Modified Projection Method Research Articles

Simultaneous competitive supply chain network design with continuous attractiveness variables

A two-stage algorithm is developed for the competitive supply chain network design problem in which two competitors simultaneously enter the market with no existing rival chain, shape their networks, and set attractiveness of their opened distribution centres to maximize their profits. The customer behaviour is based on the Huff gravity-based rule.The proposed algorithm is constructed based on the Lemke and Howson algorithm and variational inequality formulation with the help of bi-level programming, the modified projection method, and the possibility theory. We derive the equilibrium condition and establish the finite dimensional variational inequality formulation, and provide properties of the equilibrium patterns in terms of the results of existence and uniqueness. Finally, we generate some instances and use a real world study to discuss the effect of the different structures of the competitors, namely centralized, decentralized, cooperative, or unknown modes, on the equilibrium solution.

Designing an integrated multi-echelon supply chain network with competition

This paper develops a model for competitive supply chain network design which investigates competition between new and existing chains where demand is deterministic and price-dependent. The network consists of raw material suppliers, manufacturers, retailers, and demand market. The supply chains compete in a non-cooperative manner and manufacturers produce homogeneous products. The objective is to maximise the future profits of both chains. First, the problem is formulated by mathematical programming and Lagrangian theory is used for adding the constraints to the objective function. Then, the equilibrium conditions are derived and the problem is formulated as a finite-dimensional variational inequality model. Qualitative properties of the equilibrium pattern are derived to establish existence and uniqueness results under reasonable conditions. The modified projection method is used to solve the variational inequality problem. Finally, a numerical example is presented to illustrate the efficiency of the proposed model and analyse the results under different conditions.

Approximate solution of Urysohn integral equations with non-smooth kernels

Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a kernel of the type of Green's function and defined on $L^\infty [0, 1]$. For $ r \geq 0$, we choose the approximating space to be a space of discontinuous piecewise polynomials of degree $\leq r$ with respect to a quasi-uniform partition of $[0, 1]$ and consider an interpolatory projection at $r+1$ Gauss points. Previous authors have proved that the orders of convergence in the collocation and the iterated collocation methods are $ r+1$ and $r + 2 + \min \{ r, 1 \}$, respectively. We show that the order of convergence in the iterated modified projection method is $4$ if $ r = 0$ and is $ 2 r + 3$ if $ r \geq 1$. This improvement in the order of convergence is achieved while retaining the size of the system of equations that needs to be solved, the same as in the case of the collocation method. Numerical results are given for specific examples.

Modified projection method for Urysohn integral equations with non-smooth kernels

Consider a nonlinear operator equation x−K(x)=f, where K is a Urysohn integral operator with a Green’s function type kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials, previous authors have investigated approximate solution of this equation using the Galerkin and the iterated Galerkin methods. They have shown that the iterated Galerkin solution is superconvergent. In this paper, a solution obtained using the iterated modified projection method is shown to converge faster than the iterated Galerkin solution. The improvement in the order of convergence is achieved by retaining the size of the system of equations same as for the Galerkin method. Numerical results are given to illustrate the improvement in the order of convergence.

Asymptotic error analysis of projection and modified projection methods for nonlinear integral equations

Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a smooth kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree $\leq r$, previous authors have established an order $r + 1$ convergence for the Galerkin solution, $2 r + 2$ for the iterated Galerkin solution, $ 3 r + 3$ for the modified projection solution and $ 4 r + 4$ for the iterated modified projection solution. Equivalent results have also been established for the interpolatory projection at Gauss points. In this paper, the iterated Galerkin/iterated collocation solution and the iterated modified projection solution are shown to have asymptotic series expansions. The Richardson extrapolation can then be used to improve the order of convergence to $ 2 r + 4$ in the case of the iterated Galerkin/iterated collocation method and to $ 4 r + 6$ in the case of the iterated modified projection method. Numerical results are given to illustrate this improvement in the orders of convergence.

A note on economic equilibrium and financial networks

Nagurney (1999) used variational inequalities to study economic equilibrium and financial networks and applied the modified projection method to solve the problem. In this paper, we formulate the problem as a nonlinear complementarity problem. The complementarity model is just the KKT condition for the model of Nagurney (1999). It is a simpler model than that of Nagurney (1999). We also establish sufficient conditions for existence and uniqueness of the equilibrium pattern, which are weaker than those in Nagurney (1999). Finally, we apply a smoothing Newton-type algorithm to solve the problem and report some numerical results.

A modified projection method for a common solution of a system of variational inequalities, a split equilibrium problem and a hierarchical fixed-point problem

In this paper, we suggest and analyze a modified projection method for finding a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem in the setting of real Hilbert spaces. We prove the strong convergence of the sequence generated by the proposed method to a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem. Several special cases are also discussed. The results presented in this paper extend and improve some well-known results in the literature. MSC: 49J30, 47H09, 47J20.

