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Abstract: In this paper different structure of the neurons in the hidden layer of a feed-forward network, for forecasting of the dynamic systems, are proposed. Each neuron in the network is a combination of the sigmoidal activation function (SAF) and wavelet activation function (WAF). The output of the hidden neuron is the product of the output from these two activation functions. A delay element is used to feedback the output of the sigmoidal and the wavelet activation function to each other. This arrangement leads to proposed five possible configurations of recurrent neurons. Besides proposing these neuron models, the presented paper tries to compare the performance of wavelet function with sigmoid function. To guarantee the stability and the convergence of the learning process, upper bound for the learning rates has been investigated using the Lyapunov stability theorem. A two-phase adaptive learning rate ensures this upper bound. Universal approximation property of the feed-forward network with the proposed neurons has also been investigated. Finally, the applicability and comparison of the proposed recurrent networks has been weathered on two benchmark problem catering different types of dynamical systems. [Copyright &y& Elsevier]

Abstract: Daily optimum economic emission scheduling of hydrothermal systems is an important task in the operation of power systems. Many heuristic techniques such as differential evolution, and particle swarm optimization have been applied to solve this problem and found to perform better in comparison with classical techniques. But a very common problem with these methods is that they often converge to sub-optimal solution prematurely. A reliable and efficient method termed as self-organizing hierarchical particle swarm optimization technique with time-varying acceleration coefficients (SOHPSO_TVAC) is presented in this paper to avoid premature convergence. A multi-chain cascaded hydrothermal system with non-linear relationship between water discharge rate, power generation and net head is considered in this paper. The water transport delay between connected reservoirs is also taken into consideration. The problem is formulated considering both cost and emission as competing objectives. The effect of valve point loading is also taken into account in the present problem formulation. The feasibility of the proposed method is demonstrated on a sample test system. The results of the proposed technique are compared with other heuristic techniques. It is found that the results obtained by the proposed technique are superior in terms of fuel cost, emission output etc. [Copyright &y& Elsevier]

Abstract: Particle swarm optimization (PSO) is one of the most important research topics on swarm intelligence. Existing PSO techniques, however, still contain some significant disadvantages. In this paper, we present a new QBL-PSO algorithm that uses QBL (query-based learning) to improve both the exploratory and exploitable capabilities of PSO. Here, we apply a QBL method proposed in our previous research to PSO, and then test this new algorithm on a real case study on problems of power conservation. Our algorithm not only broadens the search diversity of PSO, but also improves its precision. Conventional PSO often snag on local solutions when performing queries, instead of finding better global solutions. To resolve this limitation, when particles converge in nature, we direct some of them into an “ambiguous solution space” defined by our algorithm. This paper introduces two ways to invoke this QBL algorithm. Our experimental results confirm that the proposed method attains better convergence to the global best solution. Finally, we present a new PSO model for solving multi-objective power conservation problems. Overall, this model successfully reduces power consumption, and to our knowledge, this paper represents the first attempt within the literature to apply the QBL concept to PSO. [Copyright &y& Elsevier]

Abstract: This paper presents an improved self-adaptive particle swarm optimization algorithm (ISAPSO) to solve hydrothermal scheduling (HS) problem. To overcome the premature convergence of particle swarm optimization (PSO), the evolution direction of each particle is redirected dynamically by adjusting the two sensitive parameters of PSO in the evolution process. Moreover, a new strategy is proposed to handle the various constraints of HS problem in this paper. The results solved by this proposed strategy can strictly satisfy the constraints of HS problem. Finally, the feasibility and effectiveness of proposed ISAPSO algorithm is validated by a test system containing four hydro plants and an equivalent thermal plant. The results demonstrate that the proposed ISAPSO can get a better solution in both robustness and accuracy while compared with the other methods reported in this literature. [Copyright &y& Elsevier]

Abstract: In this paper, we study a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of variational inequalities for two inverse strongly accretive mappings and the set of common fixed points of an infinite family of nonexpansive mappings in a real 2-uniformly smooth and uniformly convex Banach space. We prove some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the corresponding results announced by many others. [Copyright &y& Elsevier]

