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In this paper, we introduce an iteration process to approximate a fixed point of a contractive self-mapping. The comparison theorem indicates that our iteration process is faster than the other existing iteration processes in the literature. We also obtain convergence and stability theorems of this iterative process for a contractive self-mapping. Numerical examples show that our iteration process for approximating a fixed point of a contractive self-mapping is faster than the existing methods. Based on this process, we finally present a new modified Newton-Raphson method for finding the roots of a function and generate some nice polynomiographs.

The traditional least squares (LS) and gradient descent (GD) algorithms can estimate the parameters of the regression models. They can be inefficient when the models have complex structures: (1) the unknown parameters in the information vector make the algorithm be impossible to update the parameters; (2) the zigzagging nature of the gradient descent algorithm and the complex structures lead to slow convergence rates; and (3) the step-size and derivative function calculations may be unsolvable for complex nonlinear models. This paper proposes two kinds of algorithms for exponential nonlinear models. The first is the two-stage algorithm, which decomposes the complex model into a linear part and a nonlinear part, where the linear part is estimated using the LS algorithm and the nonlinear part is identified based on the GD algorithm. The second is the particle swarm optimization algorithm which can simultaneously obtain all the parameters. To increase the convergence rates, the Aitken method is also introduced. The simulation results demonstrate the effectiveness of the proposed algorithms.

This article deals with an efficient numerical technique to solve a class of multi-term time fractional Volterra–Fredholm partial integro-differential equations of first kind. The fractional derivatives are defined in Caputo sense. The Adomian decomposition method is used to construct the scheme. For simplicity of the analysis, the model problem is converted into a multi-term time fractional Volterra–Fredholm partial integro-differential equation of second kind. In addition, the convergence analysis and the condition for existence and uniqueness of the solution are provided. Several numerical examples are illustrated in support of the theoretical analysis.

In this work, we construct two numerical approximations methods to deal with approximations of a linear Fredholm integro-differential equation. In order to show the existence and uniqueness of the solution we start by the reformulation of the integro-differential equation to a system of integral equations. We explain the general framework of the projection method which helps to clarify the basic ideas of the Collocation and Kantorovich methods. We introduce a new iterative method to avoid inversing the block operator matrix. Next, we apply the iterative projection methods and we present theorems to show the convergence of the constructed solutions to the exact solution. Finally, to observe the error behavior of the two methods, we give two numerical examples.

This paper aims to establish higher order convergence of the (inexact) Newton’s method for solving generalized equations composed of the sum of a single-valued mapping and a set-valued mapping between arbitrary Banach spaces without Lipschitz conditions. Imposing Holder calmness property on the gradient of the single-valued mapping instead of Lipschitz continuity, by virtue of the contraction mapping principle, we establish exact relationship between the order of calmness for the gradient and the order of local convergence for the (inexact) Newton’s method. Furthermore, we extend the obtained results to a restricted version of the Newton’s method, which ensures that every sequence generated by this method converges to a solution of the generalized equation. Numerical examples are provided to illustrate the theoretical results.

The purpose of this paper is to present an inexact version of the scaled gradient projection method on a convex set, which is inexact in two sense. First, an inexact projection on the feasible set is computed, allowing for an appropriate relative error tolerance. Second, an inexact non-monotone line search scheme is employed to compute a step size which defines the next iteration. It is shown that the proposed method has similar asymptotic convergence properties and iteration-complexity bounds as the usual scaled gradient projection method employing monotone line searches., Comment: 30 pages, 14 figures

The aim of this work is to formulate and analyze nonstandard finite difference (NSFD) schemes for a recognized virus-patch dynamic model. We introduce a simple and efficient approach to study the global asymptotic stability (GAS) of the constructed NSFD schemes. This approach is based on the Lyapunov stability theorem for discrete-time dynamical systems in combination with a well-known result on the GAS of discrete-time nonlinear cascade systems. As an important consequence, we obtain dynamically consistent NSFD schemes that provide reliable numerical approximations for the virus-patch dynamic model regardless of chosen step sizes. In addition, the convergence and error bounds for the constructed NSFD schemes are also investigated. Finally, a set of numerical examples is performed to support and illustrate the theoretical results. The numerical results show that some typical standard Runge–Kutta schemes fail to preserve the positivity and GAS of the continuous model for some given step sizes, meanwhile, the NSFD schemes preserve these properties for the same step sizes.

