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This paper is concerned with the numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. The considered systems are described by second-order stochastic differential-algebraic equations involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, the drift coefficient of which is typically not globally one-sided Lipschitz continuous. We investigate a half-explicit, drift-truncated Euler scheme that fulfills the constraint exactly. Pathwise uniform |$L_p$| -convergence is established. The proof is based on a suitable decomposition of the discrete Lagrange multipliers and on norm estimates for the single components, enabling the verification of consistency, semistability and moment growth properties of the scheme. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a numerical solution for the time-fractional inverse heat equations. We focus on obtaining the unknown source term along with the unknown temperature function based on an additional condition given in an integral form. The proposed scheme is based on a spectral collocation approach to obtain the two independent variables. Our approach is accurate, efficient, and feasible for the model problem under consideration. The proposed Jacobi spectral collocation method yields an exponential rate of convergence with a relatively small number of degrees of freedom. Finally, a series of numerical examples are provided to demonstrate the efficiency and flexibility of the numerical scheme. [ABSTRACT FROM AUTHOR]

This paper presents an analysis solution for systems of partial differential equations using a new modification of the decomposition method to overcome the computational difficulties. Convergence of series solution was discussed with two illustrated examples, and the method showed a high-precision, being a fast approach to solve the non-linear system of PDEs with initial conditions. There is no need to convert the nonlinear terms into the linear ones due to the Adomian polynomials. The method does not require any discretization or assumption for a small parameter to be present in the problem. The steps of the suggested method are easily implemented, with high accuracy and rapid convergence to the exact solution, compared with other methods that can be used to solve systems of PDEs. [ABSTRACT FROM AUTHOR]

In this paper we show that it is possible to use an iterative eigensolver in conjunction with Halverson and Poirier's symmetrized Gaussian (SG) basis [T. Halverson and B. Poirier, J. Chem. Phys. 137, 224101 (2012)] to compute accurate vibrational energy levels of molecules with as many as five atoms. This is done, without storing and manipulating large matrices, by solving a regular eigenvalue problem that makes it possible to exploit direct-product structure. These ideas are combined with a new procedure for selecting which basis functions to use. The SG basis we work with is orders of magnitude smaller than the basis made by using a classical energy criterion. We find significant convergence errors in previous calculations with SG bases. For sum-of-product Hamiltonians, SG bases large enough to compute accurate levels are orders of magnitude larger than even simple pruned bases composed of products of harmonic oscillator functions. [ABSTRACT FROM AUTHOR]

One of the challenges facing on-the-fly ab initio semiclassical time evolution is the large expense needed to converge the computation. In this paper, we suggest that a significant saving in computational effort may be achieved by employing a semiclassical initial value representation (SCIVR) of the quantum propagator based on the Heisenberg interaction representation. We formulate and test numerically a modification and simplification of the previous semiclassical interaction representation of Shao and Makri [J. Chem. Phys. 113, 3681 (2000)]. The formulation is based on the wavefunction form of the semiclassical propagation instead of the operator form, and so is simpler and cheaper to implement. The semiclassical interaction representation has the advantage that the phase and prefactor vary relatively slowly as compared to the "standard" SCIVR methods. This improves its convergence properties significantly. Using a one-dimensional model system, the approximation is compared with Herman-Kluk's frozen Gaussian and Heller's thawed Gaussian approximations. The convergence properties of the interaction representation approach are shown to be favorable and indicate that the interaction representation is a viable way of incorporating on-the-fly force field information within a semiclassical framework. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give some extensions of Steffensen's method to the space ℝn. [ABSTRACT FROM AUTHOR]

In this paper, the Lax-Wendroff difference scheme has been presented for solving the one-dimensional wave equation with integral boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced by the central finite difference approximation of functional values at each grid point by using Taylor series expansion. Then, for solving the resulting second-order linear ordinary differential equation, the displacement function is discretized in the direction of a temporal variable by using Taylor series expansion and the Lax-Wendroff difference scheme is developed, then it gives a system of algebraic equations. The derivative of the initial condition is also discretized by using the central finite difference method. Then the obtained system of algebraic equations is solved by the matrix inverse method. The stability and convergent analysis of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields fourth-order convergent in spatial variable and sixth-order convergent in a temporal variable. To validate the applicability of the proposed method, three model examples are considered and solved for different values of the mesh sizes in both directions. Numerical results are presented in tables in terms of maximum absolute error, L2 and L∞ norm. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution. [ABSTRACT FROM AUTHOR]

