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Abstract: In this note, we point out an error in the recently published article [R.K. Mohanty, M.K. Jain, D. Dhall, A cubic spline approximation and application of TAGE iterative method for the solution of two-point boundary value problems with forcing function in integral form, Appl. Math. Model. 35 (2011) 3036–3047] and then correct it. [Copyright &y& Elsevier]

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]

In game theory the interaction among players obligates each player to develop a belief about the possible strategies of the other players, to choose a best-reply given those beliefs, and to look for an adjustment of the best-reply and the beliefs using a learning mechanism until they reach an equilibrium point. Usually, the behavior of an individual cost-function, when such best-reply strategies are applied, turns out to be non-monotonic and concluding that such strategies lead to some equilibrium point is a non-trivial task. Even in repeated games the convergence to a stationary equilibrium is not always guaranteed. The best-reply strategies analyzed in this paper represent the most frequent type of behavior applied in practice in problems of bounded rationality of agents considered within the Artificial Intelligence research area. They are naturally related with the, so-called, fixed-local-optimal actions or, in other words, with one step-ahead optimization algorithms widely used in the modern Intelligent Systems theory. This paper shows that for an ergodic class of finite controllable Markov games the best-reply strategies lead necessarily to a Lyapunov/Nash equilibrium point. One of the most interesting properties of this approach is that an expedient (or absolutely expedient) behavior of an ergodic system (repeated game) can be represented by a Lyapunov-like function non-decreasing in time. We present a method for constructing a Lyapunov-like function: the Lyapunov-like function replaces the recursive mechanism with the elements of the ergodic system that model how players are likely to behave in one-shot games. To show our statement, we first propose a non-converging state-value function that fluctuates (increases and decreases) between states of the Markov game. Then, we prove that it is possible to represent that function in a recursive format using a one-step-ahead fixed-local-optimal strategy. As a result, we prove that a Lyapunov-like function can be built using the previous recursive expression for the Markov game, i.e., the resulting Lyapunov-like function is a monotonic function which can only decrease (or remain the same) over time, whatever the initial distribution of probabilities. As a result, a new concept called Lyapunov games is suggested for a class of repeated games. Lyapunov games allow to conclude during the game whether the applied strategy provides the convergence to an equilibrium point (or not). The time for constructing a Potential (Lyapunov-like) function is exponential. Our algorithm tractably computes the Nash, Lyapunov and the correlated equilibria: a Lyapunov equilibrium is a Nash equilibrium, as well it is also a correlated equilibrium. Validity of the proposed method is successfully demonstrated both theoretically and practically by a simulated experiment related to the Duel game. [ABSTRACT FROM AUTHOR]

The paper studies the convergence of some parallel multisplitting block iterative methods for the solution of linear systems arising in the numerical solution of Euler equations. Some sufficient conditions for convergence are proposed. As special cases the convergence of the parallel block generalized AOR (BGAOR), the parallel block AOR (BAOR), the parallel block generalized SOR (BGSOR), the parallel block SOR (BSOR), the extrapolated parallel BAOR and the extrapolated parallel BSOR methods are presented. Furthermore, the convergence of the parallel block iterative methods for linear systems with special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper. [ABSTRACT FROM AUTHOR]