We consider a system in which the charged particle orbits under the influence of the electromagnetic field of three dipoles located on a system of three celestial bodies. Using well-known bivariate iterative scheme, known as Newton–Raphson (NR) iterative scheme, we numerically evaluated the positions of the stationary points (SPs) or equilibrium points (EPs) or libration points (LPs) and the linked basins of convergence (BoCs), and we also evaluated their linear stability. Moreover, we unveiled how the parameters, entering the effective potential function, affect the convergence dynamics of the system. Moreover, we also unveiled how the involved parameters affect the geometry of the zero velocity curves (ZVCs). Further, the correlation with the required number of iterations and the regions of convergence as well as the probability distributions associated to the BoCs is illustrated. In order to quantify the degree of final-state uncertainty of the BoCs, the basin entropy (BE) and for the fractality of boundaries of BoCs, the boundary basin entropy (BBE) are computed.

In this paper, we summarize results on convergence rates of various kernel based non- and semiparametric estimators, focusing on the impact of insufficient distributional smoothness, possibly unknown smoothness and even non-existence of density. In the presence of a possible lack of smoothness and the uncertainty about smoothness, methods of safeguarding against this uncertainty are surveyed with emphasis on nonconvex model averaging. This approach can be implemented via a combined estimator that selects weights based on minimizing the asymptotic mean squared error. In order to evaluate the finite sample performance of these and similar estimators we argue that it is important to account for possible lack of smoothness.

The moving asymptotes function is widely applied as sequential explicit convex approximation in topology optimization, due to the controllable conservatism and convergence. In virtue of these advantages, it is supposed that the efficiency of optimality criteria method will become higher if constraint functions are approximated by moving asymptotes function instead of linear function. This work presents an optimality criteria method for topology optimization under multiple constraints where the constraint functions are approximated by moving asymptotes function. The dual feasibility condition is customarily adopted to establish explicit update scheme of topological variable, where hypothesis of active topological variable set is avoided, in the case that gradient of objective function is positive. The complementary slackness condition and primal feasibility condition are combined into a simplified dual programming to solve for the Lagrange multipliers, where hypothesis of active constraint set is avoided. Three benchmark examples under multiple displacement, stress, compliance or eigenfrequency constraints are solved by the presented optimality criteria method, the results are compared to the method of moving asymptotes and the optimality criteria method with linear constraint approximation.

In this paper, we discuss the modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems under a weakened condition, and present the general convergence conditions for the method in terms of spectral radius and matrix norm, respectively. Moreover, for some special cases of the method, we propose the concrete convergence conditions and optimal parameters. These convergence theories improve the existing results to some extent. The numerical experiments verify the validity and practicality of the presented results.

In this paper, an adaptive modified weak Galerkin (AMWG) method is considered to solve second-order elliptic problem. Under the assumption of a penalty parameter, by showing reliability of error estimator, comparison of solutions and reduction of error estimator, the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops, is proved to be a contraction, namely, the adaptive algorithm is convergent. Numerical experiments are implemented to support the theoretical results.

In this paper, we consider a regularized least squares problem subject to convex constraints. Our algorithm is based on the superiorization technique, equipped with a new step size rule which uses subgradient projections. The superiorization method is a two-step method where one step reduces the value of the penalty term and the other step reduces the residual of the underlying linear system (using an algorithmic operator T). For the new step size rule, we present a convergence analysis for the case when T belongs to a large subclass of strictly quasi-nonexpansive operators. To examine our algorithm numerically, we consider box constraints and use the total variation (TV) functional as a regularization term. The specific test cases are chosen from computed tomography using both noisy and noiseless data. We compare our algorithm with previously used parameters in superiorization. The T operator is based on sequential block iteration (for which our convergence analysis is valid), but we also use the conjugate gradient method (without theoretical support). Finally, we compare with the well-known "fast iterative shrinkage-thresholding algorithm" (FISTA). The numerical results demonstrate that our new step size rule improves previous step size rules for the superiorization methodology and is competitive with, and in several instances behaves better than, the other methods.