In this paper, we suggest some new and efficient iterative methods for solving nonlinear equations f (x) = 0. These methods are free from derivatives having high order of convergence. We also give some examples to illustrate the efficiency of these methods. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the classical Steffensen's method. These new methods can be considered as alternative of existing derivative-free methods. [ABSTRACT FROM AUTHOR]

In the present paper, we estimate the rate of convergence of exponential type operators connected with x³ on functions of bounded variation. [ABSTRACT FROM AUTHOR]

In this paper, the initial-value problem for a linear second order delay differential equation is considered. To solve this problem numerically, an appropriate difference scheme is constructed by using the method of integral identities which contains basis functions and interpolating quadrature rules with weight and remainder term in integral form. Besides, the method is proved to be first-order convergent in discrete maximum norm. The numerical illustration provided support the theoretical results. Finally, the proposed method is compared with the implicit Euler method. [ABSTRACT FROM AUTHOR]

In this paper, aiming at the unconstrained optimization problem, a new nonmonotone adaptive retrospective trust region line search method is presented, which takes advantages of multidimensional filter technique to increase the acceptance probability of the trial step. The new nonmonotone trust region ratio is presented, which based on the convex combination of nonmonotone trust region ratio and retrospective ratio. The global convergence and the superlinear convergence of the algorithm are shown in the right circumstances. Comparative numerical experiments show the better effective and robustness. [ABSTRACT FROM AUTHOR]

For solving large scale linear systems, a fast convergent randomized Kaczmarz-type method was constructed in Bai and Wu (2018) [4]. In this paper, we propose a geometric probability randomized Kaczmarz (GPRK) method by introducing a new index set J k and three supervised probability criteria defined on J k from a geometric point of view. Linear convergence of GPRK is proved, and the way of argument for the analysis of GPRK also leads to new sharper upper bounds for the randomized Kaczmarz (RK) method and the greedy randomized Kaczmarz (GRK) method. In practice, GPRK is implemented with a simple geometric probability criterion, i.e., the most efficient one of the aforementioned three supervised probability criteria defined on J k. The numerical results demonstrate that GPRK is robust and efficient, and it is faster than GRK in most of the tests in the sense of computing time. [ABSTRACT FROM AUTHOR]

This paper is dealing with a class of operators fixing exponential functions. We study its convergence and an asymptotic formula is obtained. Finally, a comparison is stated, that shows that in a certain particular cases the new sequence approximates better than the classical ones. [ABSTRACT FROM AUTHOR]

The stochastic Frank–Wolfe method has recently attracted much general interest in the context of optimization for statistical and machine learning due to its ability to work with a more general feasible region. However, there has been a complexity gap in the dependence on the optimality tolerance ε in the guaranteed convergence rate for stochastic Frank–Wolfe compared to its deterministic counterpart. In this work, we present a new generalized stochastic Frank–Wolfe method which closes this gap for the class of structured optimization problems encountered in statistical and machine learning characterized by empirical loss minimization with a certain type of "linear prediction" property (formally defined in the paper), which is typically present in loss minimization problems in practice. Our method also introduces the notion of a "substitute gradient" that is a not-necessarily-unbiased estimate of the gradient. We show that our new method is equivalent to a particular randomized coordinate mirror descent algorithm applied to the dual problem, which in turn provides a new interpretation of randomized dual coordinate descent in the primal space. Also, in the special case of a strongly convex regularizer our generalized stochastic Frank–Wolfe method (as well as the randomized dual coordinate descent method) exhibits linear convergence. Furthermore, we present computational experiments that indicate that our method outperforms other stochastic Frank–Wolfe methods for a sufficiently small optimality tolerance, consistent with the theory developed herein. [ABSTRACT FROM AUTHOR]