In two recent papers Gordeliy and Peirce (2013) investigating the use of the Extended Finite Element Method (XFEM) for modeling hydraulic fractures (HF), two classes of boundary value problem and two distinct enrichment types were identified as being essential components in constructing successful XFEM HF algorithms. In this paper we explore the accuracy and convergence properties of these boundary value formulations and enrichment strategies. In addition, we derive a novel set of crack-tip enrichment functions that enable the XFEM to model HF with the full range of power law r λ behavior of the displacement field and the corresponding r λ − 1 singularity in the stress field, for 1 2 ≤ λ < 1 . This novel crack-tip enrichment enables the XFEM to achieve the optimal convergence rate, which is not achieved by existing enrichment functions used for this range of power law. The two XFEM boundary value problem classes are as follows: (i) a Neumann to Dirichlet map in which the pressure applied to the crack faces is the specified boundary condition and the XFEM is used to solve for the corresponding crack width ( P → W ); and (ii) a mixed hybrid formulation of the XFEM that makes it possible to incorporate the singular behavior of the crack width in the fracture tip and uses a pressure boundary condition away from it ( P & W ). The two enrichment schemes considered are: (i) the XFEM-t scheme with full singular crack-tip enrichment and (ii) a simpler, more efficient, XFEM-s scheme in which the singular tip behavior is only imposed in a weak sense. If enrichment is applied to all the nodes of tip-enriched elements, then the resulting XFEM stiffness matrix is singular due to a linear dependence among the set of enrichment shape functions, which is a situation that also holds for the classic set of square-root enrichment functions. For the novel set of enrichment functions we show how to remove this rank deficiency by eliminating those enrichment shape functions associated with the null space of the stiffness matrix. Numerical experiments indicate that the XFEM-t scheme, with the new tip enrichment, achieves the optimal O ( h 2 ) convergence rate we expect of the underlying piece-wise linear FEM discretization, which is superior to the enrichment functions currently available in the literature for 1 2 < λ < 1 . The XFEM-s scheme, with only signum enrichment to represent the crack geometry, achieves an O ( h ) convergence rate. It is also demonstrated that the standard P → W formulation, based on the variational principle of minimum potential energy, is not suitable for modeling hydraulic fractures in which the fluid and the fracture fronts coalesce, while the mixed hybrid P & W formulation based on the Hellinger–Reissner variational principle does not have this disadvantage. [ABSTRACT FROM AUTHOR]

In this paper, a reliable approach based on the implicit hybrid method is presented to solve first order delay problems. The main difficulty in this approach is that, some points are not grid points. We use interpolation to overcome this difficulty. Theoretical results including stability and convergence results are presented. The efficiency of the proposed approach is tested through two examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce and consider a new class of equilibrium variational inequalities, called the mixed general equilibrium bifunction variational inequalities. We suggest and analyze some proximal methods for solving mixed general equilibrium bifunction variational inequalities using the auxiliary principle technique. Convergence of these methods is considered under some mild suitable conditions. Several cases are also discussed. Results in this paper include some new and known results as special cases. [Copyright &y& Elsevier]

Abstract: Net subdivision schemes recursively refine nets of univariate continuous functions defined on the lines of planar grids, and generate as limits bivariate continuous functions. In this paper a family of interpolatory net subdivision schemes related to the family of Dubuc–Deslauriers interpolatory subdivision schemes is constructed and analyzed. The construction is based on Gordon blending interpolants to nets of univariate functions, and on a particular class of blending functions with properties related to the Dubuc–Deslauriers schemes. The general analysis tools for net subdivision schemes, developed in a previous paper by the authors, together with the properties of the blending functions, lead to the proof of the convergence of these schemes to limit functions having the same integer smoothness as the limits of the corresponding Dubuc–Deslauriers schemes. These results are proved for net subdivision schemes corresponding to the first 84 members of the Dubuc–Deslauriers family, and conjectured for the rest. A concrete example of a family of piecewise polynomial blending functions is considered, together with the corresponding family of net subdivision schemes. The performance of the first two net subdivision schemes in this family is demonstrated by two examples. [Copyright &y& Elsevier]

Let G be a locally compact abelian group and let M (G) be the measure algebra of G. Assume that μ ∈ M (G) is power bounded, that is, sup n ≥ 0 ⁡ ‖ μ n ‖ 1 < ∞. This paper is concerned mainly with finding necessary and sufficient conditions for strong convergence of iterates of the convolution operators T μ f : = μ ⁎ f in L p (G) (1 ≤ p < ∞) spaces. Some related problems are also discussed. [ABSTRACT FROM AUTHOR]