Abstract: This paper presents a novel three-dimensional finite element for gradient elasticity. The new finite element BR153L9 is a straightforward extension of the two-dimensional element QU34L4 developed by Shu et al. (1999) , which is based on a mixed-type formulation. Within this paper the derivation of the mixed-type finite element scheme is reviewed and details of the implementation are given. Finally, numerical results of an extended patch test and a benchmark test with the three-dimensional finite element are presented in order to validate the formulation and prove the convergence. [Copyright &y& Elsevier]

Abstract: There are verities of useful Krylov subspace methods to solve nonsymmetric linear system of equations. GMRES is one of the best Krylov solvers with several different variants to solve large sparse linear systems. Any GMRES implementation has some advantages. As the solution of ill-posed problems are important. In this paper, some GMRES variants are discussed and applied to solve these kinds of problems. Residual smoothing techniques are efficient ways to accelerate the convergence speed of some iterative methods like CG variants. At the end of this paper, some residual smoothing techniques are applied for different GMRES methods to test the influence of these techniques on GMRES implementations. [Copyright &y& Elsevier]

Abstract: Many Radial Basis Functions (RBFs) contain a free shape parameter that plays an important role for the application of Meshless method to the analysis of multilayered composite and sandwich plates. In most papers the authors end up choosing this shape parameter by trial and error or some other ad hoc means. In this paper a novel algorithm for shape parameter selection, based on a convergence analysis, is presented. The effectiveness of this algorithm is assessed by static analyses of laminated composite and sandwich plates. [Copyright &y& Elsevier]

Abstract: In Bukovský et al. (2001) the authors proved Theorem 5.10 saying that eleven seemingly different properties of a perfectly normal space X are equivalent. One of the properties says that every Borel image of X into is eventually bounded. B. Tsaban and L. Zdomskyy (in press) have proved that any perfectly normal topological QN-space (for the definition see Bukovský et al. (1991) ) possesses this property, therefore all properties of the theorem. In this paper we simply prove that every perfectly normal topological QN-space possesses another property of that theorem – see Theorems 1 and 3. The main tools of our proof are sequence selection principles for quasi-normal convergence introduced in the paper. Some of introduced principles are worth studying in their own right and we initiate their study. Moreover, one of our main results immediately implies Recławʼs Theorem (Recław, 1997) . [Copyright &y& Elsevier]

Highlights • Sine cosine algorithm is modified to solve high-dimensional optimization problem. • A modified position-updating equation is presented to accelerate convergence. • A novel nonlinear decreasing conversion parameter strategy is designed to balance exploration and exploitation. • The performance of ISCA is tested on 24 high-dimensional functions and three engineering design problem. • Comparisons demonstrate that ISCA is a competitive solver for high-dimensional optimization problems. Abstract The sine cosine algorithm (SCA) is a relatively novel population-based optimization technique that has been proven competitive with other algorithms and it has received significant interest from researchers in different fields. However, similar to other population-based algorithms, SCA tends to be trapped in local optima and unbalanced exploitation. Additionally, to our limited knowledge, the present SCA and its variants have not been applied to the high-dimensional global optimization problems. This paper presents an improved version of the SCA (ISCA) for solving high-dimensional problems. A modified position-updating equation by introducing inertia weight is proposed to accelerate convergence and avoid falling into the local optima. In addition, to balance the exploration and exploitation of the SCA, we present a new nonlinear conversion parameter decreasing strategy based on the Gaussian function. The effectiveness of the proposed ISCA is evaluated using 24 benchmark high-dimensional (D = 30, 100, 500, 1000, and 5000) functions, large-scale global optimization problems from the IEEE CEC2010 competition, and several real-world engineering applications. The comparisons show that the proposed ISCA can better escape from local optima with faster convergence than both the traditional SCA and other population-based algorithms. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider a coupled system of m (≥ 2) singularly perturbed semilinear reaction–diffusion equations with a discontinuous source term having a discontinuity at a point in the interior of the domain. The diffusion term of each equation is multiplied by small singular perturbation parameter, but these parameters are assumed to be different in magnitude. A numerical method is constructed on a variant of Shishkin mesh. The approximations generated by this method are shown to be almost second order uniformly convergent with respect to all perturbation parameters. Numerical results are in support of the theoretical results. Highlights • A coupled system of m (≥ 2) singularly perturbed semilinear reaction-diffusion equations is considered. • The source term is having a discontinuity at a point in the interior of the domain. • The diffusion term of each equation is multiplied by a small perturbation parameter. • All the perturbation parameters are different in magnitude. • The constructed numerical method has almost second parameter uniform convergence. [ABSTRACT FROM AUTHOR]

