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The aim of this paper is the development of a mathematical model to compute the optimal inventory mix to face stochastic demand at minimum cost in a two-level supply chain. The paper addresses a multi-product dynamic lot-sizing problem under stochastic demand subject to capacity and service level constraints. This model is executed to compare a Make To Order (MTO) strategy to a Vendor Managed Inventory (VMI) partnership between the supplier and their customers. Both strategies provide the demand order to be produced. A schedule of production orders is determined over the planning horizon in order to minimize the inventory holding costs of the supply chain, taking into consideration that the supplier is also responsible of initiating the replenishment orders and deliveries of their customers according to the VMI partnership. The simulation model is illustrated empirically using a real case study: a paper manufacturing company that pursues to improve customer service level and supply chain inventory costs through a proper production planning of their paper machines and a suitable VMI order replenishment schedule.

In this paper, we prove the existence of solutions for the following fractional boundary value problem c D α u ( t ) = f ( t , u ( t ) ) , α ∈ ( n , n + 1 ) , 0 ≤ t + ∞ , u ( 0 ) = 0 , u ′ ′ ( 0 ) = 0 , … , u ( n ) ( 0 ) = 0 , lim t → + ∞ c D α − 1 u ( t ) = β u ( ξ ) . The considerations of this paper are based on the concept of a new family of measures of noncompactness in the space of functions C n , γ ( R + ) satisfying the Holder condition and a fixed point theorem of Darbo type. We also provide an illustrative example in support of our existence theorems. Finally, to credibility, we apply successive approximation and homotopy perturbation method to find solution of the above problem with high accuracy.

In this paper, we discuss the accuracy and the stability of the CN-WSGD when it is applied to the initial–boundary value problem for a space fractional diffusion equation. In the CN-WSGD scheme, the weighted and shifted Grunwald difference (WSGD) scheme is used to discretize the space fractional derivative and the Crank–Nicolson (CN) scheme is used to discretize the temporal derivative. There is a free parameter in a CN-WSGD scheme that affects the stability and the accuracy of the scheme. In general, a CN-WSGD scheme has temporally and spatially second order accuracy, and there exists a scheme that has spatially third order accuracy. However, the third order accuracy scheme is not always stable. In this paper, we present and prove some theoretical results (old and new) on the stability of several CN-WSGD schemes, and consider choosing a value for the parameter such that the corresponding CN-WSGD scheme is unconditioned stable and has optimal upper bound for the accuracy. Numerical results are presented to verify the theoretical results.

Non-negative matrix factorization (NMF) can be used in clustering, feature representation or blind source separation. Many NMF methods have been developed including least squares (LS) error, Kullback–Leibler (KL) divergence, Itakura–Saito (IS) divergence, Bregman-divergence, α -divergence, β -divergence, γ -divergence, convex, constrained, graph-regularized NMFs. The main contribution of this paper is to develop a framework to generalize the existing NMF methods and also provide new NMF methods. This paper constructs a general optimization model and develops a generic updating rule with a simple structure using a surrogate function, which possesses similar properties as the standard NMF methods. The experimental results, obtained using several standard databases, demonstrate the power of the work in which some new methods provide performance superior to that of the other existing methods.

In this paper, we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods. It is shown that these novel methods can be of arbitrarily high order and exactly or approximately preserve first integrals or Lyapunov functions. We also consider order estimates of the new methods. Furthermore, we explore and discuss the application of our methods in important stiff gradient systems, and it turns out that our methods are unconditionally energy-diminishing and strongly damped even for very stiff gradient systems. Practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear Schrodinger equation. As a byproduct of this paper, arbitrary-order trigonometric/RKN collocation methods are also presented and analysed for second-order highly oscillatory/general systems. The paper is accompanied by numerical results that demonstrate the great potential of this work.

A general framework is given for applying the Newton–Raphson method to solve power flow problems, using power and current-mismatch functions in polar, Cartesian coordinates and complex form. These two mismatch functions and three coordinates, result in six possible ways to apply the Newton–Raphson method for the solution of power flow problems. We present a theoretical framework to analyze these variants for load (PQ)buses and generator (PV)buses. Furthermore, we compare newly developed versions in this paper with existing variants of the Newton power flow method. The convergence behavior of all methods is investigated by numerical experiments on transmission and distribution networks. We conclude that variants using the polar current-mismatch and Cartesian current-mismatch functions that are developed in this paper, performed the best result for both distribution and transmission networks.