We consider the finite element (FE) approximation of the two dimensional shallow water equations (SWE) by considering discretizations in which both space and time are established using a stable FE method. Particularly, we consider the automatic variationally stable FE (AVS-FE) method, a type of discontinuous Petrov-Galerkin (DPG) method. The philosophy of the DPG method allows us to establish stable FE approximations as well as accurate a posteriori error estimators upon solution of a saddle point system of equations. The resulting error indicators allow us to employ mesh adaptive strategies and perform space-time mesh refinements, i.e., local time stepping. We establish a priori error estimates for the AVS-FE method and linearized SWE and perform numerical verifications to confirm corresponding asymptotic convergence behavior. In an effort to keep the computational cost low, we consider an alternative space-time approach in which the space-time domain is partitioned into finite sized space-time slices. Hence, we can perform adaptive mesh refinements on each individual slice to preset error tolerances as needed for a particular application. Numerical verifications comparing the two alternatives indicate the space-time slices are superior for simulations over long times, whereas the solutions are indistinguishable for short times. Multiple numerical verifications show the adaptive mesh refinement capabilities of the AVS-FE method, as well the application of the method to some commonly applied benchmarks for the SWE.

The main purpose of this paper is to analyze nonconforming iterative finite element methods for 2D/3D stationary incompressible magneto-hydrodynamics equations. First, the Crouzeix-Raviart–type finite element is used to approximate the velocity and the conforming piecewise linear element P1 is used for the pressure. Since the finite element method for the velocity field and the pressure is unstable, a simple locally stabilization term is added to satisfy the weak inf-sup condition. Then, the well-posedness and the optimal error estimates of the continuous and discrete problems are analyzed with the nonlinear terms being iteratively updated. Three effective iterative methods are proposed and their stability and convergence analyses are carried out. Finally, the theoretical analysis presented in this paper is verified by numerical experiments.

The objective of this paper is to find the numerical solution of two-dimensional multi-term time fractional mobile-immobile equation. The proposed technique is based on the arbitrary-order orthogonal spline collocation method for the spatial discretization, the L1 approximation for the Caputo fractional derivative, and a second-order backward differentiation formula in time. The stability and convergence are proved in detail. Then, the convergence analysis is validated by a number of numerical experiments.

In this paper we suggest a theoretical framework to study the generalized split null point problem (GSNPP) via a multi-step inertial based parallel hybrid projection algorithm in Hilbert spaces. The architecture and the suitable set of control conditions ensure the strong convergence of the algorithm. The possible applications of the algorithm are illustrated via various theoretical and numerical results. The key advantages of the algorithm are highlighted in comparison to the classical and non-inertial algorithm as well as (one-step and two-step) inertial variants of the algorithm for the GSNPP.

We propose a proximal point method for quasiconvex pseudomonotone equilibrium problems. The subproblems of the method are optimization problems whose objective is the sum of a strongly quasiconvex function plus the standard quadratic regularization term for optimization problems. We prove, under suitable additional assumptions, convergence of the generated sequence to a solution of the equilibrium problem, whenever the bifunction is strongly quasiconvex in its second argument, thus extending the validity of the convergence analysis of proximal point methods for equilibrium problems beyond the standard assumption of convexity of the bifunction in the second argument.

We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions.

The application of Levy processes including major movements or jumps over a small period of time has proved to be an effective technique in financial research to catch certain unusual or extreme cases in stock price dynamics. Models that follow the Levy process are The FMLS, Kobol, and CGMY models. These models gradually grow the interest for study among researchers because of some of the best choices them. Therefore the topic of approaching these three different models has drawn yet more attention. In the current paper, we present these models’ numerical method. At first, The Riemann-Liouville tempered fractional derivative (RLTFD) with arbitrary order is approximated by using the basis function of the shifted Chebyshev polynomials of the fourth kind (SCPFK). In the second step, we gain the semi-discrete design to solve the tempered fractional B-S model (TFBSM) by applying finite difference approximation. We’re going to show that this system is stable and $$\mathcal {O}(\delta \tau )$$ is the convergence order. In fact, a fast stabilized method will obtain to reduce the time from processing and the computation time per repetition. Then to get the full scheme, we use SCPFK to approximate the spatial fractional derivative. Finally, two numerical examples are presented to demonstrate the accuracy and usefulness of the developed system.