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]

Abstract This paper presents van der Corput-type lemmas for Bessel and Airy transforms with shifted Jacobi weight functions. These results on the asymptotic orders of the highly oscillatory integrals on the frequency are optimal. Furthermore, from these estimates, the convergence rates on Filon-type methods are easily derived. [ABSTRACT FROM AUTHOR]

Abstract Ultra high precision servo control systems usually require fast sampling rate to achieve desired digital control performance, where output sampling restrictions (such as sensor bandwidth) pose major challenges for such applications. In this paper, we propose a novel sampled-data multi-rate active disturbance rejection control (ADRC) architecture under sampling rate restrictions, where a digital extended state observer (ESO) with inter-sample output prediction structure is employed to construct inter-sample dynamics between two consecutive sampling instants, such that high frequency disturbances can be better rejected. The convergence of the estimation and the stability of the closed-loop system are analyzed based on Lyapunov function method and separation principle. The proposed control architecture is implemented on a voice coil motor (VCM) actuated micro–nano servo gantry, demonstrating significant performance improvement over existing single-rate or multi-rate ADRC methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a nonconforming finite element method (NFEM) is proposed for the constrained optimal control problems (OCPs) governed by a bilinear state equation. The state and adjoint state are approximated by the nonconforming E Q 1 r o t element, and the control is approximated by the orthogonal projection through the state and adjoint state. Some superclose and superconvergence properties are obtained by full use of the distinguish characters of this E Q 1 r o t element, such as the interpolation operator equals the Ritz projection, and the consistency error is one order higher than its interpolation error in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract In this paper, by adopting the preconditioned technique for the accelerated generalized successive overrelaxation method (AGSOR) proposed by Edalatpour et al. (2015), we establish the preconditioned AGSOR (PAGSOR) iteration method for solving a class of complex symmetric linear systems. The convergence conditions, optimal iteration parameters and corresponding optimal convergence factor of the PAGSOR iteration method are determined. Besides, we prove that the spectral radius of the PAGSOR iteration method is smaller than that of the AGSOR one under proper restrictions, and its optimal convergence factor is smaller than that of the preconditioned symmetric block triangular splitting (PSBTS) one put forward by Zhang et al. (2018) recently. The spectral properties of the preconditioned PAGSOR matrix are also proposed. Numerical experiments illustrate the correctness of the theories and the effectiveness of the proposed iteration method and the preconditioner for the generalized minimal residual (GMRES) method. [ABSTRACT FROM AUTHOR]

Abstract The construction of finite element approximations in H (div , Ω) usually requires the Piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region Ω. It is known that degradation may occur in convergence order if non affine geometric mappings are used. On this point, we revisit a general procedure for the improvement of two-dimensional flux approximations discussed in a recent paper of this journal (Comput. Math. Appl. 74 (2017) 3283–3295). The starting point is an approximation scheme, which is known to provide L 2 -errors with accuracy of order k + 1 for sufficiently smooth flux functions, and of order r + 1 for flux divergence. An example is R T k spaces on quadrilateral meshes, where r = k or k − 1 if linear or bilinear geometric isomorphisms are applied. Furthermore, the original space is required to be expressed by a factorization in terms of edge and internal shape flux functions. The goal is to define a hierarchy of enriched flux approximations to reach arbitrary higher orders of divergence accuracy r + n + 1 as desired, for any n ≥ 1. The enriched versions are defined by adding higher degree internal shape functions of the original family of spaces at level k + n , while keeping the original border fluxes at level k. The case n = 1 has been discussed in the mentioned publication for two particular examples. General stronger enrichment n > 1 shall be analyzed and applied to Darcy's flow simulations, the global condensed systems to be solved having same dimension and structure of the original scheme. [ABSTRACT FROM AUTHOR]