Highlights • DCDG-EA algorithm uses reference vector decomposition to solve MaOPs. • CDOS selects an appropriate operator to generate offspring. • CDIS strategy simultaneously considers the convergence and diversity of solutions. Abstract Maintaining a good balance between the convergence and the diversity is particularly crucial for the performance of the evolutionary algorithms (EAs). However, the traditional multi-objective evolutionary algorithms, which have shown their competitive performance with a variety of practical problems involving two or three objectives, face significant challenges in case of problems with more than three objectives, namely many-objective optimization problems (MaOPs). This paper proposes a dynamic convergence–diversity guided evolutionary algorithm, namely (DCDG-EA) for MaOPs by employing the decomposition technique. Besides, the objective space of MaOPs is divided into K subspaces by a set of uniformly distributed reference vectors. Each subspace has its own subpopulation and evolves in parallel with the other subspaces. In DCDG-EA, the balance between the convergence and the diversity is achieved through the convergence–diversity based operator selection (CDOS) strategy and convergence–diversity based individual selection (CDIS) strategy. In CDOS, for each operator of the set of operators, a selection probability is assigned which is related to its convergence and diversity capabilities. Based on the attributed selection probabilities, an appropriate operator is selected to generate the offsprings. Furthermore, CDIS is used which allows to greatly overcome the inefficiency of the Pareto dominance approaches. It updates each subpopulation by using two independent distance measures that represent the convergence and the control diversity, respectively. The experimental results on DTLZ and WFG benchmark problems with up to 15 objectives show that our algorithm is highly competitive comparing with the four state-of-the-art evolutionary algorithms in terms of convergence and diversity. [ABSTRACT FROM AUTHOR]

Abstract This paper extends the matrix based approach to the setting of multiple subdivision schemes studied in Sauer (2012). Multiple subdivision schemes, in contrast to stationary and non-stationary schemes, allow for level dependent subdivision weights and for level dependent choice of the dilation matrices. The latter property of multiple subdivision makes the standard definition of the transition matrices, crucial ingredient of the matrix approach in the stationary and non-stationary settings, inapplicable. We show how to avoid this obstacle and characterize the convergence of multiple subdivision schemes in terms of the joint spectral radius of certain square matrices derived from subdivision weights. We illustrate our results with several examples. [ABSTRACT FROM AUTHOR]

Highlights • A fast successive relaxation updating method for continuous-discontinuous cellular automaton(CDCA) is proposed. • A fast CDCA is developed, and increments of displacement and nodal force are enlarged by the accelerating factor. • A new discontinuity tracking method which combines cell space cutting and cell neighbor searching is proposed. • The optimal value of the accelerating factor is studied, and an adaptive iteration scheme is proposed. Abstract Because of its local property, cellular automaton method has been widely applied in different subjects, but the main problem is that the cellular updating is time-consuming. In order to improve its calculation efficiency, a fast successive relaxation updating method is proposed in this paper. Firstly, an accelerating factor ω is defined, and a fast successive relaxation updating theory and its corresponding convergence conditions are developed. In each updating step, the displacement increment is enlarged ω times as a new increment to replace the old one, similarly, the nodal forces for its neighbors caused by this displacement increment are also enlarged by the same accelerating factor, and do those updating operations until the convergence is achieved. By this method, the convergence rate is greatly improved, by a suitable accelerating factor, 90 to 98% of iteration steps are decreased compared to that of the traditional method. Besides, the influence factors for the accelerating factor are studied, and numerical studies show that the suitable accelerating factor is 1.85 < ω < 1.99, which is greatly influenced by cell stiffness, and the optimal accelerating factor is 1.96 < ω < 1.99. Finally, numerical examples are given to illustrate that the present method is effective and high convergence rate compared to the traditional method. [ABSTRACT FROM AUTHOR]

Abstract In this paper we consider the analytical and numerical solutions for a two-dimensional multi-term time-fractional diffusion and diffusion-wave equation. We derive the analytical solution for the equation using the method of separation of variables and properties of the multivariate Mittag-Leffler function. An implicit difference approximation is constructed. Stability and convergence analysis of the numerical scheme are proved by the energy method. Numerical examples are constructed to evaluate the working of the numerical scheme as compared to theoretical analysis. Highlights • A new two-dimensional multi-term time-fractional diffusion and diffusion. • Wave equation (2D-MT-TFD-DWE) is considered. • The analytical solution for the 2D-MT-TFD-DWE is derived. • A novel implicit difference method (IDM) for 2D-MT-TFD-DWE is proposed. • The stability and convergence of the IDM are proved by the energy method. • Numerical examples are given. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce and consider a new class of mixed variational inequalities involving four operators, which are called extended general mixed variational inequalities. Using the resolvent operator technique, we establish the equivalence between the extended general mixed variational inequalities and fixed point problems as well as resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving general mixed variational inequalities. We study the convergence criteria for the suggested iterative methods under suitable conditions. Our methods of proof are very simple as compared with other techniques. The results proved in this paper may be viewed as refinements and important generalizations of the previous known results. [Copyright &y& Elsevier]

