The Colliery Model: A Computer Aid to Planning at a Colliery

Planning the operations of a colliery in the medium term requires management to produce a statement of proposed layout of faces and deployment of men. A computer program was developed to calculate the outcome of implementing a particular plan at a hypothetical colliery. This formed the basis of a competitive syndicate exercise at the Staff College of the National Coal Board. The concept was then extended to real colliery planning, to allow planners and managers to consider a wider set of options than current manual methods permit. The paper describes the common approach and discusses the modifications that were necessary to achieve results quickly enough to influence colliery planning decisions.     

On frequency estimations in phase fitted variational integrators for the general N-body problem

In the present work, the advantages of high order variational integrator methods are combined with phase lag properties for the numerical integration of the general N-body problem. Expressing the action integral at any intermediate points along the curve segment using a discrete Lagrangian that depends only on the end points of the interval, high order integrators can be obtained by defining the discrete Lagrangian in any time segment as a weighted sum on intermediate points, whose expressions for positions and velocities use Galerkin interpolation techniques. When oscillatory behavior is taken into account, the methods derived use trigonometric interpolation functions that depend on a frequency, which needs to be estimated. For that, using phase lag analysis, a new way to derive methods has been developed, that uses frequency estimation for each body at every time step. Results on special cases of the N-body problem show more stable orbits and less energy error when compared with the linear interpolation scheme. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On convergence and complexity of the modified forward‐backward method involving new linesearches for convex minimization

In optimization theory, convex minimization problems have been intensively investigated in the current literature due to its wide range in applications. A major and effective tool for solving such problem is the forward‐backward splitting algorithm. However, to guarantee the convergence, it is usually assumed that the gradient of functions is Lipschitz continuous and the stepsize depends on the Lipschitz constant, which is not an easy task in practice. In this work, we propose the modified forward‐backward splitting method using new linesearches for choosing suitable stepsizes and discuss the convergence analysis including its complexity without any Lipschitz continuity assumption on the gradient. Finally, we provide numerical experiments in signal recovery to demonstrate the computational performance of our algorithm in comparison to some well‐known methods. Our reports show that the proposed algorithm has a good convergence behavior and can outperform the compared methods.     

Two stable methods with numerical experiments for solving the backward heat equation

This paper presents results of some numerical experiments on the backward heat equation. Two quasi-reversibility techniques, explicit filtering and structural perturbation, to regularize the ill-posed backward heat equation have been used. In each of these techniques, two numerical methods, namely Euler and Crank-Nicolson (CN), have been used to advance the solution in time.Crank-Nicolson method is very counter-intuitive for solving the backward heat equation because the dispersion relation of the scheme for the backward heat equation has a singularity (unbounded growth) for a particular wave whose finite wave number depends on the numerical parameters. In comparison, the Euler method shows only catastrophic growth of relatively much shorter waves. Strikingly we find that use of smart filtering techniques with the CN method can give as good a result, if not better, as with the Euler method which is discussed in the main text. Performance of these regularization methods using these numerical schemes have been exemplified.     

Tactical design of rail freight networks. Part II: Local search methods with statistical analysis

This paper deals with the statistical study of the local search methods which we used in Part I of this work. In that part, a tactical planning model of the rail freight transportation was defined as a network design model. Different local search methods have been used to solve it: Simulated Annealing, Tabu Search and a ‘Descent’ method. The solution and method convergence depends on the initial feasible solution and the convexity of the feasible region, so the comparison among them will be made with the help of statistical theory. Assuming the hypothesis that the distribution of local minima can be represented by the Weibull distribution, it is possible to obtain an approach to the global minimum and a confidence interval of this. The global minimum estimation has been used to compare the heuristic methods and the parameters for a given heuristic, and to obtain a stopping criterion.     

The numerical solution of linear time-dependent partial differential equations by the Laplace transform and finite difference approximations

A feasible method is presented for the numerical solution of a large class of linear partial differential equations which may have source terms and boundary conditions which are time-varying. The Laplace transform is used to eliminate the time-dependency and to produce a subsidiary equation which is then solved in complex arithmetic by finite difference methods. An effective numerical Laplace transform inversion algorithm gives the final solution at each spatial mesh point for any specified set of values of t. The single-step property of the method obviates the need to evaluate the solution at a large number of unwanted intermediate time points. The method has been successfully applied to a variety of test problems and, with two alternative numerical Laplace transform inversion algorithms, has been found to give results of good to excellent accuracy. It is as accurate as other established finite difference methods using the same spatial grid. The algorithm is easily programmed and the same program handles equations of parabolic and hyperbolic type.     

Generating functions for computing power indices efficiently

TheShapley-Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. TheBanzhaf power index depends on the number of ways in which each voter can effect a swing. We introduce a combinatorial method based ingenerating functions for computing these power indices efficiently and we study thetime complexity of the algorithms. We also analyze the meet of two weighted voting games. Finally, we compute the voting power in the Council of Ministers of the European Union with the generating functions algorithms and we present its implementation in the system Mathematica. This work has been partially supported by the Spanish Ministery of Science and Technology under grant SEC2000-1243.     

