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The main focus is to realize the on-line parameter estimation of the nonlinear fractional-order model with colored noises in this paper. Firstly, an auxiliary model gradient descent algorithm is derived to synchronously produce the estimates of the parameters, including the fractional-order. In order to decrease the noise interference, the filtering-based gradient descent estimation framework by constructing a linear filter provides a feasible method. Furthermore, the forgetting factor is introduced to the proposed filtering-based algorithm for the sake of improving the convergence rate. Comparative simulation results demonstrate the proposed effectiveness and high approximation accuracy of the proposed algorithms. • An auxiliary model is built to obtain the estimates of these unmeasurable variables. • The proposed auxiliary model gradient algorithm can realize on-line identification. • A filtering-based gradient algorithm is proposed to reduce the colored noise impact. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new numerical simulation iterative algorithm to solve the split feasibility problem and the fixed point problem with demicontractive mappings. Our algorithm mainly involves primal-dual iterative, relaxed projection, inertial technique and self-adaptive step size. Under reasonable conditions, the strong convergence of our algorithm is established. Moreover, we provide some numerical simulation examples to demonstrate the efficiency of our iterative algorithm compared to existing algorithms in the other literature. [ABSTRACT FROM AUTHOR]

In this paper we are interested in gradient flow type methods for computing the ground state of nonlinear Schrödinger / Gross–Pitaevskii equations. Fractional Normalized Gradient Flow methods, involving fractional derivatives and generalizing the celebrated Normalized Gradient Flow method (see Bao and Du, 2004) are derived and analyzed. Several experiments are proposed to illustrate some convergence properties of the developed algorithms. • Derivation of novel efficient fractional normalized gradient flow algorithms. • Analysis of the algorithms at the PDE-level. • Exhaustive numerical study and comparisons. [ABSTRACT FROM AUTHOR]

In this paper, a numerical algorithm to compute the first eigenvalue and the corresponding eigenfunction of the p -Laplacian eigenvalue problem with Robin boundary conditions is developed. The algorithm generates a sequence of numbers and functions. It is established that the sequence of numbers is decreasing and converges to the first eigenvalue, while the sequence of functions converges to the corresponding eigenfunction. The method is easier to implement in comparison with the current methods.The order of convergence for the algorithm is investigated. Several numerical experiments reveal the robustness and the efficacy of the algorithm for domains of various geometries. Our results provide deeper insights for the limiting cases p → 1 and ∞. • A numerical algorithm to principle eigenvalue of p-Laplacian problem is developed. • We consider the eigenvalue problem with Robin boundary conditions. • It is proved that the method is convergent. • The method is easier to implement in comparison with the current methods. • The results provide deeper insights for the limiting cases p → 1 and p → ∞. [ABSTRACT FROM AUTHOR]

In this paper we apply Cimmino algorithm for common fixed point problems with a countable family of quasi-nonexpansive operators in an arbitrary normed space and show its convergence. Our results are an extension of the recent results by T. Y. Kong , H. Pajoohesh and G. T. Herman obtained for operators which are projections on convex closed sets in a finite-dimensional Euclidean space. • The study of common fixed point problems with a countable family of operators. • The study of common fixed point problems in infinite dimensional spaces. • The study of common fixed point problems with quasi-nonexpansive maps. [ABSTRACT FROM AUTHOR]