In this paper, we propose and examine the fractional form corresponding to the Grassi–Miller integer-order discrete-time system. We show experimental phase portraits and bifurcation diagrams to highlight the ranges of parameters and fractional orders over which chaos is observed. In addition, we propose two distinct control schemes for the proposed fractional map. The first controller stabilizes the states and forces them towards zero asymptotically. The second controller aims to synchronize a pair of maps with non-identical parameters. Throughout the paper, numerical results are presented to verify the analytic results.

This paper develops and analyzes the optimal weighted upwind finite volume method (OWUFVM) for convection–diffusion equations on nonuniform rectangular meshes. The dual meshes in this method are associated with the local Peclet numbers, which are determined by the convection and diffusion coefficients and the mesh size. We first prove the stability and H 1 -norm error estimate. Then we prove a weak error estimate, which plays a key role in deriving the optimal second-order error estimate in the L 2 norm. The L 2 -norm error estimate is the most important result of this paper. We also briefly examine the discrete maximum principle. Numerical experiments are presented to illustrate the theoretical analysis.

In this paper, we formulate and analyze a one-level Additive Schwarz preconditioner with full-domain overlap for a class of parallel adaptive finite elements, including the Bank–Holst paradigm and the local and parallel algorithms based on two-grid discretizations. We show that the generous overlap is more than enough to compensate for the single-level nature of the preconditioner, and the effective condition number of the associated preconditioned operator can be bounded from above independently of the number of the subdomains, the fine and coarse mesh sizes. The main results of this paper substantially extend those in Loisel and Nguyen (2017), in the sense that we are able to remove from the bound of the effective condition number the explicit term, 1 ∕ ( 1 − γ 2 ) , where γ is the maximum of the constants in the strengthened Cauchy–Buniakowskii–Schwarz inequalities applied to the hierarchical decompositions of full-domain local adaptive finite element spaces into the coarse space and fine spaces. In other words, our new results confirm that the proposed preconditioner is even more robust as the constant γ can be close to 1 for adaptive meshes with bad shape regularity. Numerical results are provided to confirm our theoretical finding.

Based on record data, numerous authors have discussed the estimation of two-parameter Weibull distribution using classical and Bayesian approaches. In this paper, prediction of future records based on observed ones, using the maximum likelihood method, is considered. For a restricted parametric space, the existence and uniqueness of the maximum likelihood predictors of future records as well as the predictive maximum likelihood estimators of all unknown quantities are also established. Alternative approximate methods to obtain the likelihood estimators and predictors, which always exist and are easy-to-determine, are also discussed. The alternative approximate procedures studied in this paper are transformation-based predictive likelihood function, corrected predictive likelihood function, maximum product of spacings prediction Monte Carlo simulations are performed to compare the proposed methods and one real data set is also analyzed for illustrative purposes.

This paper suggests a novel continuous-time gradient descent algorithm of a constrained convex unknown function with a stochastic noise in the observed data. The penalty function approach is employed to introduce the restrictions of the system and to provide a successful optimization process. The solution is restricted to a static scheme dealing with the class of strongly convex functions subject to a set of constraints. To estimate the stochastic gradient we employ a modified version of the synchronous detection method. All the parameters of the proposed approach are decreasing in time to both compensate noise effect in the observations and to provide the mean-square convergence of the suggested approach. We present two different numerical examples to validate the contributions of the paper.

The generalized finite difference method (GFDM) has been proved to be a good meshless method to solve several linear partial differential equations (PDEs): wave propagation, advection–diffusion, plates, beams, etc. The GFDM allows us to use irregular clouds of nodes that can be of interest for modelling non-linear parabolic PDEs. This paper illustrates that the GFD explicit formulae developed to obtain the different derivatives of the pde’s are based in the existence of a positive definite matrix that it is obtained using moving least squares approximation and Taylor series development. Criteria for convergence of fully explicit method using GFDM for different non linear parabolic PDEs are given. This paper shows the application of the GFDM to solving different non-linear problems including applications to heat transfer, acoustics and problems of mass transfer.

In this paper, a new viscosity iterative method minimization algorithm for image restoration algorithm is proposed. Based on a new viscosity iteration, the algorithm for finding the common zeros of two accretive operators in the framework of uniformly smooth Banach spaces. Moreover, the strong convergence theorems for the iterative algorithms and an example is proposed which shows the validity of main theorem are proved. The results of this paper are improved and extended of the corresponding ones announced by many others and we also applied our result to solve a convex minimization problem and Gâteaux differentiable E P s and Variational Inequality. Experiment results show that the proposed algorithms outperform some other methods. Finally, we give the numerical experiments to show the efficiency and advantage of the proposed methods and we also used our proposed algorithm for applying to solve the image deblurring and image recovery problems.

