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Particle-based methods based on material models using spheres can provide good approximations for many physical phenomena at both the micro and macroscales. The point of departure for the simulations, in general, is a dense arrangement of spherical particles (sphere pack) inside a given container. For generic domains, the generation of a sphere pack may be complex and time-consuming, especially if the pack must comply with a prescribed sphere size distribution and the stability requirements of the simulation. The primary goal of this paper is to present an efficient algorithm that is capable of producing packs with millions of spheres following a statistical sphere size distribution inside complex arbitrary domains. This algorithm uses a new strategy to ensure that the sphere size distribution is preserved even when large particles are rejected in the growing process. The paper also presents numerical results that enable an evaluation of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the matrix equation A X A H + C Y C H = F , where A , C and F are given matrices with appropriate sizes and [ X , Y ] is an unknown Hermitian and generalized skew-Hamiltonian matrix pair. Based on matrix differential calculus and projection theorem in inner product spaces, we exploit the best approximate solution [ X ̂ , Y ̂ ] in the set S to a given matrix pair [ X ∗ , Y ∗ ] , where S signifies the least-squares Hermitian and generalized skew-Hamiltonian solution set of the matrix equation A X A H + C Y C H = F . The analytical expression of the best approximate solution is presented by applying the canonical correlation decomposition and the generalized singular value decomposition. Finally, a numerical algorithm and an illustrated example are given. [ABSTRACT FROM AUTHOR]

Abstract: Many simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge–Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic order [Copyright &y& Elsevier]

This paper deals with the problem of positivity preservation in numerical algorithms for simulating fiber suspension flows. In contrast to fiber orientation models based on the Advani–Tucker evolution equations for even-order orientation tensors, the probability distribution function of fiber orientation is approximated using the Galerkin discretization of the Fokker–Planck equation with Fourier basis functions or spherical harmonics. This procedure leads to a natural generalization of orientation tensor models replacing ad hoc closure approximations by Galerkin equations for the fine-scale components. After introducing an operator splitting approach to solving the discretized Fokker–Planck equation, we present conditions and correction techniques that guarantee physically correct distribution functions. As the reader will see, the derivation of these conditions is independent of the space dimension and their applicability is not limited to the simulation of fiber suspensions. [ABSTRACT FROM AUTHOR]

Based on kernel-based approximation technique, we devise in this paper an efficient and accurate numerical scheme for solving a backward space–time fractional diffusion problem (BSTFDP). The kernels used in the approximation are the fundamental solutions of the space–time fractional diffusion equation expressed in terms of inverse Fourier transform of Mittag-Leffler functions. The use of Inverse fast Fourier transform (IFFT) technique enables an accurate and efficient evaluation of the fundamental solutions and gives a robust numerical algorithm for the solution of the BSTFDP. Since the BSTFDP is intrinsic ill-posed, we apply the standard Tikhonov regularization technique to obtain a stable solution to the highly ill-conditioned resultant system of linear equations. For choosing optimal regularization parameter, we combine the regularization technique with the generalized cross validation (GCV) method for an optimal placement of the source points in the use of fundamental solutions. Meanwhile, the proposed algorithm also speeds up the previous method given in Dou and Hon (2014). Several numerical examples are constructed to verify the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: A semi-linear elliptic problem with variable coefficient is approximated by expanded mixed formulation based on the RTN and BDM mixed finite elements. In order to solve the nonlinear approximation system of equations efficiently, a two-grid algorithm is considered and discussed in this paper. The work includes a small nonlinear system on a coarse grid space with mesh size and a linear system on a fine grid space with mesh size . It follows from error estimates that asymptotically optimal accuracy can be obtained as long as the mesh sizes satisfy in the norm. Some numerical examples are presented to illustrate the theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we prove that an existing algorithm with periodical impulses for chaos suppression in continuous systems can be further extended and successfully applied to a class of piecewise continuous systems. For this purpose, the underlying initial value problem is transformed into a continuous problem via Filippov regularization and Cellina’s approximation theorem. Next, some results on existence and continuation for ODEs with impulses are applied. As an example, the piece-wise continuous Chen system is considered. [Copyright &y& Elsevier]