In this paper, an accelerated subgradient extragradient algorithm with a new non-monotonic step size is proposed to solve bilevel variational inequality problems involving non-Lipschitz continuous operator in Hilbert spaces. The proposed algorithm with a new non-monotonic step size has the advantage of requiring only one projection onto the feasible set during each iteration and does not require prior knowledge of the Lipschitz constant of the mapping involved. Under suitable and weaker conditions, the proposed algorithm achieves strong convergence. Some numerical tests are provided to demonstrate the efficiency and advantages of the proposed algorithm against existing related algorithms. • We studied the variational inequality problem with a variational inequality constraint (BVIP). • The proposed Algorithm 3.1 uses a new non-monotonic step size criteria. • The strong convergence theorem of the proposed algorithm is established under weaker conditions. • Numerical simulation examples are given to show the application and the benefits of algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we construct two kinds of exponential SAV approach with relaxation (R-ESAV) for dissipative system. The constructed schemes are linear and unconditionally energy stable. They can guarantee the positive property of SAV without any assumption compared with R-SAV and R-GSAV approaches, preserve all the advantages of the ESAV approach and satisfy dissipation law with respect to a modified energy which is directly related to the original free energy. We also give the rigorous consistency estimates of the constructed schemes for the L 2 gradient flows. Moreover, the second version of R-ESAV approach is easy to construct high-order BDF k schemes. Especially for Navier–Stokes equations, we construct two kinds of novel schemes based on the R-ESAV method. Finally, ample numerical examples are presented to exhibit that the proposed approaches are accurate and effective. • R-ESAV schemes can improve the accuracy of the solution significantly compared with the original ESAV approach. • We construct two kinds of novel schemes based on the R-ESAV method for Navier–Stokes equations. • Only one linear system with constant coefficients at each time step needs to be solved. • The positive property of SAV without any assumption can be guaranteed. • Numerical simulations are tested to show that the modified energy equals to the original free energy at almost all times. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with two-grid fully discrete finite element algorithms for solving the nonlinear multi-term time-fractional diffusion equation with a variable coefficient. Taking into account the weak singularity of the solution at the initial time t = 0 , we employ graded meshes in time, which is an effective method to compensate for the lack of smoothness. A fully discrete implicit finite element scheme is established with the L 1 method on temporally graded meshes; the stability and optimal error estimates are theoretically proved. To improve the computational efficiency, we propose a two-grid algorithm based on the fully discrete scheme, and then the second algorithm is constructed to further improve the optimal convergence order. Similar properties as standard FEM are presented for the two algorithms. A significant fact is that this technique reduces the computation time without loss of numerical precision. Several numerical examples are provided to support our theoretical findings and verify the performance of the proposed algorithms. • A scheme is established with the L 1 method on temporally graded meshes. • A two-grid fully discrete finite element algorithm is constructed and analyzed. • The second two-grid algorithm is proposed to improve the optimal convergence order. [ABSTRACT FROM AUTHOR]

• Pattern visualization of the zeta-function zeros. • Multidimensional scaling analysis using Lorentzian metric. • Periodical patterns in the scope of Riemann hypothesis. In this paper, we analyze the nontrivial zeros of the Riemann zeta-function using the multidimensional scaling (MDS) algorithm and computational visualization features. The nontrivial zeros of the Riemann zeta-function as well as the vectors with several neighboring zeros are interpreted as the basic elements (points or objects) of a data set. Then we employ a variety of different metrics, such as the Jeffreys and Lorentzian ones, to calculate the distances between the objects. The set of the calculated distances is then processed by the MDS algorithm that produces the loci, organized according to the objects features. Then they are analyzed from the perspective of the emerging patterns. Surprisingly, in the case of the Lorentzian metric, this procedure leads to the very clear periodical structures both in the case of the objects in form of the single nontrivial zeros of the Riemann zeta-function and in the case of the vectors with a given number of neighboring zeros. The other tested metrics do not produce such periodical structures, but rather chaotic ones. In this paper, we restrict ourselves to numerical experiments and the visualization of the produced results. An analytical explanation of the obtained periodical structures is an open problem worth for investigation by the experts in the analytical number theory. [ABSTRACT FROM AUTHOR]

The primary purpose of this paper is to develop and analyze two fast and conservative algorithms for the fractional sine-Gordon equation. The numerical schemes are derived by using the second-and fourth-order difference method in space and the implicit midpoint rule in time to approximate an equivalent system obtained via the energy quadratic method. In addition, the conservation, existence and uniqueness, and convergence of the two schemes are investigated. Based on the properties of the Toeplitz matrix, a fast algorithm is given in the calculation for the proposed schemes. Numerical experiments verify the theoretical analysis of the schemes, showing their efficiency and excellent behavior. • We propose the Hamiltonian formulation of the 2D fractional sine-Gordon equation. • Two linearly implicit energy-preserving schemes are developed for the 2D fractional sine-Gordon equation. • An optimal l ∞ -error estimate for the proposed scheme is established without any restriction on the grid ratio. • A fast algorithm based on the fast Fourier transformation technique is given. [ABSTRACT FROM AUTHOR]

Artificial neural network (ANN) have shown great success in various scientific fields over several decades. Recently, one of its variants known as deep feedforward neural network (FNN) led to dramatic improvement in many tasks, including getting more accurate approximation solution for integer-order differential equations. However, its capability on solving FDEs is still remain questionable. Thus, this paper aims to design new scheme based on deep feedforward neural network (FNN) with vectorized algorithm (FNNVA) using selected first-order optimization techniques which are gradient descent (GD), momentum method (MM) and adaptive moment estimation method (Adam) to solve Caputo FDEs. At the first stage, a detailed method formulations on solving Caputo FDEs using FNN are presented. Then, a vectorized algorithm is developed for the scheme to be computationally efficient. The effectiveness and applicability of the scheme were validate on linear and nonlinear FDEs through comparison based on different number of hidden layers with varying learning rates and number of neurons. The results show that FNNVA with Adam technique with one and two hidden layers outperformed among others by appropriate selection value of learning rates and number of neurons respectively. This scheme also provide high accuracy and low computational cost compared to several existing numerical methods. • To design new scheme based on deep feedforward neural network (FNN) with vectorized algorithm (FNNVA). • To propose a detailed method formulations on solving Caputo FDEs using FNN are presented. • A vectorized algorithm is developed for the scheme to be computationally efficient. • The effectiveness and applicability of the scheme were validated on linear and nonlinear FDEs. [ABSTRACT FROM AUTHOR]