Modified projection and the iterated modified projection methods for nonlinear integral equations

Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a smooth kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree $\leq r$, previous authors have established an order $r + 1$ convergence for the Galerkin solution and $2 r + 2$ for the iterated Galerkin solution. Equivalent results have also been established for the interpolatory projection at Gauss points. In this paper, a modified projection method is shown to have convergence of order $ 3 r + 3$ and one step of iteration is shown to improve the order of convergence to $ 4 r + 4$. The size of the system of equations that must be solved, in implementing this method, remains the same as for the Galerkin method.

Modified projection method for strongly pseudomonotone variational inequalities

A modified projection method for strongly pseudomonotone variational inequalities is considered. Strong convergence and error estimates for the sequences generated by this method are studied in two versions of the method: the stepsizes are chosen arbitrarily from a given fixed closed interval and the stepsizes form a non-summable decreasing sequence of positive real numbers. We also propose some interesting examples to analyze the obtained results.

Modified projection approach for robust adaptive array beamforming

A novel modified projection method is proposed in this paper, which is robust against the signal steering vector mismatch and covariance matrix uncertainty. First, an enhanced covariance matrix estimate based on a shrinkage method is obtained. Then, the desired signal subspace is estimated from the eigenvectors of the enhanced covariance matrix, and a calibrated steering vector of the desired signal is obtained in sequence by projecting the presumed one onto the new estimated desired signal subspace. Finally, the robust adaptive beamformer weight is obtained from the estimated covariance matrix and the calibrated steering vector. Compared with the traditional projection method, the proposed method can work well even at low signal-to-noise ratio. Also, it is not necessary to estimate the number of sources. Simulation results demonstrate the efficiency of the proposed robust method compared with some of the existing ones.

An augmented immersed interface method for moving structures with mass

We present an augmented immersed interface method for simulating the dynamics of a deformable structure with mass in an incompressible fluid. The fluid is modeled by the Navier-Stokes equations in two dimensions. The acceleration of the structure due to mass is coupled with the flow velocity and the pressure. The surface tension of the structure is assumed to be a constant for simplicity. In our method, we treat the unknown acceleration as the only augmented variable so that the augmented immersed interface method can be applied. We use a modified projection method that can enforce the pressure jump conditions corresponding to the unknown acceleration. The acceleration must match the flow acceleration along the interface. The proposed augmented method is tested against an exact solution with a stationary interface. It shows that the augmented method has a second order of convergence in space. The dynamics of a deformable circular structure with mass is also investigated. It shows that the fluid-structure system has bi-stability: a stationary state for a smaller Reynolds number and an oscillatory state for a larger Reynolds number. The observation agrees with those in the literature.

A projection method for solving Cauchy singular integro-differential equations

In this work we present a modified projection method based on Legendre polynomials, for solving integro-differential equations with Cauchy kernel, in L 2 ( [ − 1 , 1 ] , C ) . The proposed numerical procedure leads to solve a system of linear equations. We prove the existence of a solution for the approximate equation, and we perform the error analysis. Numerical examples illustrate the theoretical results.

A new approach to supply chain network equilibrium models

We establish nonlinear complementarity formulations for the supply chain network equilibrium models. The formulations have simple structures and facilitate us to study qualitative properties of the models. In this setting, we obtain weaker conditions to guarantee the existence and uniqueness of the equilibrium pattern for a supply chain. A smoothing Newton algorithm that exploits the network structure is proposed for solving these models. Not only is the smoothing Newton algorithm proved to be globally convergent without requiring the assumptions of monotonicity and Lipschitz continuity, but also it can overcome the flaw that the performance of the modified projection method heavily depends on the choice of the predetermined step size. Numerical results indicate the advantages of the nonlinear complementarity formulation and the smoothing Newton algorithm.

Numerical Simulation of Macrosegregation Formation in Binary Alloy Solidification Processing Based on Modified Projection Method

A simplified mathematical model for describing the solidification processing of fluid flow, heat and solute transfer in binary alloys is developed. The conservation equations are numerically solved on a staggered grid by using the control volume-based finite difference method. As the interdendritic flow in the mushy zone is governed by Darcy’s law, the Modified Projection Method is presented for solving the momentum and continuity equations. The solidification processing of Fe-C alloy in rectangular domains is simulated and discussed. The numerical result shows that, by using the Modified Projection Method technique, the presented model is able to predict the macrosegregation formation effectively.

New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows.

