Authors:Mohamed A. Tawhid; Wei-Zhe Gu; Benjamin Tran Pages: 1 - 26 Abstract: We study an unconstrained minimization approach to the generalized complementarity problem GCP(f, g) based on the generalized Fischer-Burmeister function and its generalizations when the underlying functions are \(C^1\) . Also, we show how, under appropriate regularity conditions, minimizing the merit function corresponding to f and g leads to a solution of the generalized complementarity problem. Moreover, we propose a descent algorithm for GCP(f, g) and show a result on the global convergence of a descent algorithm for solving generalized complementarity problem. Finally, we present some preliminary numerical results. Our results further give a unified/generalization treatment of such results for the nonlinear complementarity problem based on generalized Fischer-Burmeister function and its generalizations. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0328-6 Issue No:Vol. 37, No. 1 (2018)

Authors:C. Lederman; R. Martin; J.-L. Cambier Pages: 27 - 51 Abstract: We describe an approach to solving a generic time-dependent differential equation (DE) that recasts the problem as a functional optimization one. The techniques employed to solve for the functional minimum, which we relate to the Sobolev Gradient method, allow for large-scale parallelization in time, and therefore potential faster “wall-clock” time computing on machines with significant parallel computing capacity. We are able to come up with numerous different discretizations and approximations for our optimization-derived equations, each of which either puts an existing approach, the Parareal method, in an optimization context, or provides a new time-parallel (TP) scheme with potentially faster convergence to the DE solution. We describe how the approach is particularly effective for solving multiscale DEs and present TP schemes that incorporate two different solution scales. Sample results are provided for three differential equations, solved with TP schemes, and we discuss how the choice of TP scheme can have an orders of magnitude effect on the accuracy or convergence rate. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0319-7 Issue No:Vol. 37, No. 1 (2018)

Authors:M. Li; F. F. Dou; T. Korakianitis; C. Shi; P. H. Wen Pages: 135 - 159 Abstract: Based on the interpolation of the Lagrange series and the Finite Block Method (FBM), the formulations of the Boundary Node Petrov–Galerkin Method (BNPGM) are presented in the weak form in this paper and their applications are demonstrated to the elasticity of functionally graded materials, subjected to static and dynamic loads. By introducing the mapping technique, a block of quadratic type is transformed from the Cartesian coordinate to the normalized coordinate with 8 seeds for two-dimensional problems. The first-order partial differential matrices of boundary nodes are obtained in terms of the nodal values of the boundary node, which can be utilized to determine the tractions on the boundary. Time-dependent partial differential equations are analyzed in the Laplace transformed domain and the Durbin’s inversion method is applied to determine the physical values in the time domain. Illustrative numerical examples are given and comparison has been made with the analytical solutions, the Boundary Element Method (BEM) and the Finite Element Method (FEM). PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0335-7 Issue No:Vol. 37, No. 1 (2018)

Authors:Elena-Gratiela Robe-Voinea; Raluca Vernic Pages: 205 - 219 Abstract: The Fast Fourier Transform provides an alternative approximate method to evaluate the distribution of aggregate losses in insurance and finance. The efficiency of this method has already been proved for univariate and bivariate insurance models; therefore, in this paper, we extend it to a multivariate setting by considering its application to a particular model that includes losses of different types and dependency between them. Since the Fourier transform method works with truncated claims distributions, it can generate aliasing errors by wrapping around the probability mass that lies at the truncation point below this point. To eliminate this problem, we also discuss a suitable change of measure called exponential tilting that forces the tail of the distribution to decrease at exponential rate. Other possible errors are also discussed. We also illustrate the method on several numerical examples. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0336-6 Issue No:Vol. 37, No. 1 (2018)

Authors:Ya-Jun Xie; Chang-Feng Ma Pages: 221 - 233 Abstract: In this paper, we extend the two-step matrix splitting iteration approach by introducing a new relaxation parameter. The main idea is based on the inner–outer iteration for solving the PageRank problems proposed by Gleich et al. (J Sci Comput 32(1): 349–371, 2010) and Bai (Numer Algebra Cont Optim 2(4): 855–862, 2012) and the two-step splitting iteration presented by Gu et al. (Appl Math Comput 271: 337–343, 2015). The theoretical analysis results show that the proposed method is efficient. Numerical experiments demonstrate that the convergence performances of the method are better than the existing methods. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0338-4 Issue No:Vol. 37, No. 1 (2018)

