IMSL Library Research Articles

Regressão Linear Múltipla versus Modelagem Linear Automática

ABSTRACT In this study, performances of Multiple Linear Regression and Automatic Linear Modelling are compared for different sample sizes and number of predictors. A comprehensive Monte Carlo simulation study was carried out for this purpose. Random numbers generated from multivariate normal distribution by using RNMVN function of IMSL library of Microsoft FORTRAN Developer Studio composed the material of this study. Results of the simulation study showed that the sample size and the number of predictors are the main factors that lead to produce different results. Although both methods gave very similar results especially when studied with large sample sizes (n≥100), the Automatic linear modelling is preferred for analyzing data sets due to its simplicity in analyzing data and interpreting the results, ability to present results visually and providing more detailed information especially studying large complex data sets. It will be beneficial to use the Automatic linear modelling especially in analyzing massive and complex data sets for the purposes of investigating the relationships between one continuous dependent and 10 or more predictors and determine the factors that affect the response or target variable. At the same time, it will also be possible to evaluate the effect of each predictor with a more detailed response.

Hybrid solution for the flow and heat transfer of a second-grade viscoelastic fluid on a stretching sheet with injection or suction

This paper analyzes the steady, laminar, two-dimensional flow and heat transfer of a second-grade viscoelastic fluid over a flat plate subjected to continuous stretching. Injection and suction effects are also investigated. The mathematical model of the problem was derived from the constitutive tensor for an incompressible fluid of the second grade, followed by thermodynamic considerations and usual boundary layer assumptions. The generalized integral transform technique (GITT) was used to transform the governing partial differential equations (continuity, conservation of momentum and energy) within the boundary layer hypotheses on a coupled system of ordinary differential equations. The numerical solution of the system of equations was performed using the appropriate subroutine DIVPAG from the IMSL Library, which it is well-indicated for solving initial value problems with stiff characteristics. Convergence behaviors of the numerical results for the velocity and temperature fields were discussed as functions of parameters of interest. Investigations on the influence of viscoelastic parameter (K), injection or suction parameter (F) and Prandtl number (Pr) were performed in order to evaluate the effects of such governing parameters on the velocity and temperature fields. Comparison with previous results from the literature were also carried out in order to verify the computational code developed in this work.

Dynamic analysis of a fluid-conveying pipe under axial tension and thermal loads

The risers and pipelines are widely used in offshore oil and gas transportation, and they may undergo instabilities and buckling generated by internal fluid and thermal loads. This paper conducted a dynamic analysis of the fluid-conveying pipe under axial tension and thermal loads through the generalised integral transform technique (GITT). The governing partial differential equation (PDE) is converted into ordinary differential equations (ODEs) which are then numerically handled by the subroutine DIVPAG from IMSL Library. An excellent convergence behaviour is demonstrated and the numerical results are verified with previous theoretical study. The influence of the fluid velocity, temperature and axial tension is analysed qualitatively and quantitatively; the results show that the vibration frequency decreases when the fluid velocity and temperature increase or the axial tension decreases, and the critical fluid velocity decreases when the temperature increases or the axial tension decreases. As for the vibration deflection, an increase in both the temperature and the fluid velocity will lead to an increase in maximum deflection, whereas increasing the axial tension will suppress the maximum deflection.

Exact solution of bending problem of clamped orthotropic rectangular thin plates

The generalized integral transform technique (GITT) is employed to obtain an exact solution for the bending problem of fully clamped orthotropic rectangular thin plates. The use of the GITT approach in the analysis of the transverse deflection equation leads to a coupled system of fourth order differential equations (ODEs) in the dimensionless longitudinal spatial variable. The resulting transformed ODE system is then numerically solved by making use of the subroutine DBVPFD from IMSL Library. Numerical results with automatic global accuracy control are obtained for different values of aspect ratio. Critical comparisons with previously reported numerical results are performed with excellent agreement. A set of reference results for clamped orthotropic rectangular plates is also provided for future covalidation purposes, which are verified by those obtained from the general-purpose finite element software package ABAQUS.

Integral transform solution of natural convection in a square cavity with volumetric heat generation

The generalized integral transform technique (GITT) is employed to obtain a hybrid numerical-analytical solution of natural convection in a cavity with volumetric heat generation. The hybrid nature of this approach allows for the establishment of benchmark results in the solution of non-linear partial differential equation systems, including the coupled set of heat and fluid flow equations that govern the steady natural convection problem under consideration. Through performing the GITT, the resulting transformed ODE system is then numerically solved by making use of the subroutine DBVPFD from the IMSL Library. Therefore, numerical results under user prescribed accuracy are obtained for different values of Rayleigh numbers, and the convergence behavior of the proposed eigenfunction expansions is illustrated. Critical comparisons against solutions produced by ANSYS CFX 12.0 are then conducted, which demonstrate excellent agreement. Several sets of reference results for natural convection with volumetric heat generation in a bi-dimensional square cavity are also provided for future verification of numerical results obtained by other researchers.

