General Nonlinear Functions Research Articles

Dynamics and Control of a Novel Discrete Internet Rumor Propagation Model in a Multilingual Environment

In the Internet age, the development of intelligent software has broken the limits of multilingual communication. Recognizing that the data collected on rumor propagation are inherently discrete, this study introduces a novel SIR discrete Internet rumor propagation model with the general nonlinear propagation function in a multilingual environment. Then, the propagation threshold R0 is obtained by the next-generation matrix method. Besides, the criteria determining the spread or demise of rumors are obtained by the stability theory of difference equations. Furthermore, combined with optimal control theory, prevention and refutation mechanisms are proposed to curb rumors. Finally, the validity and applicability of the model are demonstrated by numerical simulations and a real bilingual rumor case study.

Signal and image reconstruction with a double parameter Hager–Zhang‐type conjugate gradient method for system of nonlinear equations

AbstractThe one parameter conjugate gradient method by Hager and Zhang (Pac J Optim, 2(1):35–58, 2006) represents a family of descent iterative methods for solving large‐scale minimization problems. The nonnegative parameter of the scheme determines the weight of conjugacy and descent, and by extension, the numerical performance of the method. The scheme, however, does not converge globally for general nonlinear functions, and when the parameter approaches 0, the scheme reduces to the conjugate gradient method by Hestenes and Stiefel (J Res Nat Bur Stand, 49:409–436, 1952), which in practical sense does not perform well due to the jamming phenomenon. By carrying out eigenvalue analysis of an adaptive two parameter Hager–Zhang type method, a new scheme is presented for system of monotone nonlinear equations with its application in compressed sensing. The proposed scheme was inspired by nice attributes of the Hager–Zhang method and the various schemes designed with double parameters. The scheme is also applicable to nonsmooth nonlinear problems. Using fundamental assumptions, analysis of the global convergence of the scheme is conducted and preliminary report of numerical experiments carried out with the scheme and some recent methods indicate that the scheme is promising.

Schistosomiasis mathematical model in a spatially heterogeneous environment

Schistosomiasis is classified by WHO as a neglected tropical disease. Recent research works have shown that large-scale development projects involving massive population displacement and water irrigation, such as the construction of dams, lakes, and the development of agricultural areas, favour the proliferation of bilharzia. These observations motivate us to propose a reaction–diffusion model to assess the role of the displacements of humans, snails, cercaria, miracidia in the transmission dynamics of Schistosomiasis. The model incorporates a general non-linear contact functions and density-dependent parameters. The aim is to better understanding the role of spatial interactions on the spread of Schistosomiasis, in order to propose appropriate recommendations for the control of that silent threat. We characterize the basic reproduction number R0 of the model. The uniform persistence theory, the maximum principle are used to conduct an in-depth analysis of both the homogeneous and heterogeneous models. Theoretical results are illustrated through numerical simulations.

Global analysis for a modified SEIR model with general non-linear incidence function

Global analysis for a modified SEIR model with general non-linear incidence function

Properties of new even and odd nonlinear coherent states with different parameters

We construct a class of nonlinear coherent states (NLCSs) by introducing a more general nonlinear function and study their nonclassical properties, specifically the second-order correlation function [Formula: see text], Mandel parameter Q, squeezing, amplitude-squared squeezing and Wigner function of the optical field. The results indicate that the nonclassical properties of the new types of even and odd NLCSs crucially depend on the nonlinear functions. More concretely, we find that the new even NLCSs could exhibit the photon-bunching effect, whereas the new odd NLCSs could show the photon-antibunching effect. The degree of squeezing is also significantly affected by the parameter selection of these NLCSs. By employing various forms of nonlinear functions, it becomes possible to construct the NLCSs with diverse properties, thereby providing a theoretical foundation for the corresponding experimental investigations.