In this article, affine recurrent fractal interpolation function is constructed and its convergence analysis is established to understand the approximation properties. Besides, the existence of optimal recurrent fractal interpolation function for given continuous function is discussed. Further, shape preserving aspects of the recurrent fractal interpolation function are investigated by imposing the necessary conditions on the vertical scaling factors. Numerical examples are explored which support the theoretical results.

This article develops a method based on the generalized Gegenbauer–Humbert wavelets in concert with their operational matrices of fractional integration to deal with the fractional partial differential equations and find the approximate solutions of it. The goal is to show that the proposed method is appropriate for boundary and initial-boundary problems even though it is generalized form. The convergence of the method under study is investigated. The numerical results gained by the proposed method are considered and compared with other methods, to establish the effectiveness and accuracy.

This work formulates an alternating direction implicit (ADI) Galerkin scheme for the nonlocal diffusion-wave equation in three-dimensional space. The L1 discretization formula is used to approximate the fractional Caputo derivative. A fully discrete Galerkin scheme is obtained via spatial discretization based on the finite element method. Then, an ADI algorithm is designed to reduce the computational cost. The stability and convergence analyses are derived via the energy method. Numerical experiments demonstrate the theoretical results.

In this study, a closed-loop $${D^\alpha }$$ -type iterative learning control (ILC) with a proportional D-type iterative learning updating law for the initial shift is applied to nonlinear conformable fractional order system. First, the system with the initial shift is introduced. Then, fractional order ILC (FOILC) frameworks that experience the initial shift problem for the path tracking of nonlinear conformable fractional order systems are addressed. Moreover, the sufficient condition for the convergence of tracking errors is obtained in the time domain by introducing $$\lambda $$ -norm and Holder’s inequality. Lastly, numerical examples are provided to illustrate the effectiveness of the proposed methods.

The authors analyze the two-stage Popov method with Bregman divergence and a new adaptive rule for choosing the step size, which does not require the Lipschitz constants to be known and operator values at additional points to be calculated. For variational inequalities with pseudo-monotone and Lipschitz continuous operators acting in a finite-dimensional normed linear space, the convergence theorem for the method is proved.

We provide a unified analytic approach to study the asymptotic dynamics of Young differential equations, using the framework of random dynamical systems and random attractors. Our method helps to generalize recent results (Duc et al. in J Differ Equ 264:1119–1145, 2018, SIAM J Control Optim 57(4):3046–3071, 2019; Garrido-Atienza et al. in Int J Bifurc Chaos 20(9):2761–2782, 2010) on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.

The optimization literature is vast in papers dealing with improvements on the global convergence of augmented Lagrangian schemes. Usually, the results are based on weak constraint qualifications, or, more recently, on sequential optimality conditions obtained via penalization techniques. In this paper, we propose a somewhat different approach, in the sense that the algorithm itself is used in order to formulate a new optimality condition satisfied by its feasible limit points. With this tool at hand, we present several new properties and insights on limit points of augmented Lagrangian schemes, in particular, characterizing the strongest possible global convergence result for the safeguarded augmented Lagrangian method.

In this paper, we propose a new amplitude-only method for pattern synthesis of uniform linear array (ULA) based on genetic algorithm (GA) and artificial neural network (ANN), which can produce a low sidelobe pattern with deep nulls. Unlike existing schemes for pattern synthesis, our method can avoid falling into local optimum and exhibits a faster convergence speed, and is also easy to implement due to its low computational complexity. In complicated and changing environments, the proposed method can quickly and accurately synthesize the desired pattern of the ULA. Numerical results demonstrate the effectiveness of the devised algorithm.

The ill-posed problem of unknown source identification in the time-fractional radial heat conduction equation is studied. In order to overcome the ill-posedness of the problem, a stationary iterated weighted Tikhonov regularization method is proposed. The a-priori and the a-posteriori choice rules for regularization parameters are discussed and the corresponding convergence rates of both are obtained. The stationary iterated weighted Tikhonov regularization method goes beyond the saturation results of weighted Tikhonov regularization method. A numerical example shows the effectiveness of this method.