Abstract: This note points out some technical problems in the proofs of some lemmas in the above-mentioned paper, and presents their corresponding corrections. Nevertheless, the main results of that paper are still true. [ABSTRACT FROM AUTHOR]

In this paper, the generalized shift-splitting (GSS) preconditioner is implemented for solving a class of generalized saddle point problems which stem from the solution of complex symmetric linear systems. The GSS preconditioner is induced by the generalized shift-splitting iterative method. Theoretical analysis shows that the generalized shift-splitting iterative method is unconditionally convergent. Some numerical experiments are provided to show the effectiveness of the proposed preconditioner. [ABSTRACT FROM AUTHOR]

Fuzzy differential equation driven by Liu process is an important tool to deal with dynamic system in fuzzy environment. In many cases, however, it is difficult to find analytic solution of fuzzy differential equation driven by Liu process. Based on credibility theory, the Taylor series of fuzzy differential driven by Liu process is given and Euler approximation that use Taylor expansion as the recursion formula is obtained in this paper. Then the local convergence of Euler approximation is deduced. [ABSTRACT FROM AUTHOR]

Abstract This paper extends the adaptive normalized sub-band adaptive filtering (NSAF) by introducing variable error bound and memorizing the error convergence. The variable error bound attempts to vary the updating point of the filter coefficients. The error memory aids in updating the point based on the history of error rather than the previous error. The extended adaptiveness significantly improved NSAF in terms of convergence, complexity and noise robustness. The algorithm is also proved for its stability though the step-size is varied. The characteristics of the step-size are also investigated to determine its significance and nature on minimizing the error. The superiority of the MVS-SNSAF algorithm is proved against conventional algorithm using the aforesaid analysis. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present analytical pricing formulae for variance and volatility swaps, when both of the volatility and interest rate are assumed to be stochastic and follow a CIR (Cox–Ingersoll–Ross) process, forming a Heston–CIR hybrid model. The solutions are written in a series form with a theoretical proof of their convergence, ensuring the accuracy of the determined swap prices. The application of the formulae in practice is also demonstrated through the designed numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the construction and analysis of linearized numerical methods for solving the two-dimensional nonlinear time fractional Schrödinger equations. By adding different correction terms, two linearized compact alternating direction implicit (ADI) methods are proposed. Convergence of the proposed methods is obtained. Numerical results are presented to verify the accuracy and efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

Particle swarm optimization (PSO), a population-based stochastic optimization algorithm, has been successfully used to solve many complicated optimization problems. However analysis on algorithm convergence is still inadequate till now. In this paper, the martingale theory is applied to analyze the convergence of the standard PSO (SPSO). Firstly, the swarm state sequence is defined and its Markov properties are examined according to the theory of SPSO. Two closed sets, the optimal particle state set and optimal swarm state set, are then obtained. Afterwards, a supermartingale is derived as the evolutionary sequence of particle swarm with the best fitness value. Finally, the SPSO convergence analysis is carried out in terms of the supermartingale convergence theorem. Our results show that SPSO reaches the global optimum in probability. Moreover, the analysis on SPSO proves that the quantum-behaved particle swarm optimization (QPSO) is also a global convergence algorithm. The proof of the SPSO convergence in this work is new, simple and more effective without specific implementation. [ABSTRACT FROM AUTHOR]