A new hybrid conjugate gradient method close to the memoryless BFGS quasi-Newton method and its application in image restoration and machine learning

<p>A new hybrid conjugate gradient algorithm for solving the unconstrained optimization problem was presented. The algorithm could be considered as a modification of the memoryless Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method. Based on a normalized gradient difference, we introduced a new combining conjugate gradient direction close to the direction of the memoryless BFGS quasi-Newton direction. It was shown that the search direction satisfied the sufficient descent property independent of the line search. For general nonlinear functions, the global convergence of the algorithm was proved under standard assumptions. Numerical experiments indicated a potential performance of the new algorithm, especially for solving the large-scale problems. In addition, the proposed method was used in practical application problems for image restoration and machine learning.</p>

A modified type of Fletcher-Reeves conjugate gradient method with its global convergence

<p><span>The conjugate gradient methods are one of the most important techniques used to address problems involving minimization or maximization, especially nonlinear optimization problems with no constraints at all. That is because of their simplicity and low memory needed. They can be applied in many areas, such as economics, engineering, neural networks, image restoration, machine learning, and deep learning. The convergence of Fletcher-Reeves (FR) conjugate gradient method has been established under both exact and strong Wolfe line searches. However, it is performance in practice is poor. In this paper, to get good numerical performance from the FR method, a little modification is done. The global convergence of the modified version has been established for general nonlinear functions. Preliminary numerical results show that the modified method is very efficient in terms of number of iterations and CPU time.</span></p>

A descent scaled conjugate gradient method for unconstrained optimization with its applications in image restoration problems

Based on combining the conjugate gradient method proposed by Hager and Zhang with the scaled gradient idea, we presented a new scaled conjugate gradient method which satisfies the sufficient descent condition. In our method, the scaled parameter is determined so that the search direction becomes close to the three-term HS method suggested by Zhang, Zhou and Li. It is proved that the new method is globally convergent for general nonlinear functions, under some standard assumptions. Numerical comparisons on some test problems from the CUTEst library and image restoration problems illustrate the efficiency and robustness of our proposed method in practice.

On the stability analysis of a fractional order epidemic model including the most general forms of nonlinear incidence and treatment function

In this paper, we propose to study a SEIR model of fractional order with an incidence and a treatment function. The incidence and treatment functions included in the model are general nonlinear functions that satisfy some meaningful biological hypotheses. Under these hypotheses, it is shown that the disease free equilibrium point of the proposed model is locally and globally asymptotically stable when the reproduction number $R_{0} $ is smaller than 1. When $ R_{0}>1 $, it is established that the endemic equilibrium of the studied system is uniformly asymptotically stable. Finally, some numerical simulations are provided to illustrate the theory.

An accelerated descent CG algorithm with clustering the eigenvalues for large-scale nonconvex unconstrained optimization and its application in image restoration problems

Conjugate gradient methods are widely used for solving large-scale unconstrained optimization problems. Since they have the attractive practical factors of simple computation and low memory requirement, interesting theoretical features of curvature information and strong global convergence property. Based on the analysis of the minimization of the condition number and the positiveness of the corresponding matrix, we propose a choice for the parameter in Dai-Liao method and design a descent conjugate gradient algorithm which owns the sufficient descent property independent of the choices for line search techniques. Under some common conditions, the global convergence property for uniformly convex function and the general nonlinear function are established. In the numerical experiments, we firstly focus on 46 ill-conditioned matrix problems and present the corresponding primal results. Then 450 large-scale unconstrained problems are referred. Finally, we give an accelerated strategy for the proposed algorithm and apply it to some image restoration problems. Numerical results indicate that the algorithm is reliable and much more efficient and effective than the other methods for the test problems.

Algorithm 1035: A Gradient-based Implementation of the Polyhedral Active Set Algorithm

The Polyhedral Active Set Algorithm (PASA) is designed to optimize a general nonlinear function over a polyhedron. Phase one of the algorithm is a nonmonotone gradient projection algorithm, while phase two is an active set algorithm that explores faces of the constraint polyhedron. A gradient-based implementation is presented, where a projected version of the conjugate gradient algorithm is employed in phase two. Asymptotically, only phase two is performed. Comparisons are given with IPOPT using polyhedral-constrained problems from CUTEst and the Maros/Meszaros quadratic programming test set.

