complementarity assumption at the P ₀-NCP solution, the proposed algorithm is proved to be convergent globally and superlinearly under suitable assumptions. Furthermore, the algorithm has local quadratic convergence under mild conditions. Numerical experiments indicate that the proposed method is quite effective. In addition, in this paper, the regularization parameter ε in our algorithm is viewed as an independent variable, hence, our algorithm seems to be simpler and more easily implemented compared to many previous methods.     

Filtration-consistent nonlinear expectations and related <Emphasis Type="Italic">g</Emphasis>-expectations

 From a general definition of nonlinear expectations, viewed as operators preserving monotonicity and constants, we derive, under rather general assumptions, the notions of conditional nonlinear expectation and nonlinear martingale. We prove that any such nonlinear martingale can be represented as the solution of a backward stochastic equation, and in particular admits continuous paths. In other words, it is a g-martingale. Received: 2 February 2000 / Revised version: 1 June 2001 / Published online: 13 May 2002     

A new technique of using homotopy analysis method for solving high‐order nonlinear differential equations

In this paper, a new technique of homotopy analysis method (HAM) is proposed for solving high‐order nonlinear initial value problems. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM) by transform the nth‐order nonlinear differential equation to a system of n first‐order equations. Second‐ and third‐ order problems are solved as illustration examples of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal tactics for close support operations: Part IV,perfect intelligence and communications

A new generation ofC ³ (command, control, and communication) models for military cybernetics is developed. Recursive equations for the solution of theC ³ problem are derived for an amphibious campaign with linear, time-varying dynamics. Air and ground commanders are assumed to have perfect intelligence and perfect communications. Numerical results are given for the optimal decision rules.     

Maximum principle and existence of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms

We study L^p-viscosity solutions of fully nonlinear, second-order, uniformly elliptic partial differential equations (PDE) with measurable terms and quadratic nonlinearity. We present a sufficient condition under which the maximum principle holds for L^p-viscosity solution. We also prove stability and existence results for the equations under consideration.     

The Knizhnik-Zamolodchikov equations associated with the root system <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

At the end of the twentieth century, in mathematical physics, the Knizhnik-Zamolodchikov equations for the root systems A _n and their generalizations for the root systems of types B _n , C _n , and D were constructed. For the root system of type G ₂, the vector version of the Knizhnik-Zamolodchikov equations was obtained by M. P. Zamakhovskii and V. P. Leksin. However, the tensor version of these equations has remained unstudied. In this paper, the Knizhnik-Zamolodchikov equations associated with the root system of type G ₂ are considered. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Local behavior of an iterative framework for generalized equations with nonisolated solutions

 An iterative framework for solving generalized equations with nonisolated solutions is presented. For generalized equations with the structure , where is a multifunction and F is single-valued, the framework covers methods that, at each step, solve subproblems of the type . The multifunction approximates F around s. Besides a condition on the quality of this approximation, two other basic assumptions are employed to show Q-superlinear or Q-quadratic convergence of the iterates to a solution. A key assumption is the upper Lipschitz-continuity of the solution set map of the perturbed generalized equation . Moreover, the solvability of the subproblems is required. Conditions that ensure these assumptions are discussed in general and by means of several applications. They include monotone mixed complementarity problems, Karush-Kuhn-Tucker systems arising from nonlinear programs, and nonlinear equations. Particular results deal with error bounds and upper Lipschitz-continuity properties for these problems. Received: November 2001 / Accepted: November 2002 Published online: December 9, 2002 Key Words. generalized equation – nonisolated solutions – Newton's method – superlinear convergence – upper Lipschitz-continuity – mixed complementarity problem – error bounds Mathematics Subject Classification (1991): 90C30, 65K05, 90C31, 90C33     

Two‐Grid method for nonlinear parabolic equations by expanded mixed finite element methods

In this article, we develop a two‐grid algorithm for nonlinear reaction diffusion equation (with nonlinear compressibility coefficient) discretized by expanded mixed finite element method. The key point is to use two‐grid scheme to linearize the nonlinear term in the equations. The main procedure of the algorithm is solving a small‐scaled nonlinear equations on the coarse grid and dealing with a linearized system on the fine space using the Newton iteration with the coarse grid solution. Error estimation to the expanded mixed finite element solution is analyzed in detail. We also show that two‐grid solution achieves the same accuracy as long as the mesh sizes satisfy H = O(h^1/2). Two numerical experiments are given to verify the effectiveness of the algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Exact solutions of nonlinear Schrödinger equation by using symbolic computation