Authors:Tian-Hui Ma; Ting-Zhu Huang; Xi-Le Zhao Pages: 277 - 296 Abstract: We propose a novel image denoising model based on the total generalized variation (TGV) regularization. In the model, a spatially dependent regularization parameter is utilized to adaptively fit the local image features, resulting in further exploitation of the denoising potential of the TGV regularization. The proposed model is formulated under a joint optimization framework, by which the estimations of the restored image and the regularization parameter are achieved simultaneously. Furthermore, the model is general purpose that can handle various types of noise occurring in image processing. An alternating minimization-based numerical scheme is especially developed, which leads to an efficient algorithmic solution to the nonconvex optimization problem. Numerical experiments are reported to illustrate the effectiveness of our model in terms of both peak signal-to-noise ratio and visual perception. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0342-8 Issue No:Vol. 37, No. 1 (2018)

Authors:Eduard Marušić-Paloka; Igor Pažanin Pages: 297 - 305 Abstract: We consider the incompressible fluid with a pressure-dependent viscosity flowing through a multiple pipe system. The viscosity–pressure relation is given by the Barus law commonly used in the engineering applications. Assuming that the ratio between pipes thickness and its length is small, we propose a rigorous asymptotic approach based on the concept of the transformed pressure. As a result, we obtain new macroscopic model describing the effective behavior of the fluid in the system. In particular, the generalized version of the Kirchhoff’s law is derived giving the explicit formula for the junction pressure. The error estimate for the asymptotic approximation is also provided. Mathematical analysis presented here can be applied to a general viscosity–pressure relation satisfied by other empiric laws. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0345-5 Issue No:Vol. 37, No. 1 (2018)

Authors:M. Salahi; A. Taati Pages: 329 - 347 Abstract: In this paper, we study the extended trust region subproblem (eTRS) in which the trust region intersects the Euclidean ball with a single linear inequality constraint. By reformulating the Lagrangian dual of eTRS as a two-parameter linear eigenvalue problem, we state a necessary and sufficient condition for its strong duality in terms of an optimal solution of a linearly constrained bivariate concave maximization problem. This results in an efficient algorithm for solving eTRS of large size whenever the strong duality is detected. Finally, some numerical experiments are given to show the effectiveness of the proposed method. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0347-3 Issue No:Vol. 37, No. 1 (2018)

Authors:Piotr Nowak; Maciej Romaniuk Pages: 365 - 394 Abstract: Natural catastrophes lead to problems of insurance and reinsurance industry. Classic insurance mechanisms are often inadequate for dealing with consequences of catastrophic events. Therefore, new financial instruments, including catastrophe bonds (cat bonds), were developed. In this paper we price the catastrophe bonds with a generalized payoff structure, assuming that the bondholder’s payoff depends on an underlying asset driven by a stochastic jump-diffusion process. Simultaneously, the risk-free spot interest rate has also a stochastic form and is described by the multi-factor Cox–Ingersoll–Ross model. We assume the possibility of correlation between the Brownian part of the underlying asset and the components of the interest rate model. Using stochastic methods, we prove the valuation formula, which can be applied to the cat bonds with various payoff functions. We use adaptive Monte Carlo simulations to analyze the numerical properties of the obtained pricing formula for various settings, including some similar to the practical cases. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0348-2 Issue No:Vol. 37, No. 1 (2018)

Authors:Xiaoni Chi; Yang Wang; Zhibin Zhu; Zhongping Wan Pages: 439 - 455 Abstract: The Jacobian consistency of smoothing functions plays an important role for achieving the rapid convergence of Newton methods or Newton-like methods with an appropriate parameter control. In this paper, we study the properties, derive the computable formula for the Jacobian matrix and prove the Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for second-order cone complementarity problems proposed by Tang et al. (Comput Appl Math 33:655–669, 2014). Then we apply its Jacobian consistency to a smoothing Newton method with the appropriate parameter control presented by Chen et al. (Math Comput 67:519–540, 1998), and show the global convergence and local quadratic convergence of the algorithm for solving the SOCCP under rather weak assumptions. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0352-6 Issue No:Vol. 37, No. 1 (2018)