Abstract: In this paper, we present a numerical approach for solving the system of multi-pantograph equations with mixed conditions. This system is usually difficult to solve analytically. By expanding the approximate solutions by means of the Bessel functions of first kind with unknown coefficients, the proposed approach consists of reducing the problem to a linear algebraic equation system. The unknown coefficients of the Bessel functions of first kind are computed using the matrix operations of derivatives together with the collocation method. An error estimation is given. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. All of the numerical computations have been performed on a computer with the aid of a program written in Matlab. [Copyright &y& Elsevier]

Abstract: In this paper, we apply the wavelet-Galerkin method to obtain approximate solutions to linear Volterra integral equations (VIEs) of the second kind. Daubechies wavelets are used to find such approximations. In this approach, we introduce some new connection coefficients and discuss their properties and propose algorithms to evaluate them. These coefficients can be computed just once and applied for solving every linear VIE of the second kind. Convergence and error analysis are discussed and numerical examples illustrate the efficiency of the method. [Copyright &y& Elsevier]

Abstract: Elimination methods are highly effective for the solution of linear and nonlinear systems of equations, but reversal of the elimination principle can be beneficial as well: competent incorporation of additional independent constraints and variables or more generally immersion of the original computational problem into a larger task, defined by a larger number of independent constraints and variables can improve global convergence of iterative algorithms, that is their convergence from the start. A well known example is the dual linear and nonlinear programming, which enhances the power of optimization algorithms. We believe that this is just an ad hoc application of general Principle of Expansion with Independent Constraints; it should be explored systematically for devising iterative algorithms for the solution of equations and systems of equations and for optimization. At the end of this paper we comment on other applications and extensions of this principle. Presently we show it at work for the approximation of a single zero of a univariate polynomial of a degree . Empirical global convergence of the known algorithms for this task is much weaker than that of the algorithms for all zeros, such as Weierstrass–Durand–Kerner’s root-finder, which reduces its root-finding task to Viète’s (Vieta’s) system of polynomial equations with unknowns. We adjust this root-finder to the approximation of a single zero of , preserve its fast global convergence and decrease the number of arithmetic operations per iteration from quadratic to linear. Together with computing a zero of a polynomial , the algorithm deflates this polynomial as by-product, and then could be reapplied to the quotient to approximate the next zero of . Alternatively by using processors that exchange no data, one can concurrently approximate up to zeros of . Our tests confirm the efficiency of the proposed algorithms. Technically our root-finding boils down to computations with structured matrices, polynomials and partial fraction decompositions. Our study of these links can be of independent interest; e.g., as by-product we express the inverse of a Sylvester matrix via its last column, thus extending the celebrated result of Gohberg and Sementsul (1972) from Toeplitz to Sylvester matrix inverses. [Copyright &y& Elsevier]

Abstract: The problem of economic dispatch with multiple fuel units has been widely addressed via different techniques using approximate methods due to the exponential complexity of full enumeration in the underlying combinatory problem. A method has recently been outlined by Min et al. (2008), that allows the problem to be solved in an exact way in polynomial time. In this paper, we present an alternative technique and take this idea further, studying and comparing two algorithms of polynomial complexity: basic recurrence and divide-and-conquer. Moreover, we provide the exact solution to the problem by Lin and Viviani (1984), that constitutes the traditional test for all approximate methods and present a comprehensive survey of several heuristic approaches. [Copyright &y& Elsevier]

Abstract: In this paper, an algorithm for approximate calculation of the attainable sets of nonlinear control systems with integral constraint on controls is presented. A numerical example of the computer application of this algorithm is also given. [Copyright &y& Elsevier]