One of the main problems in celestial mechanics is the study of the figure adopted by large deformable bodies in slow rotation around an axis with a constant angular velocity ω → when they reach their equilibrium configuration. This figure corresponds to the lowest equipotential surface containing the entire mass and, in order to determine it, calculating its potential at an arbitrary point is required. Classical methods address this problem generally by performing a series development of the inverse-distance by using Clairaut’s coordinates. These methods show convergence problems, already in first order in ω 2 , so that to avoid them they must assume no demonstrated hypotheses. The authors of this paper warned and proved this fact in a previous work, for which they used two methods: 1. Taking into account the asymptotic properties of numeric quadrature formulas. 2. By a process similar to that used by Laplace to develop the inverse of the distance between two planets. Thus, the authors demonstrated that, although the intermediate formulas obtained by the classical methods are wrong, the self-gravitational potential, up to first order in ω 2 , obtained with them, were coincident with those obtained by other methods. Now, in this paper an extension up to second order in ω 2 of the first method mentioned above is proposed. It shows rigorously that up to second order in ω 2 , the intermediate formulas obtained by the classical methods are erroneous. In addition, the correct developments, up to second order in ω 2 , of those intermediate formulas are obtained. In spite of these discrepancies, the auto-gravitational potential, up to second order in ω 2 , obtained by the classical methods and the one obtained in this work are coincident, consequently it is proved that the classical theory is correct until second order in ω 2 .

Encouraged by the success of applying metaheuristics algorithms to other ranking-based problems (Kemeny ranking problem and parameter estimation for Mallows distributions), in this paper we deal with the rank aggregation problem (RAP), which can be viewed as a generalization of the Kemeny problem to arbitrary rankings. While in the Kemeny problem the input is a set of permutations, the RAP consists in obtaining the consensus permutation for a sample of arbitrary rankings. This is an NP-hard problem which can be approached by using greedy heuristic algorithms (e.g. Borda). Such algorithms are fast but the solutions so obtained are far to be optimal. In this paper, we propose the use of more complex search processes to deal with the RAP. In particular, we perform a comparative study among some local-based search metaheuristics: hill climbing (HC), iterated local search (ILS), variable neighborhood search (VNS) and greedy randomized adaptive search procedure (GRASP). We provide a complete analysis of the experimental study regarding accuracy and number of iterations required to reach the best solution. From the results we can conclude that the selection of a suitable neighborhood plays a key role, and that depending on the available resources (cpu time) a different algorithm (VNS, ILS or GRASP) could be the proper choice.

Strong H -tensors play an important role in the theories and applications of numerical linear algebra. It is necessary to identify whether a given tensor is a strong H -tensor or not. In this paper, we establish some iterative criteria for identifying strong H -tensors. These criteria depend on the elements of the tensors; therefore, they are easy to be verified. The results obtained in this paper extend the corresponding conclusions for strictly generalized diagonally dominant matrices. As an application, some sufficient conditions for the positive definiteness of an even-order real symmetric tensor are presented. Some numerical experiments show the feasibility and efficiency of the results which are obtained in this paper.

Piezoelectrical solid with nano-holes under time-harmonic anti-plane load is studied in the paper. The model is defined by the system of two partial differential equations and the boundary conditions which define the generalized stress. We reduce the model into an integro-differential boundary value problem and construct Cellular Nanoscale Network (CNN) architecture. Dynamical behavior of the obtained CNN model is studied by means of describing function technique. The simulations are provided which illustrate the theoretical results. The results of this paper are applicable in the field of non-destructive testing and fracture mechanics of multi-functional materials and structural elements based on them.

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

The aim of this paper is to study the convergence and smoothness of non-stationary Hermite subdivision schemes of order 2. In Conti et al. (2017) provided sufficient conditions for the convergence of a non-stationary Hermite subdivision scheme that reproduces a set of functions including exponential polynomials. The analysis has been focused on the non-stationary Hermite scheme with the order ≥ 3 , but the case of 2 (which is practically most useful) is yet to be investigated. In this regard, the first goal of this paper is to fill the gap. We analyze the convergence of non-stationary Hermite subdivision schemes of order 2. Next, we provide a tool which allows us to estimate the smoothness of a non-stationary Hermite scheme by developing a novel factorization framework of non-stationary vector subdivision operators. Using the proposed non-stationary factorization framework, we estimate the smoothness of the non-stationary Hermite subdivision schemes: the non-stationary interpolatory Hermite scheme proposed by Conti et al., (2015) and a new class of non-stationary dual Hermite subdivision schemes of de Rham-type.