In this paper, we establish the generalized viscosity implicit rules of asymptotically nonexpansive mappings in CAT(0) spaces. The strong convergence theorems of the implicit rules proposed are proved under certain assumptions imposed on the control parameters. The results presented in this paper improve and extend some recent corresponding results announced. [ABSTRACT FROM AUTHOR]

In this paper, we present a explicit viscosity technique of nonexpansive mappings in the framework of CAT(0) spaces. The strong convergence theorem of the proposed technique is proved under certain assumptions imposed on the sequence of parameters. The results presented in this paper extend and improve some recent announced in the current literature. [ABSTRACT FROM AUTHOR]

Salp Swarm Algorithm (SSA) is a novel swarm intelligent algorithm with good performance. However, like other swarm-based algorithms, it has insufficiencies of low convergence precision and slow convergence speed when dealing with high-dimensional complex optimisation problems. In response to this concerning issue, in this paper, we propose an improved SSA named as WASSA. First of all, dynamic weight factor is added to the update formula of population position, aiming to balance global exploration and local exploitation. In addition, in order to avoid premature convergence and evolution stagnation, an adaptive mutation strategy is introduced during the evolution process. Disturbance to the global extremum promotes the population to jump out of local extremum and continue to search for an optimal solution. The experiments conducted on a set of 28 benchmark functions show that the improved algorithm presented in this paper displays obvious superiority in convergence performance, robustness as well as the ability to escape local optimum when compared with SSA. [ABSTRACT FROM AUTHOR]

Highlights • Sampling-based algorithms are commonly used in motion planning problems. • The RRT* algorithm incrementally builds a tree of motion to find a solution. • Taking a shortcut to the ancestry increases the convergence rate to the optimal. • Combination with sampling strategies further improves the performance. Abstract The Rapidly-exploring Random Tree (RRT) algorithm is a popular algorithm in motion planning problems. The optimal RRT (RRT*) is an extended algorithm of RRT, which provides asymptotic optimality. This paper proposes Quick-RRT* (Q-RRT*), a modified RRT* algorithm that generates a better initial solution and converges to the optimal faster than RRT*. Q-RRT* enlarges the set of possible parent vertices by considering not only a set of vertices contained in a hypersphere, as in RRT*, but also their ancestry up to a user-defined parameter, thus, resulting in paths with less cost than those of RRT*. It also applies a similar technique to the rewiring procedure resulting in acceleration of the tendency that near vertices share common parents. Since the algorithm proposed in this paper is a tree extending algorithm, it can be combined with other sampling strategies and graph-pruning algorithms. The effectiveness of Q-RRT* is demonstrated by comparing the algorithm with existing algorithms through numerical simulations. It is also verified that the performance can be further enhanced when combined with other sampling strategies. [ABSTRACT FROM AUTHOR]

Stochastic variational inequalities (SVI) provide a unified framework for the study of a general class of nonlinear optimization and Nash-type equilibrium problems with uncertain model data. Often the true solution to an SVI cannot be found directly and must be approximated. This paper considers the use of a sample average approximation (SAA), and proposes a new method to compute confidence intervals for individual components of the true SVI solution based on the asymptotic distribution of SAA solutions. We estimate the asymptotic distribution based on one SAA solution instead of generating multiple SAA solutions, and can handle inequality constraints without requiring the strict complementarity condition in the standard nonlinear programming setting. The method in this paper uses the confidence regions to guide the selection of a single piece of a piecewise linear function that governs the asymptotic distribution of SAA solutions, and does not rely on convergence rates of the SAA solutions in probability. It also provides options to control the computation procedure and investigate effects of certain key estimates on the intervals. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the superconvergence properties of the quadratic finite volume method (FVM) on triangular meshes for elliptic equations. We proved the 3rd order superconvergence rate of the gradient approximation ‖ u h − u I ‖ 1 = O (h 3) and the 4th order superconvergence rate of the function value approximation ‖ u h − u I ‖ 0 = O (h 4) for the quadratic FVM on triangular meshes. Here u h is the FVM solution and u I is the piecewise quadratic Lagrange interpolation of the exact solution. It should be pointed out that the superconvergence phenomena of FVM strongly depends on the construction of dual mesh. Specially for quadratic FVMs, the scheme presented in this paper is the unique scheme which holds superconvergence. [ABSTRACT FROM AUTHOR]