Abstract: In this paper, a fractional order differential system for modeling human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells is studied and its approximate solution is presented using a multi-step generalized differential transform method. The method is only a simple modification of the generalized differential transform method, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding systems. The solutions obtained are also presented graphically. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new symmetric rank-one (SR1) method for the solution of unconstrained optimization problems. The proposed method involves an algorithm in which the usual SR1 Hessian is updated a number of times in a way to be specified in some iterations, to improve the performance of the Hessian approximation. In particular, we discuss how to consider a criterion for indicating at each iteration whether it is necessary to employ extra updates. However it is well known that there are some theoretical difficulties when applying the SR1 update. Even for a current positive definite Hessian approximation, it is possible that the SR1 update may not be defined or the SR1 update may not preserve positive definiteness at some iterations. We then employ a restarting procedure that guarantees that updated matrices will be well-defined while preserving positive definiteness of updates. Numerical results support these theoretical considerations. They show that the implementation of the SR1 method using extra updating techniques improves the performance of the SR1 method substantially for a number of test problems from the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we present an new approach to construct the so-called shape preserving interpolation curves. The basic idea is first to approximate the set of interpolated points with a class of MQ quasi-interpolation operators and then pass through the set with the use of multivariate interpolation by using compactly supported radial basis functions. This approach possesses the advantages of certain shape preserving and good approximation behaviors. The proposed algorithm is easy to implement. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of partial differential equations with a singularly perturbed term in a reproducing kernel space. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective. [ABSTRACT FROM AUTHOR]

Abstract: Many real problems can be modeled to the problems with a hierarchical structure, and bilevel programming is a useful tool to solve the hierarchical optimization problems. So the bilevel programming is widely applied, and numerous methods have been proposed to solve this programming. In this paper, we propose an approximate programming algorithm to solve bilevel nonlinear programming problem. Finally, the example illustrates the feasibility of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, we develop a discrete approximation method for solving continuous-time linear fractional programming problems. Our method enables one to derive a recurrence structure which shall overcome the computational curse caused by the increasing numbers of decision variables in the approximate decision problems when the subintervals are getting smaller and smaller. Furthermore, our algorithm provides estimation for the error bounds of the approximate solutions. We also establish the convergence of our approximate solutions to the continuous-time linear fractional programming problems. Numerical examples are provided to illustrate the quality of the approximate solutions. [Copyright &y& Elsevier]

Abstract: In this paper, a novel learning strategy for radial basis function networks (RBFN) is proposed. By adjusting the parameters of the hidden layer, including the RBF centers and widths, the weights of the output layer are adapted by local optimization methods. A new local optimization algorithm based on a combination of the gradient and Newton methods is introduced. The efficiency of some local optimization methods to update the weights of RBFN is studied in solving systems of nonlinear integral equations. [Copyright &y& Elsevier]

Abstract: The piecewise algebraic variety, as a set of the common zeros of multivariate splines, is a kind of generalization of the classical algebraic variety. In this paper, we present an algorithm for isolating the zeros of the zero-dimensional piecewise algebraic variety which is primarily based on the interval zeros of univariate interval polynomials. Numerical example illustrates that the proposed algorithm is flexible. [Copyright &y& Elsevier]

Abstract: In this paper we consider a conic trust-region method for unconstrained optimization problems and analyze its convergence properties. We propose a convenient curvilinear search method to approximately solve the arising conic trust-region subproblem. Note that this approximate method preserves the strong convergence properties of the exact solution methods and is easy to implement. Both the linear and superlinear convergence of the method are established under commonly used conditions. Numerical experiments are conducted to show the efficiency of the new method. [Copyright &y& Elsevier]

Abstract: This paper is devoted to construction interpolatory multiscaling functions with a general dilation factor . We first show two-scale matrix symbol associated with interpolatory multiscaling functions is reduced to a special form. Also, a characterization on approximation order for the multiscaling functions is described in terms of elements of this special two-scale matrix symbol. Then, an algorithm is provided for constructing compactly supported interpolating multiscaling functions with dilation factor and higher approximation order. Finally, the associated several families examples with one-parameter or two-parameters are explicitly presented. The optimal parameter values which make multiscaling functions provide the highest regularity are also computed. [Copyright &y& Elsevier]