On the use of homotopy analysis method for solving unsteady MHD flow of Maxwellian fluids above impulsively stretching sheets

In the present work, unsteady MHD flow of a Maxwellian fluid above an impulsively stretched sheet is studied under the assumption that boundary layer approximation is applicable. The objective is to find an analytical solution which can be used to check the performance of computational codes in cases where such an analytical solution does not exist. A convenient similarity transformation has been found to reduce the equations into a single highly nonlinear PDE. Homotopy analysis method (HAM) will be used to find an explicit analytical solution for the PDE so obtained. The effects of magnetic parameter, elasticity number, and the time elapsed are studied on the flow characteristics.     

A fair method for resetting the target in interrupted one-day cricket matches

A method is described for setting revised target scores for the team batting second when a limited-overs cricket match has been forcibly shortened after it has commenced. It is designed so that neither team benefits or suffers from the shortening of the game and so is totally fair to both. It is easy to apply, requring nothing more than a single table of numbers and a pocket calculator, and is capable of dealing with any number of interruptions at any stage of either or both innings.The method is based on a simple model involving a two-factor relationship giving the number of runs which can be scored on average in the remainder of an innings as a function of the number of overs remaining and the number of wickets fallen. It is shown how the relationship enables the target score in an interrupted match to be recalculated to reflect the relative run scoring resources available to the two teams, that is overs and wickets in combination. The method was used in several international and domestic one-day competitions and tournaments in 1997.     

Quad mesh generation for k-sided faces and hex mesh generation for trivalent polyhedra

Two mesh generation methods are presented in this paper: one for quadrilateral meshing of faces with convex vertices but an arbitrary number of sides, the other hexahedral meshing of polyhedra with convex edges, vertices connected to three edges and an arbitrary number of faces each of which has at least three sides. Efficient mapping methods for determining nodal positions are developed based on an extension to transfinite mapping and it is shown that these are identical to those obtained by iterative isoparametric smoothing.     

The design of colliery information and control systems

The paper is concerned with describing an investigation of information usage in the control of colliery operations. The premise of the work is that to make the most of new information retrieval technology currently being installed in collieries research in needed to provide compatible advances in methods of information usage. The approach adopted was to construct a continuous simulation model using system dynamics capable of providing a laboratory assessment of alternative managerial control policies based on alternative sources and levels of aggregation of information.The model developed represents a typical colliery situation composed of three working coal-faces and incorporating planning, production, development and manpower sectors. The face sectors transform coal reserves to mined coal output, under manpower constraints and geological shocks, and these are all interlinked by means of allocation policies for manpower and machine shifts.A range of policies for the exercise of control through these allocations are considered subject to a range of shocks. It is concluded that, although there are difficulties in designing single policies which are universally best, there are clear advantages associated with fully integrated colliery policies based on information inputs from all aspects of the operations.     

A Queueing Model for Spare Coal Faces

Determining the best provision of spare coal faces to allow for unexpected closures of faces can be considered as a problem in queueing theory. A machine-interference type of model is examined with random input, exponential service time and a fixed maximum number of spares. Steady state results are derived for the probability of a given number of faces being out of action, for the average number of spare faces held and for the average loss of working time. Numerical values of the average loss of working time have been computed as a guide to management and examples are given. No practical applications have yet been made.     

Quasi-Newton methods for multiobjective optimization problems

This work is an attempt to develop multiobjective versions of some well-known single objective quasi-Newton methods, including BFGS, self-scaling BFGS (SS-BFGS), and the Huang BFGS (H-BFGS). A comprehensive and comparative study of these methods is presented in this paper. The Armijo line search is used for the implementation of these methods. The numerical results show that the Armijo rule does not work the same way for the multiobjective case as for the single objective case, because, in this case, it imposes a large computational effort and significantly decreases the speed of convergence in contrast to the single objective case. Hence, we consider two cases of all multi-objective versions of quasi-Newton methods: in the presence of the Armijo line search and in the absence of any line search. Moreover, the convergence of these methods without using any line search under some mild conditions is shown. Also, by introducing a multiobjective subproblem for finding the quasi-Newton multiobjective search direction, a simple representation of the Karush–Kuhn–Tucker conditions is derived. The H-BFGS quasi-Newton multiobjective optimization method provides a higher-order accuracy in approximating the second order curvature of the problem functions than the BFGS and SS-BFGS methods. Thus, this method has some benefits compared to the other methods as shown in the numerical results. All mentioned methods proposed in this paper are evaluated and compared with each other in different aspects. To do so, some well-known test problems and performance assessment criteria are employed. Moreover, these methods are compared with each other with regard to the expended CPU time, the number of iterations, and the number of function evaluations.     

Hypergeometric Summation Techniques for High Order Finite Elements

The goal of this paper is to discuss the application of computer algebra methods in the design of a high order finite element solver. The finite element method is nowadays the most popular method for the computer simulation of partial differential equations. The performance of iterative solution methods depends on the condition number of the system matrix, which itself depends on the chosen basis functions. A major goal is to design basis functions minimizing the condition number, and which can be implemented efficiently. A related goal is the application of symbolic summation techniques to derive cheap recurrence relations allowing a simple and efficient implementation of basis functions. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

多层快速多极子方法的快速插值

多层快速多极子方法(MLFMM)可用来加速迭代求解由Maxwell方程组或Helmholtz方程导出的积分方程,其复杂度理论上是O(NlogN),N为未知量个数.MLFMM依赖于快速计算每层的转移项,以及上聚和下推过程中的层间插值.本文引入计算类似N体问题的一维快速多极子方法(FMM1D).基于FMM1D的快速Lagr...     