A class of spectral three-term descent Hestenes-Stiefel conjugate gradient algorithms for large-scale unconstrained optimization and image restoration problems

Conjugate gradient methods are much effective and widely used for large-scale unconstrained optimization problems by their simple computation, low memory requirement and strong global convergence property. Spectral gradient methods are also effective for large-scale problems. In this paper, a class of new descent spectral three-term conjugate gradient algorithms are proposed which automatically have the sufficient descent property and satisfy the Dai-Liao conjugate condition. Under the Wolfe line search technique and some standard conditions, the proposed methods are globally convergent for strongly convex functions and general nonlinear functions with the help of the modified secant equations. In numerical part, 732 problems with dimensions varying from 1500 to 150000 and three image restoration problems with three noise levels are considered. Numerical results indicate that the proposed algorithms are more efficient, reliable and robust than the other methods for the testing problems.

Dynamics of a diffusion dispersal viral epidemic model with age infection in a spatially heterogeneous environment with general nonlinear function

We propose a generalization of a model with age of infection in a heterogeneous environment. Firstly, we give the well‐posedness of the model and prove that the solutions are bounded and positive. The difficult mathematical issue in this research is that the model is partially degenerate, and the solution map is not compact. In addition, we construct a global attractor of a bounded set to establish the existence of total trajectory. Moreover, we define the principal eigenvalue associated with a principal eigenvalue problem to give a relation with the basic reproduction number and this value. By assuming that , then the infection‐free steady‐states is globally asymptotically stable. Furthermore, for and by using the persistence results, we prove the existence of endemic steady‐states , and by constructing an appropriate Lyapunov function, we show that is globally asymptotically stable. Lastly, we validate our theoretical analysis by giving some numerical graphics.

Dynamics of an age-structured HIV model with general nonlinear infection rate

Abstract In this paper, the asymptotical behaviour of an age-structured Human Immunodeficiency Virus infection model with general non-linear infection function and logistic proliferation term is studied. Based on the existence of the equilibria and theory of operator semigroups, linearized stability/instability of the disease-free and endemic equilibria is investigated through the distribution of eigenvalues of the linear operator. Then persistence of the solution semi-flow of the considered system is studied by showing the existence of a global attractor and the obtained result shows that the solution semi-flow is persistent as long as the basic reproduction number $R_{0}>1$. Moreover, the Hopf bifurcations problem around the endemic equilibrium is also considered for the situation with a specific infection function. Since the system has two different delays, four cases are discussed to investigate the influence of the time delays on the dynamics of system around the endemic equilibrium including stability and Hopf bifurcations. At last, some numerical examples with concrete parameters are provided to illustrate the obtained results.

A descent modified HS conjugate gradient method with an optimal property

In this paper, by minimizing the distance between the CG direction and the direction of the improved Perry conjugate gradient method [Yao et al., Numer. Algorithms 78 (2018) 1255–1269], we propose a descent modified HS conjugate gradient method. A remarkable property of the modified HS method is that it can produce sufficient descent property, which is independent of the line search used. Under suitable conditions, we prove that the modified HS method with the standard Armijo line search is globally convergent for uniformly convex functions and the modified HS+ method with standard Wolfe line search is globally convergent for general nonlinear functions. Extensive numerical experiments show that the proposed method is efficient.

Dynamic analysis and optimal control of a reaction-diffusion rumor propagation model in multi-lingual environments

The development of networking technologies has broken down the restrictions of language and region, greatly increasing the destructive power of rumors. To explore the influence of spatial factors on rumors, a new spatial distance on online social networks is first defined by considering friendship hops and the intimacy level among friends. On that basis, a novel reaction-diffusion SIR rumor spreading model with the general nonlinear propagation function is proposed in multi-lingual environments. Firstly, the global existence and ultimate boundedness of solutions are strictly proven. Theoretical analysis further reveals the existence of equilibrium points and the global stability of the rumor-free/prevailing equilibrium points of the proposed model. Secondly, an optimal control strategy with an education mechanism and a forced silence method is introduced to better curb rumor propagation. Moreover, the sensitivity analysis of the threshold is conducted to discuss the relative importance of model parameters on rumor propagation. Finally, some necessary numerical simulations are performed to verify the validity of the theoretical analysis.