Solution of the two dimensional diffusion and transport equations in a rectangular lattice with an elliptical fuel element using Fourier transform methods: One and two group cases

A method for solving the diffusion and transport equations in a rectangular lattice cell with an elliptical fuel element has been developed using a Fourier expansion of the neutron flux. The method is applied to a one group model with a source in the moderator. The cell flux is obtained and also the associated disadvantage factor. In addition to the one speed case, we also consider the two group equations in the cell which now become an eigenvalue problem for the lattice multiplication factor. The method of solution relies upon an efficient procedure to solve a large set of simultaneous linear equations and for this we use the IMSL library routines. Our method is compared with the results from a finite element code. The main drawback of the problem arises from the very large number of terms required in the Fourier series which taxes the storage and speed of the computer. Nevertheless, useful solutions are obtained in geometries that would normally require the use of finite element or analogous methods, for this reason the Fourier method is useful for comparison with that type of numerical approach. Extension of the method to more intricate fuel shapes, such as stars and cruciforms as well as superpositions of these, is possible.

Integral transform solutions of dynamic response of a clamped–clamped pipe conveying fluid

Analysis of dynamic response of pipe conveying fluid is an important aspect in nuclear power plant design. In the present paper, dynamic response of a clamped–clamped pipe conveying fluid was solved by the generalized integral transform technique (GITT). The governing partial differential equation was transformed into a set of second-order ordinary differential equations which is then numerically solved by making use of the subroutine DIVPAG from IMSL Library. A thorough convergence analysis was performed to yield sets of reference results of the transverse deflection at different time and spanwise position. We found good agreement between the computed natural frequencies at mode 1–3 and those obtained by previous theoretical study. Besides, modal separation analysis was carried out and the influence of mass ratio on deflection and natural frequencies was qualitatively and quantitatively assessed.

Dynamic response of clamped axially moving beams: Integral transform solution

The generalized integral transform technique (GITT) is employed to obtain a hybrid analytical–numerical solution for dynamic response of clamped axially moving beams. The use of the GITT approach in the analysis of the transverse vibration equation leads to a coupled system of second order differential equations in the dimensionless temporal variable. The resulting transformed ODE system is then solved numerically with automatic global accuracy control by using the subroutine DIVPAG from IMSL Library. Excellent convergence behavior is shown by comparing the vibration displacement of different points along the beam length. Numerical results are presented for different values of axial translation velocity and flexural stiffness. A set of reference results for the transverse vibration displacement of axially moving beam is provided for future co-validation purposes.

A Moving Balls Approximation Method for a Class of Smooth Constrained Minimization Problems

We introduce a new algorithm for a class of smooth constrained minimization problems which is an iterative scheme that generates a sequence of feasible points that approximates the constraints set by a sequence of balls and is accordingly called the Moving Balls Approximation algorithm (MBA). The computational simplicity of MBA, which uses first order data information, makes it suitable for large scale problems. Theoretical and computational properties of MBA in its primal and dual forms are studied, and convergence and global rate of convergence results are established for nonconvex and convex problems. We then introduce a variant of MBA that includes an active set technique and is particularly suitable for problems with a large number of constraints. This variant is as simple as MBA and is proven to preserve the convergence properties of MBA. Extension of MBA is also developed for a class of variational inequalities. Initial numerical experiments on quadratically constrained problems demonstrate the viability and performance of our methods when compared to some existing state-of-the-art optimization methods/software such as a sequential quadratic programming solver from the IMSL Library and the CVXOPT software package for convex optimization.

JMASM28: Gibbs Sampling for 2PNO Multi-unidimensional Item Response Theory Models (Fortran)

A Fortran 77 subroutine is provided for implementing the Gibbs sampling procedure to a multiunidimensional IRT model for binary item response data with the choice of uniform and normal prior distributions for item parameters. In addition to posterior estimates of the model parameters and their Monte Carlo standard errors, the algorithm also estimates the correlations between distinct latent traits. The subroutine requires the user to have access to the IMSL library. The source code is available at http://www.siuc.edu/~epse1/sheng/Fortran/MUIRT/GSMU2.FOR. An executable file is also provided for download at http://www.siuc.edu/~epse1/sheng/Fortran/MUIRT/EXAMPLE.zip to demonstrate the implementation of the algorithm on simulated data.