A parallel time integrator for solving the linearized shallow water equations on the rotating sphere

With the stagnation of processor core performance, further reductions in the time to solution for geophysical fluid problems are becoming increasingly difficult with standard time integrators. Parallel‐in‐time exposes and exploits additional parallelism in the time dimension, which is inherently sequential in traditional methods. The rational approximation of exponential integrators (REXI) method allows taking arbitrarily long time steps based on a sum over a number of decoupled complex PDEs that can be solved independently massively parallel. Hence, REXI is assumed to be well suited for modern massively parallel super computers, which are currently trending. To date, the study and development of the REXI approach have been limited to linearized problems on the periodic two‐dimensional plane. This work extends the REXI time stepping method to the linear shallow‐water equations on the rotating sphere, thus moving the method one step closer to solving fully nonlinear fluid problems of geophysical interest on the sphere. The rotating sphere poses particular challenges for finding an efficient solver due to the zonal dependence of the Coriolis term. Here, we present an efficient REXI solver based on spherical harmonics, showing the results of a geostrophic balance test, a comparison with alternative time stepping methods, an analysis of dispersion relations indicating superior properties of REXI, and finally, a performance comparison on the Cheyenne supercomputer. Our results indicate that REXI not only can take larger time steps but also can be used to gain higher accuracy and significantly reduced time to solution compared with currently existing time stepping methods.     

A location–allocation heuristic for the capacitated multi-facility Weber problem with probabilistic customer locations

The capacitated multi-facility Weber problem is concerned with locating m facilities in the Euclidean plane, and allocating their capacities to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a non-convex optimization problem and difficult to solve. In this work, we focus on a probabilistic extension and consider the situation where the customer locations are randomly distributed according to a bivariate distribution. We first present a mathematical programming formulation, which is even more difficult than its deterministic version. We then propose an alternate location–allocation local search heuristic generalizing the ideas used originally for the deterministic problem. In its original form, the applicability of the heuristic depends on the calculation of the expected distances between the facilities and customers, which can be done for only very few distance and probability density function combinations. We therefore propose approximation methods which make the method applicable for any distance function and bivariate location distribution.     

A Fast MPC Algorithm Using Nonfeasible Active Set Methods

Model predictive control (MPC) is an optimization-based control framework which is attractive to industry both because it can be practically implemented and it can deal with constraints directly. One of the main drawbacks of MPC is that large MPC horizon times can cause requirements of excessive computational time to solve the quadratic programming (QP) minimization which occurs in the calculation of the controller at each sampling interval. This motivates the study of finding faster ways for computing the QP problem associated with MPC. In this paper, a new nonfeasible active set method is proposed for solving the QP optimization problem that occurs in MPC. This method has the feature that it is typically an order of magnitude faster than traditional methods. This work has been supported by the Canadian NSERC under Grant A4396.     

Scheduling service in tandem queues attended by a single server

Consider a tandem queue model with a single server who can switch instantaneously from one queue to another. Customers arrive according to a Poisson process with rate λ . The amount of service required by each customer at the i^th queue is an exponentially distributed random variable with rate μ_i. Whenever two or more customers are in the system, the decision as to which customer should be served first depends on the optimzation criterion. In this system all server allocation policies in the finite set of work conserving deterministic policies have the same expected first passage times (makespan) to empty the system of customers from any initial state. However, a unique policy maximizes the first passage probability of empty-ing the system before the number of customers exceeds K, for any value of K, and it stochastically minimizes (he number of customers in the system at any time t > 0 . This policy always assigns the server to the non empty queue closest to the exit     

The Visual Design and Control of Trellis Display

Abstract

Trellis display is a framework for the visualization of data. Its most prominent aspect is an overall visual design, reminiscent of a garden trelliswork, in which panels are laid out into rows, columns, and pages. On each panel of the trellis, a subset of the data is graphed by a display method such as a scatterplot, curve plot, boxplot, 3-D wireframe, normal quantile plot, or dot plot. Each panel shows the relationship of certain variables conditional on the values of other variables. A number of display methods employed in the visual design of Trellis display enable it to succeed in uncovering the structure of data even when the structure is quite complicated. For example, Trellis display provides a powerful mechanism for understanding interactions in studies of how a response depends on explanatory variables. Three examples demonstrate this; in each case, we make important discoveries not appreciated in the original analyses. Several control methods are also essential to Trellis display. A control method is a technique for specifying information so that a display can be drawn. The control methods of Trellis display form a basic conceptual framework that can be used in designing software. We have demonstrated the viability of the control methods by implementing them in the S/S-PLUS system for graphics and data analysis, but they can be implemented in any software system with a basic capability for drawing graphs.