Abstract: Variational iteration method has been widely used to handle linear and nonlinear models. The main property of the method is its flexibility and ability to solve nonlinear equations accurately and conveniently. In this paper, we present an alternative approach of the method then we study the convergence of the method for nonlinear differential equations. Our emphasis is to address the sufficient condition for convergence and the error estimate. Simple approaches of variational iteration method to nonlinear ordinary, partial and fractional differential equations are presented and the convergence results are briefly discussed. Some examples are investigated to verify convergence results and to illustrate the efficiency of the method. The basic ideas described in this paper are expected to be further employed to handle nonlinear models. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a new and interesting system of generalized mixed quasi-variational-like inclusions with -accretive operators and relaxed cocoercive mappings which contains variational inequalities, variational inclusions, systems of variational inequalities, systems of variational-like inequalities and systems of variational inclusions in the literature as special cases. By using the resolvent technique for the -accretive operators, we prove the existence of solutions and the convergence of a new -step iterative algorithm for this system of generalized mixed quasi-variational-like inclusions in real -uniformly smooth Banach spaces. The results in this paper unifies, extends and improves some known results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, a new system of generalized nonlinear variational-like inclusions is introduced and investigated in Hilbert spaces. By means of the resolvent operator technique, the existence and uniqueness of solution for the system of generalized nonlinear variational-like inclusions is demonstrated. Moreover, a perturbed Mann iterative method with errors for approximating the solution of the system of generalized nonlinear variational-like inclusions is constructed and the convergence and stability of the iterative sequence generated by the algorithm is discussed. The results presented in this paper generalize and unify many known results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, a class of stochastic age-dependent population equations with Markovian switching is considered. The main aim of this paper is to investigate the convergence of the numerical approximation of stochastic age-dependent population equations with Markovian switching. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions. An example is given for illustration. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a robust semi-explicit difference scheme for solving the Kuramoto–Tsuzuki equation with homogeneous boundary conditions. Because the prior estimate in -norm of the numerical solutions is very hard to obtain directly, the proofs of convergence and stability are difficult for the difference scheme. In this paper, we first prove the second-order convergence in -norm of the difference scheme by an induction argument, then obtain the estimate in -norm of the numerical solutions. Furthermore, based on the estimate in -norm, we prove that the scheme is also convergent with second order in -norm. Numerical examples verify the correction of the theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this paper a class of weakly singular Volterra integral equations with an infinite set of solutions is investigated. Among the set of solutions only one particular solution is smooth and all others are singular at the origin. The numerical solution of this class of equations has been a difficult topic to analyze and has received much previous investigation. The aim of this paper is to improve the convergence rates by a graded mesh method. The convergence rates are proved and a variety of numerical examples are provided to support the theoretical findings. [Copyright &y& Elsevier]

Abstract: The paper studies the convergence of some block iterative methods for the solution of linear systems when the coefficient matrices are generalized -matrices. A truth is found that the class of conjugate generalized -matrices is a subclass of the class of generalized -matrices and the convergence results of R. Nabben [R. Nabben, On a class of matrices which arises in the numerical solution of Euler equations, Numer. Math. 63 (1992) 411–431] are then extended to the class of generalized -matrices. Furthermore, the convergence of the block AOR iterative method for linear systems with generalized -matrices is established and some properties of special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper. [Copyright &y& Elsevier]

This paper discusses the double convergence hypothesis (DCH) that the uncritical analysis of the so-called environmental Kuznets curve (EKC) implicitly implies. According to the EKC, empirical evidence would support the hypothesis that economic growth, through a deterministic sequence of phases, would produce cross-country convergence in per capita output and, as a by-product, convergence in (the demand for) environmental quality. However, factual analysis seems to reject the general hypothesis of convergence in per capita output, limiting the validity of such an assessment to the case of homogeneous groups of countries. This is why, to test the DCH, the paper focuses on the original group of OECD countries for which the economic convergence turned out to be true. This allows us to verify whether ¿green¿ ¿ and ¿ convergence follows as a consequence of economic convergence. The paper also tests for a more equitable distribution of protection policies among countries. Unlike other studies, which do not make the DCH explicit and have focused on pollutant emissions, this research explicitly tests for the double convergence using, as a proxy for the demand for environmental quality, the territory set aside for protected areas. The results confirm that, for the selected homogenous group of countries, growth has accompanied the demand for environmental quality. This happens both in terms of conditional and stochastic convergence. [Copyright &y& Elsevier]