Let H m × n be the set of all m × n matrices over the quaternion algebra H . In this paper, we construct a simultaneous decomposition for seven matrices with compatible sizes: A ∈ H m × n , B ∈ H m × p 1 , C ∈ H m × p 2 , D ∈ H m × p 3 , E ∈ H q 1 × n , F ∈ H q 2 × n and G ∈ H q 3 × n . As applications of the simultaneous matrix decomposition, we give some solvability conditions, general solutions, as well as the range of ranks of the general solutions to the following two generalized Sylvester matrix equations B X E + C Y F + D Z G = A and B X + W E + C Y F + D Z G = A , where A , B , C , D , E , F , and G are given quaternion matrices. Moreover, we provide some numerical examples to illustrate the results of this paper.

The aim of this paper is to present efficient and stable methods to compute Caputo fractional derivative (CFD) of the Muntz–Legendre polynomials based on three-term recurrence relations and Gauss–Jacobi quadrature rules. This approach with collocation method at Chebyshev–Gauss–Lobatto points has been applied for solving linear and nonlinear fractional multi-order differential equations (FDEs) described in Caputo sense. The main characteristic of spectral collocation method is that the problems reduce to linear or nonlinear systems of algebraic equations. In this work, for the first time, we present the new rates of convergence for projection error which are more accurate than the rate presented by Shen and Wang in Shen and Wang (2016). Moreover, we present convergence rate for spectral collocation method for linear FDEs with initial value on a finite interval and endpoint singularities. Also, we propose an error analysis for Jacobi–Gauss type quadrature and present a way to accelerate the convergence rate for singular integrands applied in this paper. Finally, the stability and applicability of the numerical approach and convergence analysis is demonstrated by some numerical examples.

G. Ramponi et al. introduced in Carrato et al. (1997,1998), Castagno and Ramponi (1996) and Ramponi (1995) a non linear rational interpolator of order two. In this paper we extend this result to get order four. We observe the Gibbs phenomenon that is obtained near discontinuities with its weights. With the weights we propose we obtain approximations of order four in smooth regions and three near discontinuities. We also introduce a rational nonlinear extrapolation which is also of order four in the smooth region of the given function. In the experiments we calculate numerically approximation orders for the different methods described in this paper and see that they coincide with those that have been obtained theoretically. We also present reconstructions in 1d and 2d with which we reach the same conclusions.

Since the security market is too complex, there are situations where the future security returns cannot be reflected by historical data and have to be given by experts’ estimations according to their knowledge and judgements. This paper discusses a portfolio selection problem in such an uncertain environment. In the paper, in order to reflect different attitudes towards risk that vary by goal in one portfolio investment, we apply mental accounts to the investment. Uncertain variables are introduced to describe the securities’ return, and a new uncertain portfolio selection model is developed when return risk control and liquidity risk control are considered. In addition, other important limitations, i.e., transaction cost, minimum transaction lot, bounds of holding, security diversification and sector diversification are also considered. In real life, investors face background risk which may affect their portfolio selection decisions. Thus, background risk is also considered. To solve the proposed problem, the equivalents of the model are proposed. Finally, a numerical example is given as an illustration.

This paper proposes a simplified reproducing kernel method to solve the linear Volterra integral equations with variable coefficients. The main idea of the method is to establish a reproducing kernel direct space that can be used in Volterra integral equations. And in the first time, this paper analyzes the convergence order and stability of the approximate solution. Then the uniform convergence of the numerical solution is proved, and the time consuming Schmidt orthogonalization process is avoided. The proposed method is proved to be stable and is not less than the second order convergence. The algorithm is proved to be feasible and stable through some numerical examples.

Most papers concentrate on the parameter identification of Hammerstein systems with known orders. This paper, motivated by the recent developments in sparse approximations, investigates the combined parameter and order determination of Hammerstein systems. The methodology used relies on greedy schemes—the orthogonal matching pursuit (OMP) algorithm in the compressive sensor (CS) theory. In particular, the first step recasts a bilinear Hammerstein system into two fictitious pseudo-regressive sub-systems which respectively contain the parameters of the nonlinear part or the parameters of the linear part by the hierarchical identification principle. The second step adopts a hierarchical orthogonal matching pursuit (H-OMP) selection procedure to interactively select the parameters and orders of the two sub-systems under the frame of the compressive sensor. Finally, the proposed algorithm is tested on a simulation example.