Authors:Na Wang; Maoan Han Pages: 475 - 484 Abstract: The predator–prey model with distributed delay is stated in present paper. On the basis of geometric singular perturbation theory, the transition of the solution trajectory is illuminated, and the existence of the relaxation oscillation is proved. It is indicated the characteristic of the relaxation oscillation is dependent on the structure of the slow manifold. Moreover, the approximate expression of the relaxation oscillation and its period are obtained analytically. One case study is given to demonstrate the validity of theoretical results. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0353-5 Issue No:Vol. 37, No. 1 (2018)

Authors:Zhifeng Zhang; Dahai Rao; Hanyu Zuo; Xinyu Zhang Pages: 501 - 514 Abstract: The production structure of a manufacturing system not only influences its transportation and operation cost, but also affects logistics and parts/machine assignment decisions. Based on the sending and feedback mechanisms of information, the principles of conditional entropy and classical probability are utilized to establish structure entropy models and ordered indices of the system with dynamic characteristics and generality, which provide effective solutions for the absence of quantitative method in evaluating the structural optimization of production system. In an empirical study, this paper analyzes different production structures when workpieces are processed by different working routes before and after the implementation of cellular manufacturing. Afterwards, the developed structure entropy models and ordered indices are utilized to calculate the orderliness of production structure under the two different states. The final result verifies well the validity of this quantitative method for evaluating the orderliness of production structures. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0354-4 Issue No:Vol. 37, No. 1 (2018)

Authors:Yi-Fen Ke; Chang-Feng Ma Pages: 515 - 524 Abstract: Based on the modified relaxed splitting (MRS) preconditioner proposed by Fan and Zhu (Appl Math Lett 55:18–26, 2016), a new relaxed splitting preconditioner is proposed for the generalized saddle point problems arising from the incompressible Navier–Stokes equations. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are analyzed. Numerical results show that the proposed preconditioner is superior to the MRS preconditioner when they are used as preconditioners to accelerate the convergence rate of the Krylov subspace methods, such as GMRES. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0357-1 Issue No:Vol. 37, No. 1 (2018)

Authors:O. Abidi; K. Jbilou Pages: 525 - 540 Abstract: In this paper, we consider the balanced truncation method for model reductions in large-scale linear and time-independent dynamical systems with multi-inputs and multi-outputs. The method is based on the solutions of two large coupled Lyapunov matrix equations when the system is stable or on the computation of stabilizing positive and semi-definite solutions of some continuous-time algebraic Riccati equations when the dynamical system is not stable. Using the rational block Arnoldi, we show how to compute approximate solutions to these large Lyapunov or algebraic Riccati equations. The obtained approximate solutions are given in a factored form and used to build the reduced order model. We give some theoretical results and present numerical examples with some benchmark models. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0359-z Issue No:Vol. 37, No. 1 (2018)

Authors:Zhichuan Zhu; Bo Yu Pages: 541 - 566 Abstract: In this paper, the principal-agent bilevel programming problem with integral operator is considered, in which the upper-level object is the agent that maximizes its expected utility with respect to an agreed compensation contract. The constraints are the principal’s participation and the agent’s incentive compatibility. The latter is a lower-level optimization problem with respect to its private action. To solve an equivalent single-level nonconvex programming problem with integral operator, a modified homotopy method for solving the Karush–Kuhn–Tucker system is proposed. This method requires only an interior point and, not necessarily, a feasible initial approximation for the constraint shifting set. Global convergence is proven under some mild conditions. Numerical experiments were performed by our homotopy method as well as by fmincon in Matlab, LOQO and MINOS. The experiments showed that: designing a piecewise linear contract is much better than designing a piecewise constant contract and only needs to solve a much lower-dimensional optimization problem and hence needs much less computation time; the optimal value of the principal-agent model with designing piecewise linear contract tends to a limitation, while the discrete segments gradually increase; and finally, the proposed modified homotopy method is feasible and effective. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0361-5 Issue No:Vol. 37, No. 1 (2018)