Highlights • A new variational model for image retinex is proposed. • The existence of minimizer for the variational model is proved. • An efficient algorithm is derived to solve the model by adding auxiliary variables. • The convergence of the numerical algorithm is proved under some assumptions. • The proposed method can be extended to other regularization and fidelity terms. Abstract In this paper, we propose a detail preserving variational model for Retinex to simultaneously estimate the illumination and the reflectance from an observed image. Most previous models use the log-transform as pretreatment which results in loss of details in reflectance. From this observation, a detail preserving variational method is proposed for better decomposition. Different from the log-transform based models, the proposed model performs the decomposition directly in the image domain. Mathematically, we prove the existence of a solution for the proposed model. Numerically, we derive an efficient iterative algorithm by utilizing alternating direction method of multipliers (ADMM) method. Experimental results demonstrate the effectiveness of the proposed method. Compared with other closely related Retinex methods, the proposed method achieves competitive results on both subjective and objective assessments. [ABSTRACT FROM AUTHOR]

Abstract In this paper, our main aim is to investigate the stability and strong convergence of an implicit numerical approximations for neutral-type stochastic differential equations with superlinearly growing coefficients. After providing moment boundedness and exponential stability for the exact solutions, we show that the backward Euler–Maruyama numerical method preserves stability and boundedness of moments, and the numerical approximations converge strongly to the true solutions for sufficiently small step size. [ABSTRACT FROM AUTHOR]

Abstract The spectral conjugate gradient method is an effective method for large-scale unconstrained optimization problems. In this paper, based on Quasi-Newton direction and Quasi-Newton equation, a new spectral conjugate gradient method is proposed. This method is motivated by the three-term modified Polak-Ribiyre-Polyak (PRP) method and spectral parameters. The global convergence of algorithm is proved for general functions under a strong Wolfe line search. Numerical results show that the new algorithm is superior to the three-term modified PRP method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, by constructing appropriate splitting, a generalized AOR (GAOR) method for solving a class of generalized saddle point problems has been presented, and the convergence of the method is studied. The numerical experiments results show the efficiency and robustness of the new method. [ABSTRACT FROM AUTHOR]

Abstract Variable preconditioning has earlier been developed as a realization of quasi-Newton methods for elliptic problems with uniformly bounded nonlinearities. This paper presents a generalization of this approach to strongly nonlinear problems, first on an operator level, then for elliptic problems allowing power order growth of nonlinearities. Numerical tests reinforce the convergence results. [ABSTRACT FROM AUTHOR]