This paper presents a novel approach to the Navier–Stokes equations which reformulates them in terms of a new tensor variable. In the first formulation discussed in the paper this variable is proportional to the gradient of the velocity field with the pressure added to the diagonal components. In the second formulation it is identical to the stress tensor. At first glance the resulting tensorial problem is more difficult than the problem in the primitive variables. However, if combined with a proper splitting, it yields locally one dimensional schemes with attractive properties, that are very competitive to the most widely used schemes for the formulation in primitive variables. In addition, it has an advantage if applied to fluid–structure interaction problems.

A simultaneous confidence band provides useful information about the plausible range of the unknown regression model. In several recent papers, confidence bands have been used for various inferential purposes; see, for example, Liu et al. (2004, 2005, 2009, 2010, 2012, 2014), Liu and Hayter (2007) [ 1 ], Liu and Lin (2009), Piegorsch et al. (2005), Han et al. (2015) and Zhou (2017). The construction of simultaneous confidence bands has a long history, going back to Working and Hotelling (1929) [ 2 ], and is often difficult, when the region over which a confidence band is required is restricted and the number of predictor variables is greater than one. In this paper we have developed methods to construct simultaneous confidence bands for a percentile hyper-plane. These methods allow us to construct the bands over arbitrary intervals, either finite or infinite; they also allow us to construct the Type I and Type II bands. In order to get an idea of the efficiency of these bands, we have also developed the confidence sets for the bands and have compared the bands under the minimum volume confidence set optimality criterion. The proposed methods are illustrated by an example.

[EN] In this paper, we present some results about the spectrum of the matrix associated with the computation of the Multiplex PageRank defined by the authors in a previous paper. These results can be considered as a natural extension of the known results about the spectrum of the Google matrix. In particular, we show that the eigenvalues of the transition matrix associated with the multiplex network can be deduced from the eigenvalues of a block matrix containing the stochastic matrices defined for each layer. We also show that, as occurs in the classic PageRank, the spectrum is not affected by the personalization vectors defined on each layer but depends on the parameter a that controls the teleportation. We also give some analytical relations between the eigenvalues and we include some small examples illustrating the main results. (C) 2018 Elsevier B.V. All rights reserved., We thank the two anonymous reviewers for their constructive comments, which helped us to improve the manuscript. This work has been partially supported by the projects MTM2014-59906-P, MTM2014-52470-P (Spanish Ministry and FEDER, EU, Spain), MTM2017-84194-P (AEI/FEDER, EU, Spain) and the grant URJC-Grupo de Excelencia Investigadora GARECOM (2014-2017), Spain.

Although there is substantial literature on the univariate zero-truncated Poisson (ZTP) distributions, few works have been done for developing multivariate ZTP distributions with dimension larger than two. In this paper, we first propose a new Type I multivariate ZTP distribution to model multivariate zero-truncated count data and develop its important distributional properties, while providing efficient statistical inference methods. The key idea for extending the univariate ZTP distribution to a multivariate version is to identify a new stochastic representation of the ZTP random variable, resulting in a novel expectation–maximization algorithm with a feature that the introduced latent variables are independent of the observed variables. Since the ZTP distribution is closely related with the zero-adjusted Poisson (ZAP) distribution, the second objective of this paper is to propose a new family of Type I multivariate ZAP distributions (including the Type I multivariate ZTP distribution, the Type I multivariate zero-inflated Poisson (ZIP) distribution, the Type I multivariate zero-deflated Poisson (ZDP) distribution as its special cases) by accounting for truncation, inflation and deflation at zero. The corresponding distributional properties are explored and the associated statistical methods are provided. Finally, three real data sets are used to illustrate the proposed distributions and corresponding analysis methods.

Hairer and Iserles (2016) presented a detailed study of skew-symmetric matrix approximation to a first derivative which is proved to be fundamental in ensuring stability of discretisation for evolutional partial differential equations with variable coefficients. An open problem is proposed in that paper which concerns about the existence and construction of the perturbed grid that supports high-order skew-symmetric differentiation matrix for a given grid and only the case p = 2 for this problem have been solved. This paper is an attempt to solve the problem for any p ⩾ 3 . We focus ourselves on the symmetric grid and prove the existence of the perturbed grid for arbitrarily high order p and give in detail the construction of the perturbed grid. Numerical experiments are carried out to illustrate our theory.