Abstract Twenty years after the classical book of Ortega and Rheinboldt was published, five definitions for the Q -convergence orders of sequences were independently and rigorously studied (i.e., some orders characterized in terms of others), by Potra (1989), resp. Beyer, Ebanks and Qualls (1990). The relationship between all the five definitions (only partially analyzed in each of the two papers) was not subsequently followed and, moreover, the second paper slept from the readers attention. The main aim of this paper is to provide a rigorous, selfcontained, and, as much as possible, a comprehensive picture of the theoretical aspects of this topic, as the current literature has taken away the credit from authors who obtained important results long ago. Moreover, this paper provides rigorous support for the numerical examples recently presented in an increasing number of papers, where the authors check the convergence orders of different iterative methods for solving nonlinear (systems of) equations. Tight connections between some asymptotic quantities defined by theoretical and computational elements are shown to hold. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a Strongly Stable generalized finite element method (SSGFEM) for a non-smooth interface problem, where the interface has a corner. The SSGFEM employs enrichments of 2 types: the nodes in a neighborhood of the corner are enriched by singular functions characterizing the singularity of the unknown solution, while the nodes close to the interface are enriched by a distance based function characterizing the jump in the gradient of the unknown solution along the interface. Thus nodes in the neighborhood of the corner and close to the interface are enriched with two enrichment functions. Both types of enrichments have been modified by a simple local procedure of "subtracting the interpolant." A simple local orthogonalization technique (LOT) also has been used at the nodes enriched with both enrichment functions. We prove that the SSGFEM yields the optimal order of convergence. The numerical experiments presented in this paper indicate that the conditioning of the SSGFEM is not worse than that of the standard finite element method, and the conditioning is robust with respect to the position of the mesh relative to the interface. Highlights • GFEM for 2D non-smooth interface problem. • Singular enrichment functions in addition to distance based enrichment functions. • Proof of optimal convergence of the GFEM. • Experimental study of conditioning and robustness of GFEM. • The notion of Strongly Stable GFEM (SSGFEM). [ABSTRACT FROM AUTHOR]

Highlights • The heat equation with nonlocal initial condition is considered. • Several finite difference schemes are constructed and analyzed. • The stability of the schemes is the main objective of investigation. • Time step size bound obtained in previous paper is revised. • Numerical experiments with linear and nonlinear problems confirm the theoretical results. Abstract In this paper, the two-level finite difference schemes for the one-dimensional heat equation with a nonlocal initial condition are analyzed. As the main result, we obtain conditions for the numerical stability of the schemes. In addition, we revise the stability conditions obtained in [21] for the Crank–Nicolson scheme. We present several numerical examples that confirm the theoretical results within linear, as well as nonlinear problems. In some particular cases, it is shown that for small regions of the time step size values, the explicit FTCS scheme is stable while certain implicit methods, such as Crank–Nicolson scheme, are not. [ABSTRACT FROM AUTHOR]

Abstract In this paper various hybrid hp -finite element methods are discussed for elliptic model problems. In particular, a stabilized primal-hybrid hp -method is introduced which approximatively ensures continuity conditions across element interfaces and avoids the enrichment of the primal discretization space as usually required to fulfill some discrete inf-sup condition. A priori as well as a posteriori error estimates are derived for this method. The stabilized primal-hybrid hp -method is also applied to a model obstacle problem since it admits the definition of pointwise constraints. The paper also describes some extensions of primal, primal-mixed and dual-mixed methods as well as their hybridizations to hp -finite elements. In numerical experiments the convergence properties of the stabilized primal-hybrid hp -method are discussed and compared to all the other hp -methods introduced in this paper. The applicability of the a posteriori error estimates to drive h - and hp -adaptive schemes is also investigated. [ABSTRACT FROM AUTHOR]