In this paper, we propose a deep learning method to solve high-dimensional optimal multiple stopping problems. We represent the policies of multiple stopping problems by the composition of functions. Using the new representation, we approximate the optimal stopping policy recursively with simulation samples. We also derive lower and upper bounds and confidence intervals for the values. Finally, we apply the algorithm to the pricing of swing options, and it produces accurate results in high-dimensional problems. • A deep learning-based algorithm for optimal multiple stopping problems is introduced. • Lower bounds and upper bounds for the optimal value are constructed. • Applications to high-dimensional swing options are considered. [ABSTRACT FROM AUTHOR]

In this paper, without the co-coerciveness assumption of the associated mappings, we show the iterative sequences generated by our new algorithms converge strongly to a solution of the splitting monotone variational inclusion problem in infinite dimensional real Hilbert spaces. The step sizes of the new algorithms are updated per iteration by a simple calculation without knowing the prior information of the operator norm. Some parameters are relaxed to enlarge the value range of the corresponding step sizes. Double inertial extrapolation steps are incorporated in the new algorithms to accelerate their convergence speed. As applications, the split variational inclusion problem, the monotone inclusion problem and the split variational inequality problem are studied. Some numerical experiments are performed to illustrate the computation efficiency of the new algorithms. • The new methods with double inertial steps and their strong convergence are given. • The new methods own self-adaptive step sizes. • Some parameters are relaxed to enlarge the value range of the step sizes. • Numerical experiments show that the proposed algorithms are more effective. [ABSTRACT FROM AUTHOR]

In the Social Internet of Things (SIoT), individuals often acquire and share information through search engines (SEs), which is undoubtedly conducive to the rapid diffusion of information. Unfortunately, for this reason, computer viruses will also have another way of transmission. Different from the previous work that explores virus spreading over SIoT from the perspective of network structure, this paper studies from the perspective of dynamics. The proposed dynamical model theoretically analyzes the combined impact of SEs and hierarchical individual-awareness on the viral spread and reveals the long-term propagation behavior of computer viruses. These theoretical results are also fully verified in experiments on the real Facebook and Peer-to-Peer (P2P) datasets under the designed algorithm. • A hierarchical individual-awareness dynamical model is proposed. • A new algorithm is designed based on traversal rules of virus spreading in SIoT. • The proposed model is validated on real Facebook and P2P datasets. [ABSTRACT FROM AUTHOR]

This paper develops a two-level fourth-order scheme for solving time-fractional convection–diffusion–reaction equation with variable coefficients subjects to suitable initial and boundary conditions. The basis properties of the new approach are investigated and both stability and error estimates of the proposed numerical scheme are deeply analyzed in the L ∞ (0 , T ; L 2) -norm. The theory indicates that the method is unconditionally stable with convergence of order O (k 2 − λ 2 + h 4) , where k and h are time step and mesh size, respectively, and λ ∈ (0 , 1). This result suggests that the two-level fourth-order procedure is more efficient than a large class of numerical techniques widely studied in the literature for the considered problem. Some numerical evidences are provided to verify the unconditional stability and convergence rate of the proposed algorithm. • Detailed description of a two-level fourth-order approach for time-fractional problem. • Proof of basis properties of the proposed approach. • Proof of unconditional stability and convergence rate of the new algorithm. • Some numerical experiments. [ABSTRACT FROM AUTHOR]

We present results towards a constructive approach to show the existence of quasi-periodic solutions in non-perturbative regimes of some dissipative systems, called conformally symplectic systems. Finding a quasi-periodic solution of conformally symplectic systems with fixed frequency requires to choose a parameter, called the drift parameter. The first step of the strategy is to establish a very explicit quantitative theorem in an a-posteriori format as in Calleja et al. (2013). A-posteriori theorems show that if we can find an approximate solution of an invariance equation, which is sufficiently approximate with respect to some condition numbers (algebraic expressions of derivatives of the approximate solution and estimates on the derivatives of the map), then there is a true solution. The second step in the strategy is to produce numerically a very accurate solution of the invariance equation (discretizations with 2 18 Fourier coefficients, each one computed with 100 digits of precision). The third step is to compute in a concrete example, the dissipative standard map, the condition numbers and verify numerically the conditions of the theorem in the approximate solutions. For some families which have been studied numerically, the results agree with three figures with the best numerical values. We point out however that the numerical methods developed here work also in examples which have not been accessible to other more conventional methods. The verification of the estimates presented here is not completely rigorous, since we do not control the round-off error, nor the truncation error of several operations in Fourier space. We hope that the positive step taken in this paper will stimulate the complete computer-assisted proof. Making explicit the condition numbers and verifying the conditions (even in an incomplete way) will be valuable for the computation close to breakdown. We make available the approximate solutions, the highly efficient algorithm (quadratic convergence, low storage requirements, low operation count per step) to compute them (incorporating high precision based on the MPFR library) and the routines used to verify the applicability of the theorem. • Existence of quasi-periodic solutions in conformally symplectic systems. • An explicit quantitative KAM theorem in an a-posteriori format. • KAM estimates for the dissipative standard map. [ABSTRACT FROM AUTHOR]