In this paper, we present a new parallel accurate algorithm called PAccSumK for computing summation of floating-point numbers. It is based on AccSumK algorithm. In the experiment, for the summation problems with large condition numbers, our algorithm outperforms in term of accuracy and computing time the PSumK algorithm. The reason is that our algorithm is based on a more accurate algorithm called AccSumK algorithm compared to the SumL algorithm used in PSumK. The proposed parallel algorithms in this paper is designed to compute a result as if computed internally in K-fold the working precision. Numerical results are presented showing the performance and the accuracy of our new parallel algorithm for calculating summation.

Information geometry has been attracted wide attentions in the past few decades. This paper focuses on the Bayesian duality on a statistical manifold derived from the exponential family with data from life tests. Based on life testing data, the statistical manifold is constructed with a new cumulant generating function. The Bregman divergence between two parameter points is studied. The dual coordinate system and dual function are obtained. Then, the dualistic structure on the manifold is discussed. The results show that the maximum likelihood estimate can be obtained by minimizing the Bregman divergence induced from the dual function. The Bayesian analysis and prediction are investigated based on informative and non-informative priors. Consider the gamma distribution as an example, the closed-form representations of the dual coordinate system and dual function are obtained. A real data set is employed to illustrate the methodologies and experimental designs developed in this paper.

This note demonstrates how the divided differences characterization for the moments of the integrated geometric Brownian process arises naturally from the solution to their differential equations. The characterization was introduced by Baxter and Brummelhuis in their paper (Baxter and Brummelhuis (2011)) where they demonstrate its consistency with the results found by Oshanin et al. (1993). Here we demonstrate its validity with more general expressions presented in other papers and explore the divided differences characterization of these moments further.

This paper discusses the stability of Runge–Kutta methods for differential systems with distributed delays. Weak delay-dependent stability of Pouzet-type Runge–Kutta methods is studied in this paper. It is revealed that under suitable conditions, Runge–Kutta methods preserve weak delay-dependent stability. An algorithm is employed to check weak delay-dependent stability of numerical solutions. And the effectiveness of the derived theoretical results is demonstrated with numerical examples.

This paper introduces a nonlinear multigrid solver for mixed finite element discretizations based on the Full Approximation Scheme (FAS) and element-based Algebraic Multigrid (AMGe). The AMGe coarse spaces with approximation properties used in this work enable us to overcome the difficulties in evaluating the nonlinear coarse operators and the degradation in convergence rates that characterized previous attempts to extend FAS to algebraic multilevel hierarchies on general unstructured grids. Specifically, the AMGe technique employed in this paper allows to derive stable and accurate coarse discretizations on general unstructured grids for a large class of nonlinear partial differential equations, including saddle point problems. The approximation properties of the coarse spaces ensure that our FAS approach for general unstructured meshes leads to optimal mesh-independent convergence rates similar to those achieved by geometric FAS on a nested hierarchy of refined meshes. In the numerical results, Newton’s method and Picard iterations with state-of-the-art inner linear solvers are compared to our FAS algorithm for the solution of a nonlinear saddle point problem arising from porous media flow applications. Our approach outperforms – both in terms of number of iterations and computational time – traditional methods in all the experiments.

In this paper the qualitative properties of certain spatial disease propagation models are investigated. The paper can be considered as a generalization of the papers (Faragoand Horvath, 2016; 2017). The models of these papers assume that the members of the population do not move and that the infection is localized in the sense that only the members in a certain neighbourhood of the infective member can be infected. The considered qualitative properties were: the nonnegativity and the monotonicity of the density functions, and the preservation of the amount of the members. Sufficient conditions for these properties were obtained for the mesh size and the time step in certain finite difference solutions of the model. In these works, the one-dimensional case was investigated only. The present paper extends the above result in two directions: with results for higher dimensional problems and for another disease propagation model given in Capasso (2008). Here the members are allowed to move according to some diffusion law. Similarly to the previous model, sufficient conditions are given that guarantee the validity of the qualitative properties. We focus only on the properties of the discrete models. The results are verified on test problems.

In this paper, the time discrete higher order projection method is proposed and analyzed for the time-dependent viscoelastic flow problem. Our numerical method is based on the time iterative discrete schemes. By the projection method, the considered problem is decoupled into two linear subproblems: One is for the velocity and the other is for the pressure. Unconditional stability of the numerical schemes is established. Convergence results for the velocity and pressure are also derived. Our main results of this paper are that the convergence analysis for the velocity is weakly second order and for the pressure is weakly first order. Finally, some numerical examples are provided to confirm the performances of the developed numerical algorithms.