In order to seek the numerical solution of large sparse nonlinear systems with positive definite Jacobian matrix, treating the normal and skew-Hermitian splitting(NSS) method (Bai et al., Numer Linear Algebra Appl 14:319–335, 2007) as internal iteration, we establish a new iteration method named the King-NSS method in this paper. The convergence of our new presented method is analyzed theoretically under suitable conditions. Furthermore, we compare the King-NSS method with other Newton-like methods, which take the Newton or modified Newton methods as the external iteration. Numerical experimental results corroborate the feasibility and superiority of our new method when the nonlinear systems with positive definite Jacobian matrix are large sparse.

Detection and recovery of noise corrupted signals using adaptive filters are becoming popular due to their application in several fields, including communication and biomedical. Adaptive filters broadly utilize the LMS algorithm since it has low computational complexity and is robust in behavior. However, it fails to achieve a faster speed of convergence and minimum steady-state MSE simultaneously. Therefore, the objective is to estimate the noise-free signal faster with improved accuracy and obtain a lower steady-state MSE with a higher convergence speed. This paper introduces a multi-stage adaptive filtering model wherein the noisy signal is analyzed through a cascade of stages. The proposed signal de-noising scheme utilizes the automatic selection of stages to be cascaded such that the steady-state MSE is minimum. Further, to enhance convergence speed, the LMS adaptive filter’s step-size is automatically adjusted at each stage. Hence, by controlling the number of stages to be cascaded automatically and utilizing a different step-size for each stage, we obtain a high convergence speed and minimum steady-state MSE. The proposed filter structure is tested for signal de-noising application to assess its performance concerning MSE, Signal-to-Noise Ratio (SNR), Average Noise Reduction (ANR), and convergence speed. The results attained have shown that the proposed filter structure provides remarkable performance improvement at different input noise levels. Further, the proposed Improved Variable-Stage (IVS) cascaded adaptive filter model employs LMS adaptive algorithm, hence offering a cost-effective and straightforward hardware implementation of ANC.

We present a new algorithm for large-scale unconstrained minimization that, at each iteration, minimizes, approximately, a quadratic model of the objective function plus a regularization term, not necessarily based on a norm. We prove convergence assuming only gradient continuity and complexity results assuming Lipschitz conditions. For solving the subproblems in the case of regularizations based on the 3-norm, we introduce a new method that quickly obtains the approximate solutions required by the theory. We present numerical experiments.

The magnetotelluric (MT) method has been widely used in geophysical electromagnetic (EM) exploration. However, complex geometries and anisotropic structures still pose challenges for the simulations of 3D MT problems. This study presents a nodal finite-element (FE) solution to simulate 3D MT responses in 3D conductivity structures with general anisotropy. The method is based on the $${\mathbf{A}} - \psi$$ decomposition of the electric field. The computational domain is discretized into hexahedral elements. The linear system equations that result from the FE discretization are solved by iterative solvers. We designed three examples to test the performance of the algorithm in this study. For the first example, we compare our results with those of other scholars to validate the effectiveness of the procedure. For the second example, the convergence behaviors of different iterative solvers with different preconditioners are tested. For the third example, a complex model is designed to demonstrate the robustness and effectiveness of the proposed code. Numerical experiments show that the convergence rate of the iterative solver of the $${\mathbf{A}} - \psi$$ method is very fast, especially at low frequencies.

The four-dimensional (4D) printing technology, as a combination of additive manufacturing and smart materials, has attracted increasing research interest in recent years. The bilayer structures printed with smart materials using this technology can realize complicated deformation under some special stimuli due to the material properties. The deformation prediction of bilayer structures can make the design process more rapid and thus is of great importance. However, the previous works on deformation prediction of bilayer structures rarely study the complicated deformations or the influence of the printing process on deformation. Thus, this paper proposes a new method to predict the complicated deformations of temperature-sensitive 4D printed bilayer structures, in particular to the bilayer structures based on temperature-driven shape-memory polymers (SMPs) and fabricated using the fused deposition modeling (FDM) technology. The programming process to the material during printing is revealed and considered in the simulation model. Simulation results are compared with experiments to verify the validity of the method. The advantages of this method are stable convergence and high efficiency, as the three-dimensional (3D) problem is converted to a two-dimensional (2D) problem. The simulation parameters in the model can be further associated with the printing parameters, which shows good application prospect in 4D printed bilayer structure design.