Conventional sliding-modes based differentiators make it possible to estimate successive derivatives of a given time-varying signal in finite-time and with exact convergence in noise free case. In general, the convergence time is an unbounded increasing function of initial estimation errors. Most already proposed solutions guarantee a convergence in a maximum time independent of initial conditions. In this paper, novel sliding mode differentiators with a prescribed convergence time are proposed. The convergence time can be chosen arbitrary whatever large initial estimation errors. The proposed key solution is based on a time-dependent transformation using modulating functions which make it possible to cancel the effect of initial conditions on the convergence time. New arbitrary order differentiators including the super-twisting algorithm based on modulating functions are introduced. Lyapunov functions and homogeneity tools are used to prove the convergence of the proposed first-order and arbitrary order differentiators, respectively. Robustness with respect to measurement noise is also addressed. • Development of innovative sliding-modes-based arbitrary order differentiators. • The predefined convergence time is chosen whatever initial estimation errors. • Time dependant transformation is used to annihilate initial estimation errors. • Convergence is based on Lyapunov theory and homogeneity properties. [ABSTRACT FROM AUTHOR]

• Both conforming and fully non-conforming VEMs are devised to solve an obstacle problem of Kirchhoff-Love plate. • Optimal order error estimates are derived in the discrete energy norm, under certain solution regularity assumptions. • The primal-dual active algorithm is applied to solve the discrete problems and the numerical results are in agreement with the theoretical analysis. This paper is devoted to the numerical solution of a fourth-order elliptic variational inequality of the first kind by the virtual element method (VEM). The variational inequality models an obstacle problem for the Kirchhoff-Love plate. Both conforming and fully nonconforming VEMs are studied to solve the fourth-order elliptic variational inequality. Optimal order error estimates are derived in the discrete energy norm, under certain solution regularity assumptions. The primal-dual active algorithm is applied to solve the discrete problems. Numerical examples are reported to show the performance of the numerical methods and to illustrate the convergence orders of the numerical solutions. [ABSTRACT FROM AUTHOR]

• A new algorithm (CS-ESN) is proposed to optimize the ESN reserve pool using compressed sensing to improve the prediction speed and accuracy. • We compare and analyze ESN model and CS-ESN model. • We simulate the ESN model and CS-ESN model with chaotic data and stock data, and analyze simulation results. Echo State Network (ESN) is often used for time series prediction and chaotic series prediction. But in the training process of ESN model, the reserve pool will involve a large number of calculations, so the reserve pool may have node redundancy, resulting in inaccurate training model. It is very important to select the active nodes in the reserve pool because the predicted nodes are very few compared with all the nodes in the reserve pool. Based on the characteristics of the original vector, compressed sensing can reduce the dimension of the vector so as to obtain the activated nodes. In this paper, the ESN model is added with compressed sensing, which simplifies the reserve pool part of the ESN model. The simulation results show that our method can not only reduce the calculation amount of training effectively, but also improve the accuracy. [ABSTRACT FROM AUTHOR]

• The exponential synchronization of coupled reaction-diffusion stochastic neural networks with time-varying delay is studied. • Two new delay-dependent IPC mechanisms are presented, where distributed and discrete TVDs are both considered. • Several new sufficient criteria under the developed control strategies are established. • Two numerical examples are provided to verify the feasibility and effectiveness of the achieved theoretical results. This paper studies the issue of the exponential synchronization for coupled reaction-diffusion stochastic neural networks with time-varying delay (TVD). Two new delay-dependent impulsive pinning control mechanisms are presented, where distributed and discrete TVDs are both considered. Through utilizing the Lyapunov function approach, several sufficient criteria under the developed control strategies are established. Our results display that exponential synchronization of the coupled reaction-diffusion stochastic neural networks can be realized via controlling a small number of the network nodes with delayed impulses. Then, the effectiveness and feasibility of the proposed method are demonstrated by several numerical examples. [ABSTRACT FROM AUTHOR]