In this paper, we are interested in the displacement of two incompressible phases in a Darcy–Brinkman flow in a porous media. The equations are obtained by the conservation of the mass and by considering the Brinkman regularization velocity of each phase. This model is treated in its general form with the whole nonlinear terms. This paper deals with construction and convergence analysis of a combined finite volume–nonconforming finite element scheme together with a phase-by-phase upstream according to the total velocity.

This paper is concerned with the problem of finding singular points of vector fields on Riemannian manifolds. We, using non trivial techniques and results of differential geometry, will exhibit a new method defined on complete Riemannian manifolds, which generalizes the important Newton and Chebyshev–Halley’s methods. Moreover, a characterization of the convergence under Kantorovich-type conditions and error estimates are also given in this study. Using the method introduced in this paper we give an algorithm which will allow us to find singular points of a vector field on the two-dimensional sphere S 2 . Finally, in order to illustrate our method and its relevance, we develop an example which allows us to find a singularity of a vector field on S 2 , such a singularity is not possible to find it using the classical numerical methods on R 3 .

In this paper, an inverse problem to identify parameters for the multi-term time fractional partial differential equation with Caputo fractional derivative is considered. The numerical solution of the direct problem is obtained by the modified fractional predictor–corrector method, while the inverse problem is conducted by the modified hybrid Nelder–Mead simplex search and particle swarm optimization (MH-NMSS–PSO) algorithm, which is expanded to estimate parameters for fractional differential equations. To verify the efficiency and accuracy of the proposed methods, three numerical examples with experimental data are conducted. Furthermore, in order to improve the practical significance of the research on inverse problems, the predictive value of parameter estimation is also testified. This paper provides effective numerical methods for parameter estimation in practical applications involving multi-term time fractional constitutive equations.

In this paper, the method of particular solutions (MPS) using trigonometric functions as the basis functions is proposed to solve two-dimensional elliptic partial differential equations. The inhomogeneous term of the governing equation is approximated by Fourier series and the closed-form particular solutions of trigonometric functions are derived using the method of undetermined coefficients. Once the particular solutions for the trigonometric basis functions are derived, the standard MPS can be applied for solving partial differential equations. In comparing with the use of radial basis functions and polynomials in the MPS, our proposed approach provides another simple approach to effectively solving two-dimensional elliptic partial differential equations. Five numerical examples are provided in this paper to validate the merits of the proposed meshless method.

This paper considers the regularization parameter determination of l 1 -regularized minimization problem. We solve the l 1 -regularized problem using iterative reweighted least squares (IRLS) which involves solving a linear system whose coefficient matrix has the form α M + ( 1 − α ) N ( α ∈ ( 0 , 1 ) ). The aim of this paper is to find an efficient and computationally inexpensive algorithm to both choose the regularization parameter and solve the l 1 -regularized problem. In order to achieve this, we propose an IRLS algorithm with adaptive regularization parameter selection based on a heuristic parameter determination rule—de Boor’s parameter selection criterion. Compared with some of the state-of-the-art algorithms and parameter selection rules, the numerical experiments show the efficiency and robustness of the proposed method.

This paper presents a computational scheme to solve nonlinear logarithmic singular boundary integral equations. These types of integral equations arise from boundary value problems of Laplace’s equations with nonlinear Robin boundary conditions. The discrete Galerkin method together with the (inverse) multiquadric radial basis functions established on scattered points is utilized to approximate the solution. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals in the method. The proposed scheme uses a special accurate quadrature formula via the nonuniform Gauss–Legendre integration rule to compute logarithm-like singular integrals appeared in the scheme. Since the numerical method developed in the current paper does not require any mesh generations on the boundary of the domain, it is meshless and does not depend to the domain form. We also investigate the error analysis of the proposed method. Illustrative examples show the reliability and efficiency of the new scheme and confirm the theoretical error estimates.

In this paper, we introduce and analyze a weak Galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes equations in primal velocity–pressure formulation and Darcy equation in the second order primary formulation, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. By using the weak Galerkin approach, we consider the two-dimensional problem with the usual polynomials of degree k ≥ 1 for the velocity and hydraulic head, while polynomials of degree k − 1 for the pressure, the velocity and hydraulic head is enhanced by polynomials of degree k − 1 on the edge of a finite element partition. This new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained. Moreover, numerical experiences are presented to illustrate the good performance, confirm the optimal order of convergence and verify the efficiency of the proposed weak Galerkin method in this paper. We also deal with finite element partitions consisting of general meshes, such as triangular mesh, quadrilateral mesh, hexagonal-dominant mesh and Voronoi mesh for the numerical weak Galerkin approximation.