In this recent work, a new two-step iterative method for solving nonlinear equations that have a fifth-order convergence is suggested and analyzed. This new iterative method is free from second derivative of functions and based on Halley’s method and Taylor’s expansion together by using Hermite orthogonal polynomials basis to implement a suitable approximation of second derivative of functions. In addition, the order of convergence and the corresponding error equations of the new method are proved. Finally, some numerical examples are given to show the efficiency and the performance of the new method as well as a comparison with the original well-known Newton’s method and some other relevant methods are illustrated.

In this study, two fractional-based stochastic gradient (FSG) algorithms for time-delayed auto-regressive exogenous (ARX) models are proposed. By combining momentum and adaptive methods, a momentum-based FSG and an adaptive-based FSG algorithms are developed. These two FSG algorithms have faster convergence rates when compared with the stochastic gradient algorithm. The mechanism of the convergence is proved in theory. Furthermore, two simulated examples are presented to illustrate the efficiency of the new proposed algorithms.

This paper deals with the convergence analysis of a shrinking approximants for the computation of a common solution associated with the fixed point problem of $$\eta $$ -demimetric operator and the generalized split null point problem in Hilbert spaces. The considered sequence of approximants is a variant of the parallel hybrid shrinking projection algorithm that converges strongly to the optimal common solution under suitable set of control conditions. The viability of the approximants with parallel implementation is demonstrated for various theoretical as well as numerical results. The results presented in this paper improve various existing results in the current literature.

In Li and Ren (Int. J. Numer. Methods Fluids 70: 742–763, 2012), a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain, in which the high-order numerical accuracy and the oscillations-free property can be achieved. In this paper, the method is extended to solve steady state problems imposed in a curved physical domain. The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations, and a geometrical multigrid method to solve the derived linear system. To achieve high-order non-oscillatory numerical solutions, the classical k-exact reconstruction with $$k=3$$ and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables. The non-uniform rational B-splines (NURBS) curve is used to provide an exact or a high-order representation of the curved wall boundary. Furthermore, an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state. A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.

We prove a limit theorem on the convergence of mathematical expectations of functionals of sums of independent random variables to the Cauchy problem solution for the nonstationary Schr¨odinger equation with a symmetric fractional derivative operator of order $$ \upalpha\ \upepsilon \bigcup_{m=3}^{\infty}\left(m-1,m\right) $$ in the righthand side.

Two improved nonlinear conjugate gradient methods are proposed by using the second inequality of the strong Wolfe line search. Under usual assumptions, we proved that the improved methods possess the sufficient descent property and global convergence. By testing the unconstrained optimization problems which taken from the CUTE library and other usual test collections, some large-scale numerical experiments for the presented methods and their comparisons are executed. The detailed results are listed in tables and their corresponding performance profiles are reported in figures, which show that our improved methods are superior to their comparisons.

The paper concerns with a new iterative method for solving a monotone variational inclusion problem in a Hilbert space. The method is of the proximal contraction type incorporated with the regularization technique. Under the prediction stepsize conditions, we establish the strong convergence of the iterative sequences generated by the method to a particular solution of the problem satisfying a variational inequality problem. Finally, we give some numerical examples to illustrate the behavior of the new method in comparison with existing ones.

In this paper, Lie classical approach is utilized for the symmetry reduction of time-fractional variable coefficient Caudrey–Dodd–Gibbon–Sawada–Kotera equation. The obtained symmetries and Erd $$\acute{e}$$ lyi–Kober fractional differential operator are used to reduce the original nonlinear partial differential equation into nonlinear ordinary differential equation. The generalized Noether operator and new conservation theorem are exploited to obtain conservation laws of the governing equation. The power series solution is also derived for the considered equation. The obtained power series solution is investigated for the convergence and the obtained power series solution is convergent.