In this paper, new finite difference methods based on the augmented immersed interface method (IIM) are proposed for simulating an inextensible moving interface in an incompressible two-dimensional flow. The mathematical models arise from studying the deformation of red blood cells in mathematical biology. The governing equations are incompressible Stokes or Navier-Stokes equations with an unknown surface tension, which should be determined in such a way that the surface divergence of the velocity is zero along the interface. Thus, the area enclosed by the interface and the total length of the interface should be conserved during the evolution process. Because of the nonlinear and coupling nature of the problem, direct discretization by applying the immersed boundary or immersed interface method yields complex nonlinear systems to be solved. In our new methods, we treat the unknown surface tension as an augmented variable so that the augmented IIM can be applied. Since finding the unknown surface tension is essentially an inverse problem that is sensitive to perturbations, our regularization strategy is to introduce a controlled tangential force along the interface, which leads to a least squares problem. For Stokes equations, the forward solver at one time level involves solving three Poisson equations with an interface. For Navier-Stokes equations, we propose a modified projection method that can enforce the pressure jump condition corresponding directly to the unknown surface tension. Several numerical experiments show good agreement with other results in the literature and reveal some interesting phenomena.

An efficient algorithm for solving supply chain network equilibria and equivalent supernetwork based traffic network equilibria

This paper is concerned with the algorithm of the supply chain network equilibrium model and its equivalent supernetwork based traffic network equilibrium model with elastic demands. Both models are further written as nonlinear complementarity problems. Semismooth least squares reformulations of the complementarity problems are presented and their convergence properties are investigated. Considering the drawbacks of Quasi-Newton method (using the Fischer-Burmeister function), a semi-smooth Levenberg-Marquardt-type method is proposed to solve the problems. Numerical examples show that the Levenberg-Marquardt-type method can solve the supply chain network equilibrium model and its equivalent supernetwork based traffic network equilibrium model significantly, and is more efficient than the Quasi Newton method and the modified projection method. Furthermore, the Levenberg-Marquardt-type method with the equivalent supernetwork based complementarity formulation can be implemented faster than with the supply chain network equilibrium complementarity formulation.

A modified projection method for solving co-coercive variational inequalities

This paper presents a modified projection method for solving variational inequalities, which can be viewed as an improvement of the method of Yan, Han and Sun [X.H. Yan, D.R. Han, W.Y. Sun, A modified projection method with a new direction for solving variational inequalities, Applied Mathematics and Computation 211 (2009) 118–129], by adopting a new prediction step. Under the same assumptions, we establish the global convergence of the proposed algorithm. Some preliminary computational results are reported.

Projection method with self‐adaptive time steps for LES of ignition and extinction in non‐premixed jet flames

AbstractIn this paper, a modified projection method is combined with a self‐adaptive time step procedure to develop numerical scheme for large eddy simulation (LES) of low Ma number turbulent reactive flows. The projection method introduced by Chorin is modified in this study to satisfy the simulation requirement of low Ma number reactive flow. The time step in this computation is automatically determined according to the time scales of both chemical reaction and turbulent fluctuations. This enables the simulation to capture detailed flow structures with less computational time. Numerical simulation of methane–air jet flames is carried out as an example to validate the developed numerical scheme. The mechanism of a simplified 4‐step chemical kinetics is applied for the methane–air reaction. The dynamic model is adopted for the turbulent motion of sub‐grid scale (SGS). The dynamic similarity model is used as the SGS model for the reaction rate. The LES results satisfactorily depict the ignition process of the turbulent jet flames and illustrate lucid and detailed coherent structures of the fully developed turbulent reactive jet flow. The LES results also exhibit the mechanism and characteristics of local extinction. The method developed in this study provides an effective way to capture more flow details with less computational time. In addition the method also helps one to investigate the mechanism of ignition and local extinction in jet flames. Copyright © 2008 John Wiley & Sons, Ltd.

Dendritic solidification under natural and forced convection in binary alloys: 2D versus 3D simulation

A model of both equiaxed and columnar dendritic growth was developed that incorporates thermal, solutal and fluid flow effects in either two or three dimensions. The model solves the momentum, mass and energy transport equations, including phase change. An imposed anisotropy algorithm, combined with a modified projection method solution of the Navier–Stokes equations, allows a relative coarse mesh and hence excellent computational efficiency. The model was used to study the effect of dimensionality (2D versus 3D) on dendritic growth with and without convection. The influence of forced convection on unconstrained equiaxed growth was studied first. In 3D, the upstream boundary layer is much thinner with a lower concentration than in 2D. This increases tip undercooling, accelerating upstream tip growth and promoting secondary branching. The influence of natural convection on constrained, columnar dendritic, growth was then studied. The 2D flow is blocked by the primary dendrite arms (which are effectively plates), while the 3D flow can wrap around the primaries. This change in flow strongly alters solute distribution and consequently the developing dendritic microstructure. 3D simulations are required to correctly predict unconstrained solidification microstructures.

Solving variational inequalities by a modified projection method with an effective step-size

In this paper, we propose a new projection method for the solution of variational inequality problems. The method is simple, which uses only function evaluations and projections onto the feasible set. We adopt a new step-size rule and a new search direction in the new method. Under the mild conditions, we prove the proposed method is globally convergent. Preliminary numerical results are reported.