BMDS: A Collection of R Functions for Bayesian Multidimensional Scaling

Bayesian MDS has recently attracted a great deal of researchers’ attention because (1) it provides a better fit than classical MDS and ALSCAL, (2) it provides estimation errors of the distances, and (3) the Bayesian dimension selection criterion, MDSIC, provides a direct indication of optimal dimensionality; see the original paper by Oh & Raftery (2001). However, Bayesian MDS is not yet widely applied in practice. One of the reasons can be attributed to the apparent lack of software: there is none except for the original Oh & Raftery’s code, which requires good experience in Fortran programming and the IMSL library, which is a commercial library for numerical calculation. It may be difficult to require such environment for many researchers. Considering this situation, we propose a set of R functions, BMDS, to perform Bayesian MDS and to evaluate the results. Using BMDS, researchers can (1) perform Bayesian estimation in MDS, (2) check the convergence of Markov chain Monte Carlo (MCMC) estimation, (3) evaluate the optimal number of dimensions, (4) evaluate the estimation errors and (5) plot the resultant configurations. Also, using BMDS users can comparatively evaluate the result of Bayesian and classical MDS in terms of the value of stress and the plot of observed and estimated distances. In our functions, we made use of WinBUGS (Spiegelhalter, Thomas, Best, & Lunn, 2007) via R2WinBUGS package (Sturtz, Ligges, & Gelman, 2005) for MCMC estimation. Because the Bayesian MDS model is rather complex and it is impossible to use single WinBUGS script for any model, our bmds() function automatically produces a BUGS script that is adequate for the current data every time we run the R function. By using WinBUGS in this way we can speed-up the MCMC estimation while maintaining the readability of the code, which tends to be complex in Bayesian estimation.

COMPUTATIONAL OPTIMIZATION FOR INTEGRAL TRANSFORM ALGORITHMS APPLIED TO THE LID-DRIVEN CAVITY FLOW PROBLEM

An analysis is presented of the computational optimization for integral transform algorithms using the streamfunction only formulation of the Navier-Stokes equations, as applied to the steady incompressible laminar flow of a Newtonian fluid in two-dimensional formulation. The classical lid-driven rectangular cavity flow problem is considered in order to revise the conventional development of the Generalized Integral Transform Technique (GITT). The GITT is applied transforming the partial differential equation into a system of coupled ordinary differential equations, which is numerically solved by a general algorithm for boundary value problems, using a subroutine from the IMSL library with automatic error control. The conventional algorithm written in FORTRAN language is modified and tested seeking its optimization. A few different strategies of applying the technique are considered to achieve improved computational performance and allowing the inspection of convergence rates in the eigenfunction expansion of the original potentials for high Reynolds numbers. These different algorithm alternatives are analyzed and the relative merits are discussed. Results for different values of the Reynolds number and cavity aspect ratios are presented in tabular and graphical forms and fully converged results are critically compared against previously published findings.

JMASM27: An Algorithm for Implementing Gibbs Sampling for 2PNO IRT Models (Fortran)

A Fortran 77 subroutine is provided for implementing the Gibbs sampling procedure to a normal ogive IRT model for binary item response data with the choice of uniform and normal prior distributions for item parameters. The subroutine requires the user to have access to the IMSL library. The source code is available at http://www.siu.edu/~epse1/sheng/Fortran/, along with a stand alone executable file.

Performance analysis and optimization of a thermally non-symmetric annular fin

Considering thermally non-symmetric convective boundary conditions, optimum dimensions of an annular fin which has a rectangular cross-section are investigated. Two-dimensional heat diffusion equation is solved analytically to obtain temperature distribution and heat transfer rate. In this work, fin volume is fixed to obtain the dimensionless geometrical parameters of the fin with maximum heat transfer rates. The optimum geometry which maximizes the heat transfer rate for a given fin volume has been found employing NCONF routine in the IMSL Library. The derived condition of optimality gives an open choice to the designer.

Application of goal programming in a multi-objective reservoir operation model in Tunisia

This paper is concerned with the operation and the control of multi-purpose reservoir systems. The problem involves finding appropriate releases from various reservoirs in the system in order to satisfy multiple conflicting objectives, such as the satisfaction of demands with the required salinity, the minimization of the pumping cost, etc. To solve this problem, we considered two characteristics: multiple objectives and stochastic parameters. The method used is based on the application of stochastic goal programming approach. The implemented model was developed under FORTRAN PowerStation and IMSL library and applied to a multi-reservoir system in the north of Tunisia.