In the paper the author presents a novel kind of the WZ factorization algorithm, namely a block WZ factorization algorithm. The aim of this new algorithm is to utilize the computational power of contemporary computers with hierarchical memory. In the paper, some properties of the matrix Z are given and analyzed. Next, a version of the block WZ factorization is presented. The author shows that such a block WZ factorization exists for strictly diagonally dominant matrices. The computational cost of this block algorithm is presented. The time and the accuracy of proposed block WZ factorization algorithm for random dense square diagonally dominant matrices are reported. The block algorithm turned out to be faster even up to 300 times than the original WZ factorization.

[EN] This paper is aimed to extend, the non-autonomous case, the results recently given in the paper [1] for solving autonomous linear and quadratic random matrix differential equations. With this goal, important deterministic results like the Abel-Liouville-Jacobi's formula, are extended to the random scenario using the so-called $\mathrm{L}_{p}$-random matrix calculus. In a first step, random time-dependent matrix linear differential equations are studied and, in a second step, random non-autonomous Riccati matrix differential equations are solved using the hamiltonian approach based on dealing with the extended underlying linear system. Illustrative numerical examples are also included. [1] M.-C. Casabán, J.-C. Cortés, L. Jódar, Solving linear and quadratic random matrix differential equations: A mean square approach, Applied Mathematical Modelling 40 (2016) 9362-9377., This work has been partially supported by the Spanish Ministerio de Economia y Competitividad grant MTM2013-41765-P and by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement no. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance).

This paper is concerned with the convergence properties of Chebyshev spectral collocation method when used to approximate the solution of multidimensional nonlinear Volterra integral equation of the second kind with a weakly singular kernel. We consider the case that the underlying solution is sufficiently smooth. The Chebyshev collocation discretization is proposed for this equation. In the present paper, we provide a rigorous error analysis which justifies that the errors of approximate solution decay exponentially in weighted L 2 norm and L ∞ norm. Numerical results are presented to demonstrate the effectiveness of the spectral method.

Este artículo se encuentra disponible en la página web de la revista en la siguiente URL: https://www.sciencedirect.com/science/article/abs/pii/S0377042717301188 Este es el pre-print del siguiente artículo: Candela, V., Falcó, A. & Romero, PD. (2018). A general framework for a class of non-linear approximations with applications to image restoration. Journal of Computational and Applied Mathematics, vol. 330 (mar.), pp. 982-994, que se ha publicado de forma definitiva en https://doi.org/10.1016/j.cam.2017.03.008 This is the pre-peer reviewed version of the following article: Candela, V., Falcó, A. & Romero, PD. (2018). A general framework for a class of non-linear approximations with applications to image restoration. Journal of Computational and Applied Mathematics, vol. 330 (mar.), pp. 982-994, which has been published in final form at https://doi.org/10.1016/j.cam.2017.03.008 In this paper, we establish sufficient conditions for the existence of optimal nonlinear approximations to a linear subspace generated by a given weakly-closed (non-convex) cone of a Hilbert space. Most non-linear problems have difficulties to implement good projection-based algorithms due to the fact that the subsets, where we would like to project the functions, do not have the necessary geometric properties to use the classical existence results (such as convexity, for instance). The theoretical results given here overcome some of these difficulties. To see this we apply them to a fractional model for image deconvolution. In particular, we reformulate and prove the convergence of a computational algorithm proposed in a previous paper by some of the authors. Finally, some examples are given. Preprint

In this paper, a modified weak Galerkin method is proposed for the Stokes problem. The numerical scheme is based on a novel variational form of the Stokes problem. The degree of freedoms in the modified weak Galerkin method is less than that in the original weak Galerkin method, while the accuracy stays the same. In this paper, the optimal convergence orders are given and some numerical experiments are presented to verify the theory.

In this paper, meshless weak form techniques are applied to find the numerical solution of nonlinear biharmonic Sivashinsky equation arising in the alloy solidification problem. Stability and convergence analysis of time-discrete scheme are proved. An error analysis of meshless global weak form method based on radial point interpolation technique is proposed for this nonlinear biharmonic equation. In addition, a comparison between meshless global and local weak form methods is done from the perspective of accuracy and efficiency. The main purpose of this paper is to show that the meshless weak form techniques can be used for solving the nonlinear biharmonic partial differential equations especially Sivashinsky equation. The numerical results confirm the good efficiency of the proposed methods for solving this nonlinear biharmonic model.