Highlights • A fast recommendation approach is proposed. • The approach is based on self-adaptation and multi-thresholding. • An automatic convergence procedure is proposed to improve the performance of our approach. • The MAE is improved between 1.02% and 12.93%, and the computational time between 25.38% and 54.83%. Abstract Finding relevant and interesting items on e-commerce websites is a major challenge in the information era. Recommender systems are widely used to help users deal with the information overload problem by giving them personalized recommendations. Furthermore, they help businesses make more profits. Collaborative filtering represents one of the most successful recommendation approaches. In general, several existing recommendation methods have demonstrated good performance in terms of prediction quality. However, they may require prohibitive computational times, and they often confront the sparsity problem, which negatively affects the efficiency of the system. In this paper, we propose a fast recommendation algorithm based on self-adaptation and multi-thresholding. It is designed to deal with the mentioned drawbacks, and improve the prediction quality. In addition, it is able to converge automatically. Extensive experiments on two real-world data sets demonstrate that our proposed method can achieve significantly better performance than other state-of-the-art methods. Particularly, it improves the MAE between 1.02% and 12.93%, and the computational time between 25.38% and 54.83%. [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 In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, generalizing the classical Biot equations. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the use of separate simulators. It is not met by the classical Newton method. We propose three different linearization schemes incorporating the fixed-stress splitting scheme, coupled with an L-scheme, Modified Picard and Newton linearization of the flow equations. All schemes allow the efficient and robust decoupling of mechanics and flow equations. In particular, the simplest scheme, the Fixed-Stress-L-scheme, employs solely constant diagonal stabilization, has low cost per iteration, and is very robust. Under mild, physical assumptions, it is theoretically shown to be a contraction. Due to possible break-down or slow convergence of all considered splitting schemes, Anderson acceleration is applied as post-processing. Based on a special case, we justify theoretically the general ability of the Anderson acceleration to effectively accelerate convergence and stabilize the underlying scheme, allowing even non-contractive fixed-point iterations to converge. To our knowledge, this is the first theoretical indication of this kind. Theoretical findings are confirmed by numerical results. In particular, Anderson acceleration has been demonstrated to be very effective for the considered Picard-type methods. Finally, the Fixed-Stress-Newton scheme combined with Anderson acceleration shows the best performance among the splitting schemes. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the Galerkin method to approximate the solution of Fredholm–Hammerstein integral equations of second kind with weakly singular kernels, using Legendre polynomial bases. We prove that for both the algebraic and logarithmic kernels, the Legendre Galerkin method has order of convergence O (n − r) , whereas the iterated Legendre Galerkin method converges with the order O (n − r − α + 1 2 ) for the algebraic kernel, and order O (log n n − r − 1 2 ) for logarithmic kernel in both L 2 -norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the solution. We also propose the Legendre multi-Galerkin and iterated Legendre multi-Galerkin methods. We prove that iterated Legendre multi-Galerkin method has order of convergence O ((1 + c l o g n) n − r − 2 α + 1 2 ) for the algebraic kernel, and order of convergence O ( (log n) 2 (1 + c l o g n) n − r − 3 2 ) for logarithmic kernel in both L 2 -norm and infinity norm. Numerical examples are given to illustrate the theoretical results. [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]

Abstract This paper is concerned with general symmetric 2 n -point interpolatory subdivision scheme (ISS) with polynomial reproduction of arbitrary order m ≤ n. An explicit formulation is derived for the corresponding refinement symbol, and convergence intervals are obtained for the one-parameter case, the left hand endpoints of which improve on previous such lower parameter convergence bounds. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a conditional disturbance negation (CDN) based active disturbance rejection control (ADRC) scheme for the velocity and altitude tracking control of flexible air-breathing hypersonic vehicles (FAHVs) in the presence of various uncertainties and disturbances. A pioneering CDN module is proposed to selectively compensate the total disturbances according to the characteristics of the total disturbances and the FAHV dynamics, which is aimed at canceling some detrimental disturbances while profiting from other beneficial disturbances. Firstly, model-based extended state observers are applied to estimate the total disturbances in FAHVs in an efficient way. Based on the Lyapunov theory and the estimated disturbances, a novel disturbance characterization indicator (DCI) is then designed to show whether the total disturbance harms or benefits the tracking performance in engineering practice. Then, DCI is utilized to construct the CDN module. In consequence, the CDN based FAHV tracking control is implemented in an ADRC framework, yielding the CDN based ADRC scheme. The stability analysis is conducted to show the convergence of the closed-loop system. Finally, the effectiveness and superiority of the proposed control scheme are validated by some representative simulations in various flight conditions. [ABSTRACT FROM AUTHOR]

Abstract Artificial Bee Colony (ABC) algorithm inspired from honeybee optimization in food search, provides an optimum reduced order model for complex grid-tied microgrid system. The paper aims to develop the best second-order model for a fifteenth order microgrid system in a grid-tied mode such that it closely replicates the original system. The comparative study of ABC reduced order model along with some other reduced order models of grid-tied microgrid system in time and frequency domain proves its suitability particularly in terms of least error in reduction and faster convergence. Eigenvalue stability has been discussed in different cases of perturbation and participation analysis. [ABSTRACT FROM AUTHOR]