The (G′/G,1/G)‐expansion method and (1/G′)‐expansion method are interesting approaches to find new and more general exact solutions to the nonlinear evolution equations. In this paper, these methods are applied to construct new exact travelling wave solutions of nonlinear Schrödinger equation. The travelling wave solutions are expressed by hyperbolic functions, trigonometric functions and rational functions. It is shown that the proposed methods provide a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

In this paper, an effective numerical approach based on a new two‐dimensional hybrid of parabolic and block‐pulse functions (2D‐PBPFs) is presented for solving nonlinear partial quadratic integro‐differential equations of fractional order. Our approach is based on 2D‐PBPFs operational matrix method together with the fractional integral operator, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro‐differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h³) . The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier.     

On some exact solutions in <Emphasis Type="Italic">p</Emphasis>-adic open-closed string theory

The paper is concerned with the construction of exact solutions for nonlinear pseudodifferential equations which describe tachyon dynamics of open-closed p-adic strings. Existence of continuous solutions and their properties are discussed.     

A new non-interior continuation method for <Emphasis Type="Italic">P</Emphasis><Subscript>0</Subscript>-NCP based on a SSPM-function

In this paper, we consider a new non-interior continuation method for the solution of nonlinear complementarity problem with P ₀-function (P ₀-NCP). The proposed algorithm is based on a smoothing symmetric perturbed minimum function (SSPM-function), and one only needs to solve one system of linear equations and to perform only one Armijo-type line search at each iteration. The method is proved to possess global and local convergence under weaker conditions. Preliminary numerical results indicate that the algorithm is effective.     

Adaptive numerical components for PDE-based simulations

Numerical simulations based on nonlinear partial differential equations (PDEs) using Newton-based methods require the solution of large, sparse linear systems of equations at each nonlinear iteration. Typically in large-scale parallel simulations such linear systems are solved by using preconditioned Krylov methods. In many cases, especially in time-dependent problems, the attributes of the linear systems can change throughout the stimulation, potentially leading to varying times for solving the linear systems during different nonlinear iterations. We present an approach to characterizing the nonlinear and linear system solution and using the resulting application performance information to dynamically select linear solver methods, with the goal of reducing the total time to solution. We discuss the effect of these adaptive heuristics on fluid dynamics and radiation transport codes. We also introduce general component infrastructure to support dynamic algorithm selection and adaptation in applications involving the solution of nonlinear PDEs. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A unified approach to global convergence of trust region methods for nonsmooth optimization

This paper investigates the global convergence of trust region (TR) methods for solving nonsmooth minimization problems. For a class of nonsmooth objective functions called regular functions, conditions are found on the TR local models that imply three fundamental convergence properties. These conditions are shown to be satisfied by appropriate forms of Fletcher's TR method for solving constrained optimization problems, Powell and Yuan's TR method for solving nonlinear fitting problems, Zhang, Kim and Lasdon's successive linear programming method for solving constrained problems, Duff, Nocedal and Reid's TR method for solving systems of nonlinear equations, and El Hallabi and Tapia's TR method for solving systems of nonlinear equations. Thus our results can be viewed as a unified convergence theory for TR methods for nonsmooth problems.Research supported by AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Corresponding author.     

Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth‐order integro‐differential equation

This paper presents general framework for solving the nth‐order integro‐differential equation using homotopy analysis method (HAM) and optimal homotopy asymptotic method (OHAM). OHAM is parameter free and can provide better accuracy over the HAM at the same order of approximation. Furthermore, in OHAM the convergence region can be easily adjusted and controlled. Comparison, via two examples, between our solution using HAM and OHAM and the exact solution shows that the HAM and the OHAM are effective and accurate in solving the nth‐order integro‐differential equation. Copyright © 2011 John Wiley & Sons, Ltd.     

Inexact implicit methods for monotone general variational inequalities

Solving a variational inequality problem is equivalent to finding a solution of a system of nonsmooth equations. Recently, we proposed an implicit method, which solves monotone variational inequality problem via solving a series of systems of nonlinear smooth (whenever the operator is smooth) equations. It can exploit the facilities of the classical Newton–like methods for smooth equations. In this paper, we extend the method to solve a class of general variational inequality problems Moreover, we improve the implicit method to allow inexact solutions of the systems of nonlinear equations at each iteration. The method is shown to preserve the same convergence properties as the original implicit method. Received July 31, 1995 / Revised version received January 15, 1999? Published online May 28, 1999