We propose in this paper a unifying scheme for several algorithms from the literature dedicated to the solving of monotone inclusion problems involving compositions with linear continuous operators in infinite dimensional Hilbert spaces. We show that a number of primal-dual algorithms for monotone inclusions and also the classical ADMM numerical scheme for convex optimization problems, along with some of its variants, can be embedded in this unifying scheme. While in the first part of the paper, convergence results for the iterates are reported, the second part is devoted to the derivation of convergence rates obtained by combining variable metric techniques with strategies based on suitable choice of dynamical step sizes. The numerical performances, which can be obtained for different dynamical step size strategies, are compared in the context of solving an image denoising problem. [ABSTRACT FROM AUTHOR]

In this paper, we present a explicit viscosity technique of nonexpansive mappings in the framework of CAT(0) spaces. The strong convergence theorem of the proposed technique is proved under certain assumptions imposed on the sequence of parameters. The results presented in this paper extend and improve some recent announced in the current literature. [ABSTRACT FROM AUTHOR]

In this paper, we trace back the genesis of Aitken's Δ2 process and Shanks' sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Padé approximants. We go back to the roots of these methods and analyze the original contributions. New and detailed explanations on the building and properties of Shanks' transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we establish the convergence of the proximal alternating direction method of multipliers (ADMM) and block coordinate descent (BCD) method for nonseparable minimization models with quadratic coupling terms. The novel convergence results presented in this paper answer several open questions that have been the subject of considerable discussion. We firstly extend the 2-block proximal ADMM to linearly constrained convex optimization with a coupled quadratic objective function, an area where theoretical understanding is currently lacking, and prove that the sequence generated by the proximal ADMM converges in point-wise manner to a primal-dual solution pair. Moreover, we apply randomly permuted ADMM (RPADMM) to nonseparable multi-block convex optimization, and prove its expected convergence for a class of nonseparable quadratic programming problems. When the linear constraint vanishes, the 2-block proximal ADMM and RPADMM reduce to the 2-block cyclic proximal BCD method and randomly permuted BCD (RPBCD). Our study provides the first iterate convergence result for 2-block cyclic proximal BCD without assuming the boundedness of the iterates. We also theoretically establish the expected iterate convergence result concerning multi-block RPBCD for convex quadratic optimization. In addition, we demonstrate that RPBCD may have a worse convergence rate than cyclic proximal BCD for 2-block convex quadratic minimization problems. Although the results on RPADMM and RPBCD are restricted to quadratic minimization models, they provide some interesting insights: (1) random permutation makes ADMM and BCD more robust for multi-block convex minimization problems; (2) cyclic BCD may outperform RPBCD for "nice" problems, and RPBCD should be applied with caution when solving general convex optimization problems especially with a few blocks. [ABSTRACT FROM AUTHOR]

Abstract We present a new framework for expressing finite element methods on multiple intersecting meshes: multimesh finite element methods. The framework enables the use of separate meshes to discretize parts of a computational domain that are naturally separate; such as the components of an engine, the domains of a multiphysics problem, or solid bodies interacting under the influence of forces from surrounding fluids or other physical fields. Such multimesh finite element methods are particularly well suited to problems in which the computational domain undergoes large deformations as a result of the relative motion of the separate components of a multi-body system. In the present paper, we formulate the multimesh finite element method for the Poisson equation. Numerical examples demonstrate the optimal order convergence, the numerical robustness of the formulation and implementation in the face of thin intersections and rounding errors, as well as the applicability of the methodology. In the accompanying paper (Johansson et al., 2018), we analyze the proposed method and prove optimal order convergence and stability. [ABSTRACT FROM AUTHOR]