Abstract: In this paper, a numerical research on the sensitivity of nonmonotone trust region algorithms to their parameters is presented. We compare the numerical efficiency of two classes of nonmonotone trust region (NTR) algorithms in the context of unconstrained optimization. We examine the sensitivity of the algorithms to the parameters related to the nonmonotone technique and the initial trust region radius. We show that the numerical efficiency of nonmonotone trust region algorithms can be improved by choosing appropriate parameters. Based on extensive numerical tests, some efficient ranges of these parameters for nonmonotone trust region algorithms are recommended. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and study a new system of variational inclusions involving -monotone operators in Banach space. Using the resolvent operator associated with -monotone operators, we prove the existence and uniqueness of solutions for this new system of variational inclusions. We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm. [Copyright &y& Elsevier]

This paper focuses on the application of the techniques of discretization to obtain an approximated probability density function (pdf) for the completion time of large-size projects, in which we allow any type of pdf for the duration of activities. In this study, we improve the techniques of discretization in the following two ways: first, we propose to replace the max operation with an approximation procedure to save significant computational loading; and second, to reduce the error from assuming independence between paths using a simple heuristic rule. To evaluate the performance of our proposed algorithm, we randomly generated 20 sets of 100-node instances in our numerical experiments. Taking the results from a Monte Carlo simulation using 20,000 samples as a benchmark, we demonstrate that the proposed algorithm significantly outperforms the PERT model and Dodin’s [B.M. Dodin, Approximating the distribution function in stochastic networks, Comput. Oper. Res. 12 (3) (1985) 251–264] algorithm in both the running time and the precision aspects. [Copyright &y& Elsevier]

Abstract: In this paper, we propose novel finite element A - θ approaches to realize the approximation of a three-dimensional eddy current problem in a bounded domain containing a simply-connected conductor region where the eddy current is formed. These algorithms are based on solving a vector potential in the whole domain by using the edge finite-element method and a local scalar potential only in the conductor region by using the nodal finite-element method. Further, error estimates for coupled and decoupled A - θ schemes are given. [Copyright &y& Elsevier]

Abstract: This paper deals with the construction of approximate solution of first-order matrixlinear differential equations using matrix cubic splines. An estimation of the approximation error, an algorithm for its implementation and some illustrative examples are included. [Copyright &y& Elsevier]

Abstract: The second-order extension of Godunov''s method for hyperbolic conservation laws, known as MUSCL schemes, is studied in this paper. We present a new type of Eulerian MUSCL scheme and oscillation-free algorithms. The intercell flux is computed from difference approximations of characteristic equations with the data influencing the values at half-time step. The data for characteristic equations are classified in several different types by the domain of dependence. The oscillation-free algorithm, based on TVD method, gives a general formulation for previously used method and provides a class of limiters including existing limiters. To validate the new TVD-MUSCL scheme, numerical experiments are performed for several model problems in one and two dimensions. [Copyright &y& Elsevier]

Abstract: In this paper, we propose an algorithm for solving the obstacle problem. We try tofind the approximated region of the contact in the obstacle problem by iteration. Numerical examples are given for the obstacle problem for a membrane and the elastic-plastic torsion problem. [Copyright &y& Elsevier]

This paper considers ordinal algorithms for parallel machine scheduling with nonsimultaneous machine available times. Two objects of minimizing the latest job completion time and minimizing the latest machine completion time are studied. For the first objective, we present the optimal algorithms for m = 2, 3, 4 machine cases. For m ≥ 5, we propose an algorithm with competitive ratio 2 - 1/(m - 1) while the lower bound is SHAPE="SOL" ALIGN="C" STYLE="S">53. For the second objective, the optimal algorithm is also given. Furthermore, for a special case, an algorithm with significantly improved competitive ratio is given. [Copyright &y& Elsevier]