Authors:Maitri Verma; A. K. Misra Pages: 605 - 626 Abstract: The anthropogenic emission of carbon dioxide ( \(\mathrm{CO}_{2}\) ) is the prime culprit for the menace of global warming. To achieve the goal of mitigation of global warming, it is crucial to curb the anthropogenic carbon dioxide emissions. The prime anthropogenic source of \(\mathrm{CO}_{2}\) is fossil fuel burning. In this paper, we propose a nonlinear mathematical model to study the impact of technological options, used for the reduction of \(\mathrm{CO}_{2}\) emissions from fossil fuel burning and industrial processes, on the control of atmospheric \(\mathrm{CO}_2\) . In the modeling process, it is assumed that the technological options are implemented to curb the \(\mathrm{CO}_{2}\) emissions from the source at a rate proportional to the anthropogenic \(\mathrm{CO}_{2}\) emissions. Model analysis reveals that the atmospheric level of \(\mathrm{CO}_{2}\) can be effectively reduced by increasing the implementation rate of technological options and their efficiency. The strategies which optimally reduce atmospheric \(\mathrm{CO}_{2}\) levels while minimizing the cost associated with the implementation of technological options are identified using optimal control theory. Numerical simulation has been carried out to illustrate theoretical results. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0364-2 Issue No:Vol. 37, No. 1 (2018)

Authors:M. Moradipour; S. A. Yousefi Pages: 627 - 639 Abstract: Under the Black–Scholes model, the value of an American option solves a free boundary problem which is equivalent to a variational inequality problem. Using positive definite kernels, we discretize the variational inequality problem in spatial direction and derive a sequence of linear complementarity problems (LCPs) in a finite-dimensional euclidean space. We use special kind of kernels to impose homogeneous boundary conditions and to obtain LCPs with positive definite coefficient matrices to guarantee the existence and uniqueness of the solution. The LCPs are then successfully solved iteratively by the projected SOR algorithm. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0351-7 Issue No:Vol. 37, No. 1 (2018)

Authors:Enrique Fernández-Cara; Laurent Prouvée Pages: 745 - 762 Abstract: This paper deals with the optimal control of a mathematical model of glioma progression incorporating the basic facts of the evolution of primary brain tumors. We will consider a model of the simplest possible kind, based on the Fischer-Kolmogorov equations, using ideas from Pérez-García (Mathematical models for the radiotherapy of gliomas (preprint), 2016). The control is the 2n-tuple \((t_1, \ldots , t_n; d_1, \ldots , d_n)\) , where \(d_i\) is the i-th applied radiotherapy dose and \(t_i\) is the i-th administration for \(1 \le i \le n\) . We search for controls that maximize the survival time, that is, the time at which the tumor mass reaches a critical value \(M_{*}\) , over the class of admissible radiation times and doses. We present theoretical and numerical results that justify the relevance of the model and the existence of potential medical applications. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0366-0 Issue No:Vol. 37, No. 1 (2018)

Authors:Dinesh Kumar; Siddhartha P. Chakrabarty Pages: 763 - 784 Abstract: In this paper, we present a modified ratio-dependent model by incorporating the supply of additional food to the predators. We analyze the dynamics of this modified model and as well as the ecological consequences of the additional food supply. The model exhibits a way of control over the predator–prey system by the appropriate handling of the quantity and quality of the additional food. An appropriate choice of these could lead to conservation of the species as well as biological control of a pest population. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0369-x Issue No:Vol. 37, No. 1 (2018)

Authors:A. K. B. Chand; P. Viswanathan; N. Vijender Pages: 785 - 804 Abstract: This paper investigates some univariate and bivariate constrained interpolation problems using fractal interpolation functions. First, we obtain rational cubic fractal interpolation functions lying above a prescribed straight line. Using a transfinite interpolation via blending functions, we extend the properties of the univariate rational cubic fractal interpolation function to generate surfaces that lie above a plane. In particular, the constrained bivariate interpolation discussed herein includes a method to construct fractal interpolation surfaces that preserve positivity inherent in a prescribed data set. Uniform convergence of the bivariate fractal interpolant to the original function which generates the data is proven. PubDate: 2018-03-01 DOI: 10.1007/s40314-016-0373-1 Issue No:Vol. 37, No. 1 (2018)