William F. Mitchell

Applied and Computational Mathematics Division National Institute of Standards and Technology Gaithersburg, MD, USA 20899-8910

Abstract: One aspect of adaptive mesh refinement in the finite element method for solving partial differential equations is the method by which elements are refined. In the early 1980’s the dominant method for refining triangles was the red-green algorithm of Bankand Sherman. The red refinements are the desired refinements, but will result in an incompatible mesh when used alone. The green refinements are used to recover compatibility for stability of the finite element discretization, and are removed before the next adaptive step. Prof. Bob Skeel raised the question as to whether it is possible to perform adaptive refinement of triangles without this complicated patching/unpatching process. As a result, a new triangle refinement method, called newest vertex bisection, was devised as an alternative to red-green refinement in the mid 1980’s. The new approach is simpler and maintains compatibility of the mesh at all times, avoiding the patching/unpatching of the green refinement. In this historical paper we review the development of the newest vertex bisection method for adaptive refinement, and subsequent extensions of the method.

Keywords: finite elements; adaptive mesh refinement; newest vertex bisection
Mathematics Subject Classification: 65-03, 65N30, 65N50
PACS: 02.60.Lj, 02.70.Dh

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Gaussian Scale Mixtures Miguel Martins Felgueiras ^{1 *}, João Paulo Martins^**, Rui Filipe Santos^**

^* CEAUL Lisbon and ESTG, CIGS, Polytechnic Institute of Leiria, Portugal
^** CEAUL Lisbon and ESTG, Polytechnic Institute of Leiria, Portugal

Abstract: In this paper we present a parsimonious approximation of a Gaussian mixture when its components share a common mean value, i.e. a scale mixture. We show that a shifted and scaled Student’s t-distribution can be approximated to this type of mixture, and use the result to develop a hypothesis test for the equality of the components mean value. A simulation study to check the quality of the approximation is also provided, together with an application to logarithmic daily returns.

Keywords: Gaussian scale mixture, Student’s t-distribution, log-returns, simulation.
Mathematics Subject Classification: 60E05; 62P20.

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Eva Volna and Martin Kotyrba¹

Faculty of Science Department of Informatics and Computers, University of Ostrava, 30 dubna 22, 70103 Ostrava, Czech Republic

¹Corresponding author. E-mail: martin.kotyrba@osu.cz

Abstract: The Vehicle Routing Problem (VRP) is one of the most challenging combinatorial optimization tasks. This problem consists in designing an optimal set of routes for a fleet of vehicles in order to serve a given set of customers. Vehicle routing problem forms an integral part of the supply chain management, which plays a significant role in productivity improvement in organizations through an efficient and effective delivery of goods/services to customers. This problem is known to be NP-hard; hence many heuristic procedures for its solution have been suggested. For such problems, it is often desirable to obtain approximate solutions, so they can be found fast enough and are sufficiently accurate for the purpose. In this paper, we have performed an experimental study that indicates a suitable use of genetic algorithms for the vehicle routing problem. We tested instances from Capacitated Vehicle Routing Problem Library (CVRPLIB) series A, B, E, M and X. The obtained experimental outputs were compared with the following heuristics: the Clarke and Wright heuristic, sweep algorithm, and Taillard's algorithm.

Keywords VRP - Vehicle Routing Problem, combinatorics, Clarke and Wright heuristic, Sweep algorithm,Taillard's algorithm, Genetic algorithm.
Mathematics Subject Classification: 97K20, 90C59

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S. Iqbal^a, A. R. Ansari^b1, A. Javed^c and A. M. Siddiqui^d

^aDepartment of Computer Science, COMSATS Institute of Information Technology, Sahiwal Campus, Pakistan.
^bCentre for Advance Studies in Engineering (CASE), 19-Attaturk Avenue, G-5/1, Islamabad, Pakistan.
^cDepartment of Mathematics & Natural Sciences, Gulf University for Science & Technology, P.O. Box 7207, Hawally 32093, Kuwait
^dDepartment of Mathematics, York Campus, Pennsylvania State University, York,PA 17403, USA

Abstract: A weighted-residual based a posteriori error estimation formulation in Galerkin's finite element fashion using quadratic Lagrange polynomials has been formulated to find numerical solutions of obstacle, unilateral and contact second-order boundary-value problems. The approach having piecewise quadratic shape functions has been utilized for checking ">the approximate solutions for spatially adaptive finite element grids. The local element balance based on the residual has been considered as an error assessment criterion. Numerical testing indicates that local errors are large at the interface regions where the gradients are large. A comparison of an adaptive refined grid with that of a uniform mesh for second order obstacle boundary value problems, confirms the superiority of the adaptive scheme without increasing the number of unknown coefficients.

Keywords: Adaptive grid renement scheme, Quadratic Lagrange polynomials, Galerkin's method, Finite element method, Boundary-value problems
Mathematics Subject Classication: 65M60

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Yu.E. Gorbachev ¹ , E.G. Kolesnichenko ^2
1Research Department, Coddan Technologies LLC, 197342 St. Petersburg, Russia
²Gas Kinetics Lab, Moscow State University, Institute for Mechanics, 117192 Moscow, Russia

Abstract: One of the main problems of the non-equilibrium physical-chemical gas-dynamics
is considered: derivation of gas-dynamics equations for reactive gas mixtures. By non-
equilibrium effects we mean all kinds of effects caused by deviation of the distribution
function from its quasi-equilibrium value. The method which is used to obtain the normal
solution for the generalized Boltzmann equation is discussed. As opposed to the tradi-
tional approach, it permits a description of experimentally observed pressure dependence
of the reaction rates (low- and high-pressure limits) and to generalize the theory of ther-
mal dissociation for the arbitrary reaction case and for spatially inhomogeneous systems.
Conclusions concerning the necessity of revising the traditional approach to the chemical
reaction description and of developing the non-equilibrium chemistry are argued. In the
framework of the non-equilibrium chemistry, chemical reactions are no longer independent
of each other (Guldberg and Waage law is not applicable under the non-equilibrium condi-
tions) and corresponding reaction rates are functions of the reacting mixture composition.
Exact expressions for reaction and relaxation rates for the cut-off harmonic oscillator model
are obtained. Only the case of hard potentials in Grad’s meaning is considered, while soft
potentials remain an issue.