Summary For highly nonlinear problems, the objective function f(x) may have multiple local optima and it is desired to locate all of them. Analytical or adjoint-based derivatives may not be available for most real optimization problems, especially, when responses of a system are predicted by numerical simulations. The distributed-Gauss-Newton (DGN) optimization method performs quite efficiently and robustly for history-matching problems with multiple best matches. However, this method is not applicable for generic optimization problems, e.g., life-cycle production optimization or well location optimization. In this paper, we generalized the distribution techniques of the DGN optimization method and developed a new distributed quasi-Newton (DQN) optimization method that is applicable to generic optimization problems. It can handle generalized objective functions F(x,y(x))=f(x) with both explicit variables x and implicit variables, i.e., simulated responses, y(x). The partial derivatives of F(x,y) with respect to both x and y can be computed analytically, whereas the partial derivatives of y(x) with respect to x (sensitivity matrix) is estimated by applying the same efficient information sharing mechanism implemented in the DGN optimization method. An ensemble of quasi-Newton optimization tasks is distributed among multiple high-performance-computing (HPC) cluster nodes. The simulation results generated from one optimization task are shared with others by updating a common set of training data points, which records simulated responses of all simulation jobs. The sensitivity matrix at the current best solution of each optimization task is approximated by either the linear-interpolation (LI) method or the support-vector-regression (SVR) method, using some or all training data points. The gradient of the objective function is then analytically computed using its partial derivatives with respect to x and y and the estimated sensitivities of y with respect to x. The Hessian is updated using the quasi-Newton formulation. A new search point for each distributed optimization task is generated by solving a quasi-Newton trust-region subproblem for the next iteration. The proposed DQN method is first validated on a synthetic history matching problem and its performance is found to be comparable with the DGN optimizer. Then, the DQN method is tested on different optimization problems. For all test problems, the DQN method can find multiple optima of the objective function with reasonably small numbers of iterations (30 to 50). Compared to sequential model-based derivative-free optimization methods, the DQN method can reduce the computational cost, in terms of the number of simulations required for convergence, by a factor of 3 to 10.

The finite-volume theory is an equilibrium-based approach characterized by the equilibrium equations' local satisfaction and the imposition of continuity conditions in a surface-averaged sense through the subvolume interfaces. Previous investigations include stress and displacement fields convergence and computational cost, showing the approach's efficiency, especially in heterogeneous materials and structures. However, those investigations did not include an energy analysis, which is especially important in compliance minimization problems. As the finite element method, energy-based approaches impose energy balance, which guarantees a monotonic energy convergence. This contribution addresses a numerical investigation of the main mechanical energy aspects involving the generalized finite-volume theory for continuum elastic structures in quasi-static analyses. The obtained results are verified with analytical and finite element-based analyses, showing the monotonic energy convergence for the three versions of the finite-volume theory and the energy balance's satisfaction for the higher-order versions when a sufficiently refined mesh is employed.

The time-fractional Black-Scholes model governing European options in which the temporal derivative is focused on the Caputo fractional derivative with $$0

In this work, we study splitting methods for inclusion problems in real Hilbert space. The algorithms are inspired by forward-backward splitting method, projection and contraction method, inertial method and a self-adaptive step size. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish convergence results of the proposed algorithms. Finally, we present the application of the proposed algorithms for solving convex minimization problems and variational inequalities.

A convergent algorithm for nonnegative matrix factorization with orthogonality constraints imposed on both factors is proposed in this paper. This factorization concept was first introduced by Ding et al. with intent to further improve clustering capability of NMF. However, as the original algorithm was developed based on multiplicative update rules, the convergence of the algorithm cannot be guaranteed. In this paper, we utilize the technique presented in our previous work to develop the algorithm and prove that it converges to a stationary point inside the solution space., Comment: arXiv admin note: substantial text overlap with arXiv:1010.5290

We prove that the implicit time Euler scheme coupled with finite elements space discretization for the 2D Navier–Stokes equations on the torus subject to a random perturbation converges in $$L^2(\varOmega )$$ , and describe the rate of convergence for an $$H^1$$ -valued initial condition. This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier–Stokes equations and convergence of a localized scheme, we can prove strong convergence of this space-time approximation. The speed of the $$L^2(\varOmega )$$ -convergence depends on the diffusion coefficient and on the viscosity parameter. In case of Scott–Vogelius mixed elements and for an additive noise, the convergence is polynomial.