Abstract: Splitting methods are used to solve most of the linear systems, when the conventional method of Gauss is not efficient. These methods use the factorization of the square matrix into two matrices and as where is nonsingular. Basic iterative methods such as Jacobi or Gauss–Seidel define the iterative scheme for matrices that have no zeros along its main diagonal. This paper is concerned with the development of an iterative method to approximate solutions when the coefficient matrix has some zero diagonal entries. The algorithm developed in this paper involves the analysis of a discrete-time descriptor system given by the equation , being the error vector. [Copyright &y& Elsevier]

In this paper we introduce and study a new system of nonlinear variational inclusions with-accretive mappings in Banach spaces. By using the resolvent operator associated with-accretive mappings, we construct some new iterative algorithms for approximating the solution of this system of variational inclusions. We also prove the existence of solutions and the convergence of the sequences generated by the algorithm in Banach spaces. The results presented in this paper extend and improve some known results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a new class of equilibrium problems, which is called the generalized mixed quasi-equilibrium problems with trifunction. Using the auxiliary principle technique, we suggest and analyze a proximal point method for solving the generalized mixed quasi-equilibrium problems. It is shown that the convergence of the proposed method requires only pseudomonotonicity, which is a weaker condition than monotonicity. Our results represent an improvement and refinement of previously known results. Since the generalized mixed quasi-equilibrium problems include equilibrium problems and variational inequalities as special cases, results proved in this paper continue to hold for these problems. [Copyright &y& Elsevier]

In this paper, we discuss the convergence of the DFP algorithm with revised search direction. Under some inexact line searches, we prove that the algorithm is globally convergent for continuously differentiable functions and the rate of convergence of the algorithm is one-step superlinear and n-step second order for uniformly convex objective functions.From the proof of this paper, we obtain the superlinear and n-step second-order convergence of the DFP algorithm for uniformly convex objective functions. [Copyright &y& Elsevier]

Generalized hill climbing (GHC) algorithms provide a general local search strategy to address intractable discrete optimization problems. GHC algorithms include as special cases stochastic local search algorithms such as simulated annealing and the noising method, among others. In this paper, a proof of convergence of GHC algorithms is presented, that relaxes the sufficient conditions for the most general convergence proof for stochastic local search algorithms in the literature. Note that classical convergence proofs for stochastic local search algorithms require either that an exponential distribution be used to model the acceptance of candidate solutions along a search trajectory, or that the Markov chain model of the algorithm must be reversible. The proof in this paper removes these limitations, by introducing a new path concept between global and local optima. Convergence is based on the asymptotic behavior of path probabilities between local and global optima. Examples are given to illustrate the convergence conditions. Implications of this result are also discussed. [Copyright &y& Elsevier]

In this paper, we consider the numerical solution of the time fractional Fokker-Planck equation. We propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of Jacobi interpolation polynomial for this equation. The convergence of the method is rigorously established. Numerical tests are carried out to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Highlights • The balance of convergence and diversity are analyzed and divided into horizontal imbalance and vertical imbalance. • The one-to-one correspondence between the region and the subproblem is established by region division. • Adaptive Region Adjustment (ARA) strategy is proposed to improve the balance between convergence and diversity. • A comprehensive experiment is designed on UF and MOP to prove the effectiveness of our proposed algorithm. Abstract The multiobjective evolutionary algorithm based on decomposition (MOEA/D), which decomposes a multiobjective optimization problem (MOP) into a number of optimization subproblems and optimizes them in a collaborative manner, becomes more and more popular in the field of evolutionary multiobjective optimization. The mechanism of balance convergence and diversity is very important in MOEA/D. In the process of optimization, the chosen solutions must be distinctive and as close as possible to the Pareto front. In this paper, we first explore the relation between subproblems and solutions. Then we propose the adaptive region adjustment strategy to balance the convergence and diversity based on the objective region partition concept. Finally, this strategy is embedded in the MOEA/D framework and then a simple but efficient algorithm is proposed. To demonstrate the effectiveness of the proposed algorithm, comprehensive experiments have been designed. The simulation results show the effectiveness of our proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, a closed-form pricing formula for European options in the form of an infinite series is derived under the Heston model with the interest rate being another random variable following the CIR (Cox–Ingersoll–Ross) model. One of the main advantages for the newly derived series solution is that we can provide a radius of convergence, which is complemented by some numerical experiments demonstrating its speed of convergence. To further verify our formula, option prices calculated through our formula are also compared with those obtained from Monte Carlo simulations. Finally, a set of pricing formulae are derived with the series expanded at different points so that the entire time horizon can be covered by converged solutions. [ABSTRACT FROM AUTHOR]