The paper considers the problem of finding a common solution of a pseudomonotone and Lipschitz-type equilibrium problem and a fxed point problem for a quasi nonexpansive mapping in a Hilbert space. A new hybrid algorithm is introduced for approximating a solution of this problem. The presented algorithm can be considered as a combination of the extragradient method (two-step proximal- like method) and a modifed version of the normal Mann iteration. It is well known that the normal Mann iteration has the weak convergence, but in this paper we has obtained the strong convergence of the new algorithm under some mild conditions on parameters. Several numerical experiments are reported to illustrate the convergence of the algorithm and also to show the advantages of it over existing methods. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a neurodynamic approach to nonlinear optimization problems with affine equality and convex inequality constraints. The proposed neural network endows with a time-varying auxiliary function, which can guarantee that the state of the neural network enters the feasible region in finite time and remains there thereafter. Moreover, the state with any initial point is shown to be convergent to the critical point set when the objective function is generally nonconvex. Especially, when the objective function is pseudoconvex (or convex), the state is proved to be globally convergent to an optimal solution of the considered optimization problem. Compared with other neural networks for related optimization problems, the proposed neural network in this paper has good convergence and does not depend on some additional assumptions, such as the assumption that the inequality feasible region is bounded, the assumption that the penalty parameter is sufficiently large and the assumption that the objective function is lower bounded over the equality feasible region. Finally, some numerical examples and an application in real-time data reconciliation are provided to display the well performance of the proposed neural network. [ABSTRACT FROM AUTHOR]

Abstract In this paper, by combining the p -order block boundary value methods with the m -th Lagrange interpolation, a class of new numerical methods for solving nonlinear fractional differential equations with the γ -order (0 < γ < 1) Caputo derivatives are obtained. It is proved under some appropriate conditions that the induced methods are convergent of order min ⁡ { p , m − γ + 1 } and globally stable. Several numerical examples are given to illustrate the theoretical results and the computational effectiveness and accuracy of the methods. Highlights • This paper deals with numerical computation and analysis for nonlinear fractional differential equations (FDEs) with γ -order Caputo derivatives. • A class of extended block boundary value methods (EBBVMs) for solving the FDEs are obtained. • The EBBVMs are proved to be convergent of order min ⁡ { p , m − γ + 1 } and globally stable under the appropriate conditions. • Several numerical examples are given to illustrate the theoretical results and the computational effectiveness and accuracy of the methods. [ABSTRACT FROM AUTHOR]

Rigid registration is a fundamental process in many applications that require alignment of different datasets. Iterative closest point (ICP) is a widely used algorithm that iteratively finds point correspondences and updates the rigid transformation. One of the key variants of ICP to its success is the selection of points, which is directly related to the convergence and robustness of the ICP algorithm. Besides uniform sampling, there are a number of normal-based and feature-based approaches that consider normal, curvature, and/or other signals in the point selection. Among them, normal-space sampling (NSS) is one of the most popular techniques due to its simplicity and low computational cost. The rationale of NSS is to sample enough constraints to determine all the components of transformation, but this paper finds that NSS actually can constrain the translational normal space only. This paper extends the fundamental idea of NSS and proposes Dual NSS (DNSS) to sample points in both translational and rotational normal spaces. Compared with NSS, this approach has similar simplicity and efficiency without any need of additional information, but has a much better effectiveness. Experimental results show that DNSS can outperform the normal-based and feature-based methods in terms of convergence and robustness. For example, DNSS can achieve convergence from an orthogonal initial position while no other methods can achieve. Note to Practitioners—ICP is commonly used to align different data to a same coordination system. While NSS is often used to speed up the alignment process by down-sampling the data uniformly in the normal space. The implementation of NSS only has three steps: 1) construct a set of buckets in the normal-space; 2) put all points of the data into buckets based on their normal direction; and 3) uniformly pick points from all the buckets until the desired number of points is selected. The algorithm is simple and fast, so that it is still the common practice. However, the weakness of NSS comes from the reason that it cannot handle rotational uncertainties. In this paper, a new algorithm called DNSS is developed to constrain both translation and rotation at the same time by introducing a dual-normal space. With a new definition of the normal space, the algorithm complexity of DNSS is the same as that of NSS, and it can be readily implemented in all types of application that are currently using ICP. The experimental results show that DNSS has better efficiency, quality, and reliability than both normal-based and feature-based methods have. [ABSTRACT FROM AUTHOR]

This paper introduces a new mapping that is weaker than nonexpansive mapping. The new mapping is different from the Suzuki's generalized nonexpansive mapping. This paper introduces a new iteration process for the fixed point, and gives fixed point theorems and Convergence theorems for a new generalized nonexpansive mapping in Banach space, insteading of uniformly convex Banach space. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the generalized viscosity implicit rules of one asymptotically nonexpansive mapping in the intermediate sense in Hilbert spaces. We obtain some strong convergence theorems under certain assumptions imposed on the parameters. We also give a numerical example to support our main results. The results obtained in this paper improve and extend many recent ones in this culture. [ABSTRACT FROM AUTHOR]