Abstract This paper presents numerical studies with three classes of quadratic Generalized FEM (GFEM) approximations and shows that all of them lead to errors that are orders of magnitude smaller than the FEM with quarter-point elements, provided that appropriate enrichments are selected. However, all of them lead to severely ill-conditioned systems of equations, with a condition number up to O (h − 10) in the case of the GFEM based on a quadratic partition of unity. Enrichment modifications able to address the ill-conditioning of quadratic GFEM approximations while preserving their optimal convergence are proposed. A robust enrichment modification strategy based on a discontinuous FE interpolant is proposed to control the conditioning of branch function enrichment. The discontinuous FE interpolant is a generalization of the continuous one used with the Stable GFEM (SGFEM). We show that SGFEM spaces based on p -hierarchical FEM enrichments are the same as their GFEM counterparts. This guarantees that both GFEM and SGFEM spaces will lead to the same solution. This is not the case for the other classes of second-order spaces. The robustness of the proposed approximation spaces with respect to the crack location in the mesh is also demonstrated. Highlights • First-order GFEM is not competitive with quarter-point FEMs. • Quadratic GFEMs are more accurate than FEM but can be severely ill-conditioned. • Proposed SGFEM provides same conditioning as FEM and same accuracy as GFEM. • III-conditioning of Heaviside functions can be controlled without a FE interpolant. • SGFEM with discontinuous FE interpolant is robust with respect to crack position in the mesh. [ABSTRACT FROM AUTHOR]

Abstract Error estimation for the pointwise error of partial expansion of the Legendre series of a given function is a classical problem. Asymptotic error estimates exist expressed by the modulus of continuity of the given function, for functions of bounded variation, and for both analytic and piecewise analytic functions. These estimates are all asymptotic in the sense that the order of the partial expansion tends to infinity. It is interesting that in spite of many attempts the estimates are not optimal. This fact is demonstrated here, and the optimal estimate for the piecewise analytic functions is formulated as a hypothesis, which in turn is validated computationally. Further, in most applications the polynomial orders are relatively small compared to those in the asymptotic range. The existing theory does not address the behaviour of the preasymptotic error. Inherent difficulties addressing this question are discussed and illustrated here. Highlights • This paper extends the theory of pointwise Legendre expansions. • The new theory sharpens the existing results. • We show the existence of preasymptotic and asymptotic ranges, and underline the difficulty of identifying these ranges a priori. [ABSTRACT FROM AUTHOR]

Abstract We prove a central limit theorem for stationary multiple (random) fields of martingale differences f ∘ T i ̲ , i ̲ ∈ Z d , where T i ̲ is a Z d action. In most cases the multiple (random) fields of martingale differences is given by a completely commuting filtration. A central limit theorem proving convergence to a normal law has been known for Bernoulli random fields and in Volný (2015) this result was extended to random fields where one of generating transformations is ergodic. In the present paper it is proved that a convergence takes place always and the limit law is a mixture of normal laws. If the Z d action is ergodic and d ≥ 2 , the limit law need not be normal. For proving the result mentioned above, a generalisation of McLeish's CLT for arrays (X n , i) of martingale differences is used. More precisely, sufficient conditions for a CLT are found in the case when the sums ∑ i X n , i 2 converge only in distribution. The CLT is followed by a weak invariance principle. It is shown that central limit theorems and invariance principles using martingale approximation remain valid in the non-ergodic case. [ABSTRACT FROM AUTHOR]

Abstract We study the convergence of bivariate order statistics, and get the almost sure convergence and convergence rate, generalizing one of main results in Huang et al. (2013). To be more precise, we prove the almost sure convergence of bivariate order statistics without the positive quadrant dependent (PQD) condition mentioned in Huang et al. (2013), and extend this result to a more general case. We also give the bound on the Kolmogorov distance between the distributions of bivariate order statistics and its limit. Our results provide an approximate algorithm for sampling from complicated structures, verified by some examples in the last section of this paper. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the local superconvergence of the discontinuous Galerkin (DG) solutions on quasi-graded meshes for nonlinear delay differential equations with vanishing delay. It is shown that the optimal order of the DG solution at the mesh points is O (h 2 m + 1). By analyzing the supercloseness between the DG solution and the interpolation Π h u of the exact solution, we get the optimal order O (h m + 2) of the DG solution at characteristic points. We then extend the convergence results of DG solutions to state dependent delay differential equations. Numerical examples are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new stationary iterative method based on two double splittings of coefficient matrix is proposed to solve linear system A x = b. Some convergence results and comparison theorems of the new method are provided. Theoretical analysis shows that this method is superior to some existing ones under certain conditions. Numerical examples are given to illustrate the theoretical results and examine the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we establish a central limit theorem and a moderate deviation for two-dimensional stochastic primitive equations driven by multiplicative noise. This is the first result about the limit theorem and the moderate deviations for stochastic primitive equations. The proof of the results relies on the weak convergence method and some delicate and careful a p r i o r i estimates. [ABSTRACT FROM AUTHOR]