Improved lumped-differential formulations in hyperbolic heat conduction

The present work aims at applying the ideas in the so-called Coupled Integral Equations Approach (CIEA) to the one-dimensional thermal wave propagation problem in a finite solid medium, leading to improved lumped-differential formulations through Hermite-type approximations for integrals. The application of CIEA methodology makes it possible to reduce the partial differential equation governing the hyperbolic heat conduction problem to a simple system consisting of two or three ordinary differential equations for the average and surface temperatures, depending on the choice of the Hermite-type approximation. The Runge-Kuta method, enclosed in the IVPRK routine from IMSL Library [1], is used to solve the system of ordinary differential equations. Results for the average and surface temperatures are computed for different thermal relaxation times.

Application of Hybrid Differential Evolution to Fuzzy Dynamic Optimization of a Batch Fermentation

Fuzzy optimization was applied to obtain the cooling rate and fermentation time of the batch beer fermentation process. The objective function and the product quality constraints under the crisp environment were reasonably softened into an optimization problem in the fuzzy environment. By eliciting the assigned membership function for the objective function and constraints, the fuzzy optimization problem could be converted into a maximizing decision problem. In order to obtain a global optimal solution, the hybrid differential evolution was introduced to solve the maximizing decision problem. The hybrid method can use the least population size of three to obtain a more satisfactory solution, as compared to the original algorithm of differential evolution, the BCPOL solver in the IMSL library, and a simple genetic algorithm solver.

An Efficient Method for Mold Thermal Cycle Analysis in Repeated Forming Process of TV Glass

An efficient method is developed for plunger thermal cycle analysis in repeated forming process of the TV glass. The plunger undergoes temperature fluctuation during a cycle due to the repeated contact and separation from the glass, which attains a cyclic steady state having same temperature history at every cycle. Straightforward analysis of this problem brings about more than 80 cycles to get reasonable solution, and yet hard to setup stopping criteria due to extremely slow convergence. An exponential fitting method is proposed to overcome the difficulty, which finds exponential function to best approximate temperature values of 3 consecutive cycles, and new cycle is restarted with the fitted value at infinite time. Numerical implementation shows that it reduces the number of cycles dramatically to only 6-18 cycles to reach convergence within 10 accuracy. A system for the analysis is constructed, in which the thermal analysis is performed by commercial software ANSYS, and the fitting of the result is done by IMSL library. From the parametric studies, one reveals some important facts that although the plunger cooling or the glass thickness is increased, its counter part in contact is not much affected, duo to the low thermal conductance of the glass.

Ordinal comparison of heuristic algorithms using stochastic optimization

The performance of heuristic algorithms for combinatorial optimization is often sensitive to problem instances. In extreme cases, a specialized heuristic algorithm may perform exceptionally well on a particular set of instances while fail to produce acceptable solutions on others. Such a problem-sensitive nature is most evident in algorithms for combinatorial optimization problems such as job shop scheduling, vehicle routing, and cluster analysis. The paper proposes a formal method for comparing and selecting heuristic algorithms (or equivalently, different settings of a same algorithm) given a desired confidence level and a particular set of problem instances. We formulate this algorithm comparison problem as a stochastic optimization problem. Two approaches for stochastic optimization, ordinal optimization and optimal computing budget allocation are applied to solve this algorithm selection problem. Computational testing on a set of statistical clustering algorithms in the IMSL library is conducted. The results demonstrate that our method can determine the relative performance of heuristic algorithms with high confidence probability while using a small fraction of computer times that conventional methods require.

Synthesis of heat‐exchanger network by simulated annealing and NLP procedures

AbstractA two‐level procedure for heat‐exchanger network synthesis problems, which is related to structural optimization, is carried out by a simulated‐annealing procedure. The main control parameters, that is, the initial annealing temperature, the temperature length, and the temperature‐decreasing scheme, are extensively studied on the well‐known 10SP1 problem, and their values are used for solving a 5SP1 problem with forbidden matches and a larger scale problem, the 20SP1. In all cases, the structural search is initialized with the network involving only utilities. For each generated network, the operating conditions (temperatures and split rates) are optimized by an NLP procedure to minimize the sum of investment and operating costs. Particular attention is given to the size reduction of the NLP problem. The proposed procedure leads to a reduction by a factor of 3 of the problem size in terms of number of variables and constraints. This slave problem is solved by an SQP procedure from the IMSL Library.