In this paper, a high-order compact multistep alternating direction implicit (ADI) difference scheme is constructed and analyzed for two-dimensional (2D) nonlinear delayed reaction–diffusion equations (NDREs) with variable coefficients. By the discrete energy method, it is presented that it achieves the convergence rate of O ( τ 2 + h x 4 + h y 4 ) with respect to H 1 - and L 2 -norms in non-constrained temporal grids, and is unconditionally stable. Finally, numerical results demonstrate the effectiveness of our ADI solver and exhibit the correctness of theoretical results. [ABSTRACT FROM AUTHOR]

Boundary integral method has been implemented successfully in practice for simulating problems with free boundaries. Though the method produces accurate and efficient numerical results, its convergence study is usually limited to numerical demonstrations by successively reducing time step and increasing resolution for a test problem. In this paper, we present a rigorous convergence and error analysis of the boundary integral method for a free boundary system. We focus our study on a nonlinear tumor growth problem. The boundary integral formulation yields a Fredholm type integral equation with moving boundaries. We show that in two dimensions, the convergence of the scheme in the L ∞ norm has first order accuracy on the time direction and Δ θ α on the spatial direction. [ABSTRACT FROM AUTHOR]

Particle swarm optimization (PSO), a population-based stochastic optimization algorithm, has been successfully used to solve many complicated optimization problems. However analysis on algorithm convergence is still inadequate till now. In this paper, the martingale theory is applied to analyze the convergence of the standard PSO (SPSO). Firstly, the swarm state sequence is defined and its Markov properties are examined according to the theory of SPSO. Two closed sets, the optimal particle state set and optimal swarm state set, are then obtained. Afterwards, a supermartingale is derived as the evolutionary sequence of particle swarm with the best fitness value. Finally, the SPSO convergence analysis is carried out in terms of the supermartingale convergence theorem. Our results show that SPSO reaches the global optimum in probability. Moreover, the analysis on SPSO proves that the quantum-behaved particle swarm optimization (QPSO) is also a global convergence algorithm. The proof of the SPSO convergence in this work is new, simple and more effective without specific implementation. [ABSTRACT FROM AUTHOR]

In this paper, based on a convergence splitting of the matrix A , we present an inner–outer iteration method for solving the linear system A x = b . We analyze the overall convergence of this method without any other restriction on its parameters. Moreover, we give the accelerated inner–outer iteration method, and discuss how to apply the inner–outer iterations as a preconditioner for the Krylov subspace methods. The inner–outer iteration method can also be used for the solution of A X B = C . Finally, several numerical examples are given to validate the performance of our proposed algorithms. [ABSTRACT FROM AUTHOR]

It is known that the concept of optimality is not defined for multidimensional iterative methods for solving nonlinear systems of equations. However, usually optimal fourth-order schemes (extended to the case of several variables) are employed as starting steps in order to design higher order methods for this kind of problems. In this paper, we use a non-optimal (in scalar case) iterative procedure that is specially efficient for solving nonlinear systems, as the initial steps of an eighth-order scheme that improves the computational efficiency indices of the existing methods, as far as the authors know. Moreover, the method can be modified by adding similar steps, increasing the order of convergence three times per step added. This kind of procedures can be used for solving big-sized problems, such as those obtained by applying finite differences for approximating the solution of diffusion problem, heat conduction equations, etc. Numerical comparisons are made with the same existing methods, on standard nonlinear systems and Fisher’s equation by transforming it in a nonlinear system by using finite differences. From these numerical examples, we confirm the theoretical results and show the performance of the proposed schemes. [ABSTRACT FROM AUTHOR]