Abstract Convergent cross-mapping (CCM), has been largely implemented for variable causality detection in complex systems like chemical process. However, this method is susceptible to problems regarding parameter selection and threshold determination. The synchronization phenomenon and the Moran effect, which are two interference terms in causality detection, must also be addressed. Therefore, an improved CCM is proposed to overcome these limitations in this paper. In the improved CCM, the optimal embedding dimension is selected based on the pseudo-nearest-neighbor theory. Also, Monte Carlo simulation is adopted to evaluate the convergence threshold. Next, by using the defined time delay detection function, the synchronization phenomenon and the Moran effect are identified to reduce the interference terms and further improve the accuracy. Finally, the improved CCM method is applied in a numerical example and a hydrocracking process to demonstrate its feasibility and superior performance than other methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we employ matrix LSQR algorithm to deal with quaternionic least squares problem in order to find the minimum norm solutions with kinds of special structures, and propose a strategy to accelerate convergence rate of the algorithm via right–left preconditioning of the coefficient matrices. We mainly focus on analyzing the minimum norm η -Hermitian solution and the minimum norm η -biHermitian solution to the quaternionic least squares problem, η ∈ { i , j , k }. Other structured solutions also can be obtained using the proposed technique. A number of numerical experiments are performed to show the efficiency of the preconditioned matrix LSQR algorithm. [ABSTRACT FROM AUTHOR]

Abstract In the paper, we present an efficient two grid method for the miscible displacement problem which discretized by mixed finite element methods for the pressure equation and concentration equation at the same time, and then analyzed the error estimate of the two-gird algorithm. At last, the numerical experiment presented confirmed the theoretical results. Compared with the standard mixed finite element methods, this two-grid scheme based on the mixed methods can keep the same convergence order and cost much less work. [ABSTRACT FROM AUTHOR]

Abstract This paper establishes a spectral conjugate gradient method for solving unconstrained optimization problems, where the conjugate parameter and the spectral parameter satisfy a restrictive relationship. The search direction is sufficient descent without restarts in per-iteration. Moreover, this feature is independent of any line searches. Under the standard Wolfe line searches, the global convergence of the proposed method is proved when | β k | ≤ β k F R holds. The preliminary numerical results are presented to show effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we discuss the time-dependent generalized membrane shell model, which has not been addressed numerically in literature. We show that the solution of this model exists and is unique. We first provide a numerical method for the time-dependent generalized membrane shell. Concretely, we semi-discretize the space variable and fully discretize the problem using time discretization by the Newmark scheme. The corresponding numerical analyses of existence, uniqueness, stability and convergence with a priori error estimates are given. Finally, we present numerical experiments with a portion of the conical shell and a portion of the hyperbolic shell to verify theoretical convergence results and demonstrate the effectiveness of the numerical scheme. Highlights • Existence and uniqueness of the time-dependent generalized membrane shell. • Newmark scheme for the time-dependent generalized membrane shell. • Analyses of existence, uniqueness, stability, convergence and a priori error estimate for elastodynamic problem. [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 We pursue the investigation of generalizations of the Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a finite word appears as a subsequence of another finite word. The finite words occurring in this paper belong to the language of a Parry numeration system satisfying the Bertrand property, i.e., we can add or remove trailing zeroes to valid representations. It is a folklore fact that the Sierpiński gasket is the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from the classical Pascal triangle modulo 2. In a similar way, we describe and study the subset of [ 0 , 1 ] × [ 0 , 1 ] associated with the latter generalization of the Pascal triangle modulo a prime number. [ABSTRACT FROM AUTHOR]