In this paper, we establish the truncated Euler-Maruyama (EM) method for stochastic functional differential equation (SFDE) and consider the strong convergence theory for the numerical solutions of SFDEs under the local Lipschitz condition plus Khasminskii-type condition instead of the linear growth condition. The type of convergence specifically addressed in this paper is strong- convergence for , and p is a parameter in Khasminskii-type condition. We also discussed the rates of -convergence for the truncated EM method. [ABSTRACT FROM AUTHOR]

Abstract In this paper we investigate the problem of influence of parametric perturbations on the Lyapunov spectrum of the discrete linear time-varying system. The main result of the paper is that for any two sequences of positive real numbers and any rate of convergence there exist a discrete linear time-varying system and a perturbation tending to zero with the given rate of convergence such that the spectra of the perturbed and unperturbed systems coincide with the a priori given sequences. Moreover, we show that this phenomenon is possible even when the perturbations are different from zero, rarely, in a certain sense. Finally, as a conclusion from the main result, we obtain that the separation type and the index of exponential stability may vary arbitrarily under the influence of exponentially decreasing perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special kind of nonlinear integral equations, Fredholm‐Hammerstein equations. As done before we use an equivalent reformulation, for which collocation methods are more efficient. For the alternative equation, we use a collocation method based on Haar wavelets to approximate the solution. We discuss the convergence and error analysis of the method, showing that, at the collocation nodes, the order of convergence is higher than expected. The paper concludes with numerical examples, which illustrate the good approximations, conclusions, and ideas for future work in this area. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we will investigate the approximations of inverse moments for double-indexed weighted sums of random variables under sub-linear expectations. The asymptotic approximations of inverse moments and the convergence rate of approximations are established under the meaning of upper expectation E ˆ and the lower expectation ε ˆ , respectively. Moreover, we carry out some simulations to verify the validity of our theoretical results. The results obtained in the paper improve and extend the corresponding ones of Yang et al. [24] from independent random variables in the classical probability space to END random variables in sub-linear expectation space. [ABSTRACT FROM AUTHOR]

Abstract: Since the invention by Peaceman and Rachford, more than 60 years ago, of the well celebrated ADI finite difference scheme for parabolic initial-boundary problems on rectangular regions, many papers have been concerned with prescribing the boundary values for the intermediate approximations at half time levels in the case of nonzero Dirichlet boundary conditions. In the present paper, for variable coefficient parabolic problems and time-stepsize sufficiently small, we prove second order accuracy in the discrete norm of the ADI finite difference scheme in which the intermediate approximations do not involve the so called "perturbation term". As a byproduct of our stability analysis we also show that, for variable coefficients and time-stepsize sufficiently small, the ADI scheme with the perturbation term converges with order two in the discrete norm. Our convergence results generalize previous results obtained for the heat equation. [ABSTRACT FROM AUTHOR]

A cell growth model for a size‐structured cell population with a stochastic growth rate for size and division into two daughter cells of unequal size is studied in this paper. The model entails an initial boundary value problem that involves a second‐order parabolic partial differential equation with two nonlocal terms, the presence of which is a consequence of asymmetry in the cell division. The solution techniques for solving such problems are rare due to the nonlocal terms. In this paper, we solve the initial boundary value problem for arbitrary initial distributions. We obtain a separable solution, as well as the general solution to the partial differential equation, and show that the solutions converge to the separable solution for large time. As in the symmetric division case, the dispersion term does not affect the rate of convergence to the separable solution. [ABSTRACT FROM AUTHOR]