Keywords: Physical-chemical gas-dynamics, kinetic theory, Boltzmann equation, reactive
mixture, transport equations, non-equilibrium effects, reaction rates
Mathematics Subject Classification: 76A99, 82B40
PACS: 05.20.Dd, 05.60 +w, 51.10+y, 82.20. Mj

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V.V. Aristov, A.A. Frolova, S.A. Zabelok

Dorodnicyn Computing Centre of Russian Academy of Sciences Vavilova str., 40, 119333, Moscow, Russia 

Abstract: Simulations of flows on the basis of kinetic equations for mixtures with chemical reactions are performed. The Nonuniform Relaxation Problems (NRP) are formulated and solved. Unified Flow Solver (UFS) is used for 1D and 2D NRP. The nonequilibrium kinetics provides results outside the traditional theory of macroscopic phenomena based on the Navier-Stokes equations. Nonequilibrium flows with different properties in relaxation zones are described. Complex processes including model anabolic and catabolic chemical reactions are also considered.

Keywords: Boltzmann equation, kinetic chemical reactions, the nonuniform relaxation problem
Mathematics Subject Classification: 76P05, 82C40
PACS: 05.20.Dd, 47.45.Ab, 51.10.+y 

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Lokendra K. Balyan^a,1, Subir Singh Lamba^b

^a,bDepartment of Mathematics, IIIT-DM Jabalpur, India
¹Corresponding author: E-mail: lokendra.balyan@gmail.com

Luisa D’Amore², Valeria Mele^a and Almerico Murli^b
aUniversity of Naples Federico II, Naples, ITALY,
^bSouthern Partnership for Advanced Computational Infrastructures (SPACI), c/o
University of Naples, Federico II, Naples
and CMCC - Centro Euro-Mediterraneo per i Cambiamenti Climatici, Lecce, ITALY
Received 29 October, 2012; accepted in revised form 24 April, 2013

Abstract: A detailed rounding errors analysis for the computation of Taylor expansion
coefficients of an analytic real function with respect to one variable, as implemented by
TADIFF, a software package written in C++ specialized for computing Taylor expansion
coefficients using Algorithmic Differentiation, is performed. The error analysis is carried
out in a finite precision arithmetic system satisfying the IEEE standard 754. Furthermore,
time and space complexity of such a computation is discussed. Experimental results aimed
to validate both the accuracy and the complexity estimates are presented.

Keywords: Algorithmic Differentiation, Taylor expansion coefficients, TADIFF
Mathematics Subject Classification: AMS: 65G50; 65Y20; 68Q25

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L. D’Amore¹ , R. Arcucci² , L. Marcellino³ and A. Murli²
1. Department of Mathematics and Application, University of Naples Federico II, Italy.
2. Centro Euro-Mediterraneo per i Cambiamenti Climatici (CMCC), Italy.
3. Department of Applied Sciences, University of Naples Parthenope, Italy.
Received 30 January, 2012; accepted in revised form 22 December, 2012
Abstract: The most significant features of Data Assimilation (DA) are that both the models
and the observations are very large and non-linear (of order at least O(108 )). Further, DA
is an ill-posed inverse problem. Such properties make the numerical solution of DA very
difficult so that, as stated in [19], ”solving this problem in ”real-time” it is not always pos-
sible and many different approximations to the basic assimilation schemes are employed”.
Thus, the exploitation of advanced computing environments is mandatory, reducing the
computational cost to a suitable turnaround time. This activity should be done according
to a co-design methodology where software requirements drive hardware design decisions
and hardware design constraints motivate changes in the software design to better fit within
those constraints.
In this paper, we address high performance computation issues of the three dimensional
DA scheme underlying the oceanographic 3D-VAR assimilation scheme, named Ocean-
VAR, developed at CMCC (Centro Euro Mediterraneo per i Cambiamenti Climatici), in
Italy. The aim is to develop a parallel software architecture which is able to effectively
take advantage of the available high performance computing resources.
c 2012 European Society of Computational Methods in Sciences, Engineering and Technology

keywords: Data Assimilation, Inverse Problem, Parallel Software, Oceanography
MSC: 65Y05, 65F22, 65Z05
PACS: 02.30.Zz

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Devendra Kumar, M.K. Kadalbajoo
Received 5 August, 2011; accepted in revised form 22 July, 2012
Abstract: This paper is devoted to the numerical study for a class of boundary value
problems of second-order differential equations in which the highest order derivative is
multiplied by a small parameter ǫ and both the convection and reaction terms are with
negative shift. To obtain the parameter-uniform convergence, a piecewise uniform mesh
(Shishkin mesh) is constructed, which is dense in the boundary layer region and coarse in
the outer region. Parameter-uniform convergence analysis of the method has been given.
The method is shown to have almost second-order parameter-uniform convergence. The
effect of small delay δ on the boundary layer has also been discussed. To demonstrate the
performance of the proposed scheme several examples having boundary layers have been
carried out. The maximum absolute errors are presented in the tables.

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