It is generally accepted that conflicts between convergence and distribution deteriorate with an increase in the number of objectives. Furthermore, Pareto dominance loses its effectiveness in many-objectives optimization problems (MaOPs), which have more than three objectives. Therefore, a more valid selection method is needed to balance convergence and distribution. This paper presents a many-objective evolutionary algorithm, called Adaptive Neighborhood Selection for Many-objective evolutionary algorithm (ANS-MOEA), to deal with MaOPs. This method defines the performance of each individual by two types of information, convergence information (CI) and distribution information (DI). In the critical layer, a well-converged individual is selected first from the population, and its neighbors, calculated by DI, are pushed into neighbor collection (NC) soon afterwards. Then, the proper distribution of the population is ensured by competition individuals with large DI go back to the population and individuals with small DI remain in the collection. Four state-of-the-art MaOEAs are selected as the competitive algorithms to validate ANS-MOEA. The experimental results show that ANS-MOEA can solve a MaOP and generate a set of remarkable solutions to balance convergence and distribution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical methods for stationary Stokes equations with damping. The mark and cell(MAC) method has been applied to discretize the problem on non-uniform grids. We establish the LBB condition and the stability for the MAC scheme. Then we obtain the second order super-convergence in L2 norm for both velocity and pressure on non-uniform grids. We also obtain the second order convergence for some terms of H1 norm of the velocity, and the other terms of H1 norm are second order convergence on uniform grids. Numerical experiments using the MAC scheme show agreement of the numerical results with theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, based on the positive-definite and positive-semidefinite splitting (PPS) iteration scheme, we establish a class of Uzawa-PPS iteration methods for solving nonsingular and singular non-Hermitian saddle point problems with the (1,1) part of the coefficient matrix being non-Hermitian positive definite. Theoretical analyses show that the convergence and semi-convergence properties of the proposed methods can be guaranteed under suitable conditions. Furthermore, we consider acceleration of the Uzawa-PPS methods by Krylov subspace (like GMRES) methods and discuss the spectral properties of the corresponding preconditioned matrix. Numerical experiments are given to confirm the theoretical results which show that the feasibility and effectiveness of the proposed methods and preconditioners. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modified generalized shift-splitting (denoted by MGSSP) preconditioned method for solving large sparse saddle-point problems. By theoretical analyses, we verify the MGSSP iteration method unconditionally converges to the unique solution of the saddle point problems, estimate the sharp eigenvalue bounds of the related iteration matrix and point out the corresponding preconditioned matrix is positive real. Finally, we perform some numerical computations to show the efficiency and the feasibility of the MGSSP preconditioner. [ABSTRACT FROM AUTHOR]

One of the most important subject in financial mathematics is the option pricing. The most famous result in this area is Black–Scholes formula for pricing European options. This paper is concerned with a method for solving a generalized Black–Scholes equation in a reproducing kernel Hilbert space. Subsequently, the convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method. Furthermore, the error estimates for obtained approximation in reproducing kernel Hilbert space are presented. Finally, a numerical example is considered to illustrate the computation efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we use a Block Nyström Method (BNM) to obtain the numerical solution for one-dimensional Bratu’s problem. The convergence analysis of the method is discussed and it is shown that the theoretical order of the method is consistent with its numerical rate of convergence. The accuracy benefit of the BNM is demonstrated by comparing it to several other known methods given in the literature. It is demonstrated that the BNM can also be used to solve Bratu’s problem associated with initial conditions by simply adjusting the boundary conditions in the algorithm. [ABSTRACT FROM AUTHOR]

Backpropagation (BP) algorithm is the typical strategy to train the feedforward neural networks (FNNs). Gradient descent approach is the popular numerical optimal method which is employed to implement the BP algorithm. However, this technique frequently leads to poor generalization and slow convergence. Inspired by the sparse response character of human neuron system, several sparse-response BP algorithms were developed which effectively improve the generalization performance. The essential idea is to impose the responses of hidden layer as a specific L 1 penalty term on the standard error function of FNNs. In this paper, we mainly focus on the two remaining challenging tasks: one is to solve the non-differential problem of the L 1 penalty term by introducing smooth approximation functions. The other aspect is to provide a rigorous convergence analysis for this novel sparse response BP algorithm. In addition, an illustrative numerical simulation has been done to support the theoretical statement. [ABSTRACT FROM AUTHOR]