Abstract A fundamental algorithm for selecting ranks from a finite subset of an ordered set is Radix Selection. This algorithm requires the data to be given as strings of symbols over an ordered alphabet, e.g., binary expansions of real numbers. Its complexity is measured by the number of symbols that have to be read. In this paper the model of independent data identically generated from a Markov chain is considered. The complexity is studied as a stochastic process indexed by the set of infinite strings over the given alphabet. The orders of mean and variance of the complexity and, after normalization, a limit theorem with a centered Gaussian process as limit are derived. This implies an analysis for two standard models for the ranks: uniformly chosen ranks, also called grand averages, and the worst case rank complexities which are of interest in computer science. For uniform data and the asymmetric Bernoulli model (i.e. memoryless sources), we also find weak convergence for the normalized process of complexities when indexed by the ranks while for more general Markov sources these processes are not tight under the standard normalizations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a method to prove the total variation convergence of approximation of Markov semigroups with singularities. In particular our approach is adapted to the study of numerical schemes for Stochastic Differential Equation (SDE) with simply locally smooth coefficients. First we present this method and then, we apply it to the CIR process. In particular, we consider the weak second order scheme introduced in Alfonsi (2010) and we prove that it also converges towards the CIR diffusion process for the total variation distance. This convergence occurs with almost order two. [ABSTRACT FROM AUTHOR]

Abstract PageRank problem is the cornerstone of Google search engine and is usually stated as solving a huge linear system. Moreover, when the damping factor approaches 1, the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve. In this paper, we demonstrate that the coefficient matrix of this system can be transferred into a block form by partitioning its rows into special sets. In particular, the off-diagonal part of the block coefficient matrix can be compressed by a simple low-rank factorization, which can be beneficial for solving the PageRank problem. Hence, a matrix partition method is proposed to discover the special sets of rows for supporting the low-rank factorization. Then a preconditioner based on the low-rank factorization is proposed for solving difficult PageRank problems. Numerical experiments are presented to support the discussions and to illustrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract This paper derives a superconvergence result for the modified weak Galerkin (MWG) finite element method of the second order elliptic problem. The convergence rate of the MWG approximation is improved by 30% after applying a low cost L 2 projection post-processing technique. These superconvergence phenomena are proved theoretically and confirmed numerically. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the convergence of numerical scheme for combined nonlinear radiation–conduction heat transfer system in a gray, absorbing and non-scattering two-dimensional medium. The radiative transfer equation is solved using a Discontinuous Galerkin method with upwind fluxes. The conductive equation is discretized using the finite element method. Moreover, the Crank–Nicolson scheme is applied for time discretization of the semi-discrete nonlinear coupled system. Existence and uniqueness of the solution for the continuous and full discrete system are presented. The convergence proof follows from the application of a discrete fixed-point theorem, involving only the temperature fields at each time step. The order of approximation error, stability, and order of convergence are investigated. Finally, the theoretical stability and convergence results are supported with numerical examples. [ABSTRACT FROM AUTHOR]

Highlights • Flow condensation. • Heat transfer coefficient. • Two phase flow. • Experimental data. Abstract In this paper a new model is presented for heat transfer calculation during film condensation inside pipes. This new model has been verified by comparison with available experimental data of a total of 22 different fluids, including water, various refrigerants and organic substances, which condense inside horizontal, vertical and inclined tubes. The model is valid for a range of internal diameters ranging from 2 mm to 50 mm, reduced pressure values ranging from 0.0008 to 0.91, Pr values for the liquid portion of the condensate from 1 to 18, values of Reynolds number for the liquid portion between 68 and 84827, and for the portion of the steam between 900 and 594373, steam quality from 0.01 to 0.99 and mass flux rates in the ranges of 3–850 kg/(m2 s). The mean deviation found for the data analyzed for vertical and inclined tubes was 13.0%, while for the horizontal tube data the mean deviation was 11.8%. In all cases, the agreement of the proposed model for horizontal, vertical and inclined tubes is good enough to be considered satisfactory for practical design. [ABSTRACT FROM AUTHOR]