Global dynamic behavior of a vaccination–age SVIR model with treatment and general nonlinear incidence rate

In this paper, we propose a vaccination–age SVIR model with general nonlinear incidence rate and general treatment function. Firstly, we establish the existence, uniqueness and positivity of solutions for the considered SVIR model in the case where the linear part is nondensely defined and satisfies the Hille–Yosida condition by the use of the integrated semigroup theory. Secondly, based on the basic reproduction number R0, we investigate the existence and the local stability of the equilibrium points. Moreover, we discuss the global stability of the equilibrium points under the light of the Lyapunov–LaSalle approach. Finally, an application with numerical simulations is done to check the validity of our theoretical results and to exemplify the impact of the treatment and the vaccination on the disease propagation.

Global stability of a general HTLV-I infection model with Cytotoxic T-Lymphocyte immune response and mitotic transmission

Mathematical models have been considered as a robust tool to support biological and medical studies of human viral infections. The global stability of viral infection models remains an important and largely open research problem. Such results are necessary to evaluate treatment strategies for infections and to establish thresholds for treatment rates. Human T-lymphotropic virus class I (HTLV-I) is a retrovirus which infects the CD4+T cells and causes chronic and deadly diseases. In this article, we developed a general nonlinear system of ODEs which describes the within-host dynamics of HTLV-I under the effect Cytotoxic T-Lymphocytes (CTLs) immunity. The mitotic division of actively infected cells are modeled. We consider general nonlinear functions for the generation, proliferation and clearancerates for all types of cells. The incidence rate of infection is also modeled by a general nonlinear function. These general functions are assumed to satisfy a set of suitable conditions and include several forms presented in the literature. We determine a bounded domain for the system’s solutions. We prove the existence of the system’s equilibrium points and determine two threshold numbers, the basic reproductive number R0 and the CTL immunity stimulation number R1. We establish the global stability of all equilibrium points by constructing Lyapunov function and applying Lyapunov-LaSalle asymptotic stability theorem. Under certain conditions it is shown that ifR0≤1, then the infection-free equilibrium point is globally asymptotically stable (GAS) and the HTLV-I infection is cleared. If R1≤1<R0, then the infected equilibrium point without CTL immunity is GAS and the HTLV-I infection becomes chronic with no sustained CTL immune response. If R1>1, then the infected equilibrium point with CTL immunity is GAS and the infection becomes chronic with persistent CTL immune response. We present numerical simulations for the system by choosing special shapes of the general functions. The effect of Crowley-Martin functional response and mitotic division of actively infected cells on the HTLV-I progression are studied. Our results cover and improve several ones presented in the literature.

Maximum Spectral Measures of Risk with Given Risk Factor Marginal Distributions

We consider the problem of determining an upper bound for the value of a spectral risk measure of a loss that is a general nonlinear function of two factors whose marginal distributions are known but whose joint distribution is unknown. The factors may take values in complete separable metric spaces. We introduce the notion of Maximum Spectral Measure (MSM), as a worst-case spectral risk measure of the loss with respect to the dependence between the factors. The MSM admits a formulation as a solution to an optimization problem that has the same constraint set as the optimal transport problem but with a more general objective function. We present results analogous to the Kantorovich duality, and we investigate the continuity properties of the optimal value function and optimal solution set with respect to perturbation of the marginal distributions. Additionally, we provide an asymptotic result characterizing the limiting distribution of the optimal value function when the factor distributions are simulated from finite sample spaces. The special case of Expected Shortfall and the resulting Maximum Expected Shortfall is also examined. Funding: M. Ghossoub and D. Saunders acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada in the form of Discovery Grants [Grants 2018-03961 and 2017-04220, respectively].

Global convergence of a modified spectral three-term CG algorithm for nonconvex unconstrained optimization problems

Spectral conjugate gradient methods are an efficient family for solving unconstrained optimization problems that have been widely studied in recent decades. In this regard, Li et al. (2019) proposed a spectral three-term conjugate gradient method and proved the global convergence of this algorithm for uniformly convex functions. Our main motivation in this paper is to develop the convergence properties of this method such that the new method possesses suitable convergence for general nonlinear functions. To do end, we introduce a modified spectral conjugate gradient method based on the CG method by Li et al. (2019). We show that the new method fulfills the sufficient descent property without any line search. The new algorithm is globally convergent for general nonlinear functions without the convexity assumption on the objective function. The numerical results indicate that the behavior of the new algorithm is not only effective, but also promising versus other conjugate gradient methods dealing with unconstrained optimization problems of the CUTEst library.