This work focuses on the influence of a simple pole on a new function representing method we called separate node ascending derivatives expansion (SNADE) convergence characteristics. Even though SNADE can be somehow considered as a new Taylor series expansion, it is distincted from Taylor series expansion by its different nodal points using features. This paper focuses on certain details of SNADE about convergence and its validity. Beyond that, this research focusing on the influence of a simple pole location in the function on the convergence characteristics involves completely new original findings and reported in this paper where a specific importance is given to the case where the SNADE nodal sequence is composed of elements alternating between two nodal values. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we discuss quantization effects in rigid formation control systems when target formations are described by interagent distances. Because of practical sensing and measurement constraints, we consider in this paper distance measurements in their quantized forms. We show that under gradient‐based formation control, in the case of uniform quantization, the distance errors converge locally to a bounded set whose size depends on the quantization error, while in the case of logarithmic quantization, all distance errors converge locally to zero. A special quantizer involving the signum function is then considered with which all agents can only measure coarse distances in terms of binary information. In this case, the formation converges locally to a target formation within a finite time. Lastly, we discuss the effect of asymmetric uniform quantization on rigid formation control. [ABSTRACT FROM AUTHOR]

Nuit debout represents one of the main mobilizations in France in recent years and the most important anti-austerity movement in the country since the financial crisis. Based on document analysis and fieldwork, this paper addresses the development of master frames within the context of free spaces. The introduction of the parliamentarian debate on the French Jobs Act can be understood as a suddenly imposed grievance that triggered the emergence in France of a movement against austerity and the perceived retrenchment of democratic life. This happened as the grievance was framed within the French left-wing movements through the adoption and adaptation of ideas coming from a movement cascade that started in 2008 in Iceland, peaked in 2011 in Spain, Greece and the US and continued in countries as Turkey in 2013. Moving from structure to action, the paper highlights how Nuit debout provided a platform for the convergence of previously disconnected mobilizations. In particular, the movement’s self-characterization as a ‘convergence of struggles’ and as a movement ‘against the Jobs Act and its world’ developed within free spaces in which contentious but also deliberative practices were accommodated. [ABSTRACT FROM AUTHOR]

From spline theory it is well-known that univariate cubic spline interpolation, if carried out in its natural Hilbert space W 2 2 [ a , b ] and on point sets with fill distance h , converges only like O ( h 2 ) in L 2 [ a , b ] if no additional assumptions are made. But superconvergence up to order h 4 occurs if more smoothness is assumed and if certain additional boundary conditions are satisfied. This phenomenon was generalized in 1999 to multivariate interpolation in Reproducing Kernel Hilbert Spaces on domains Ω ⊂ R d for continuous positive definite Fourier-transformable shift-invariant kernels on R d . But the sufficient condition for superconvergence given in 1999 still needs further analysis, because the interplay between smoothness and boundary conditions is not clear at all. Furthermore, if only additional smoothness is assumed, superconvergence is numerically observed in the interior of the domain, but a theoretical foundation still is a challenging open problem. This paper first generalizes the “improved error bounds” of 1999 by an abstract theory that includes the Aubin–Nitsche trick and the known superconvergence results for univariate polynomial splines. Then the paper analyzes what is behind the sufficient conditions for superconvergence. They split into conditions on smoothness and localization , and these are investigated independently. If sufficient smoothness is present, but no additional localization conditions are assumed, it is numerically observed that superconvergence always occurs in the interior of the domain, and some supporting arguments are provided. If smoothness and localization interact in the kernel-based case on R d , weak and strong boundary conditions in terms of pseudodifferential operators occur. A special section on Mercer expansions is added, because Mercer eigenfunctions always satisfy the sufficient conditions for superconvergence. Numerical examples illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

Genetic algorithm (GA) is a random search algorithm, which has been commonly used to solve optimization problems. A new encoding method of GA, adaptive degressive ary number encoding, is proposed in this paper. This paper firstly introduces the N-ary encoding genetic algorithm and then defines the degressive ary number encoded genetic algorithm. Based on degressive ary number encoded genetic algorithm, this paper proposes a feasible adaptive change rule of the ary number encoding, which can change ary number with the fitness function’s value. All the parameters are selected according to the parameters’ experiments. The proposed algorithm is used to solve the function optimization problems to test the performance. The performances of the proposed algorithm are compared with GA, some classic algorithms and some latest algorithms. The experiments show that the improved adaptive degressive ary number encoded genetic algorithm has a better searching ability and a faster rate of convergence. [ABSTRACT FROM AUTHOR]