The cyclic Barzilai--Borwein method for unconstrained optimization

^** Email: dyh{at}lsec.cc.ac.cn^*** Email: hager{at}math.ufl.edu^**** Email: klaus.schittkowski{at}uni-bayreuth.de^***** Email: hzhang{at}math.ufl.edu In the cyclic Barzilai–Borwein (CBB) method, the sameBarzilai–Borwein (BB) stepsize is reused for m consecutiveiterations. It is proved that CBB is locally linearly convergentat a local minimizer with positive definite Hessian. Numericalevidence indicates that when m > n/2 3, where n is the problemdimension, CBB is locally superlinearly convergent. In the specialcase m = 3 and n = 2, it is proved that the convergence rateis no better than linear, in general. An implementation of theCBB method, called adaptive cyclic Barzilai–Borwein (ACBB),combines a non-monotone line search and an adaptive choice forthe cycle length m. In numerical experiments using the CUTErtest problem library, ACBB performs better than the existingBB gradient algorithm, while it is competitive with the well-knownPRP+ conjugate gradient algorithm.     

On computing eigenvalues of second-order linear pencils

^** Email: mhannaby{at}yahoo.com^*** Email: zahraa26{at}yahoo.com In this paper, we use sinc techniques to compute the eigenvaluesof a second-order operator pencil of the form Q – P approximately.Here Q and P are self-adjoint differential operators of thesecond and first order, respectively. Also the eigenparameterappears in the boundary conditions linearly.     

A stabilized finite-element method for the Stokes problem including element and edge residuals

^** Email: raraya{at}ing-mat.udec.cl^*** Corresponding author. Email: gbarrene{at}ing-mat.udec.cl^**** Email: valentin{at}lncc.br A new stabilized finite-element method is presented for theStokes problem. The method is of a Douglas–Wang type,and includes a positive jump term controlling the residual ofthe Cauchy stress tensor on the internal edges of the triangulation.A priori error estimates are obtained in the natural norms ofthe unknowns and an a posteriori error estimator is proposed,analysed and tested through numerical experiments.     

Finite element approximation of the Neumann eigenvalue problem in domains with multiple cracks

^** Email: belhach{at}poncelet.univ-metz.fr^*** Email: bucur{at}math.univ-metz.fr^**** Email: jmse{at}math.univ-metz.fr We study the Neumann–Laplacian eigenvalue problem in domainswith multiple cracks. We derive a mixed variational formulationwhich holds on the whole geometric domain (including the cracks)and implements efficient finite-element discretizations forthe computation of eigenvalues. Optimal error estimates aregiven and several numerical examples are presented, confirmingthe efficiency of the method. As applications, we numericallyinvestigate the behaviour of the low eigenvalues in domainswith a large number of cracks.     

A coupled multivalued model for ice streams and its numerical simulation

^** Email: nati{at}dma.uvigo.es^*** Email: durany{at}dma.uvigo.es^**** Email: anaisabel.munoz{at}urjc.es^***** Email: emanuele.schiavi{at}urjc.es^****** Email: carlosv{at}udc.es This paper deals with the numerical solution of a non-linearmodel describing a free-boundary problem arising in modern glaciology.Considering a shallow, viscous ice sheet flow along a soft,deformable bed, a coupled non-linear system of differentialequations can be obtained. Particularly, an obstacle problemis then deduced and solved in the framework of its complementarityformulation. We present the numerical solution of the resultingmultivalued system modelling the ice sheet non-Newtonian dynamicsdriven by the underlying drainage system. Our numerical resultsshow the existence of fast ice streams when positive wave-likeinitial conditions are considered. The solutions are numericallycomputed with a decoupling iterative method and finite-elementtechnique. A duality algorithm and a projected Gauss–Seidelmethod are the alternatives used to cope with the resultingvariational inequality while the explicit treatment, Newtonmethod or a duality method are proposed to deal with the non-linearsource term. Finally, the numerical solutions are physicallyinterpreted and some comparisons among the numerical methodsare then discussed.     

Error estimates for stochastic differential games: the adverse stopping case

^** Email: frederic.bonnans{at}inria.fr^*** Email: stefania.maroso{at}inria.fr^**** Email: zidani{at}ensta.fr We obtain error bounds for monotone approximation schemes ofa particular Isaacs equation. This is an extension of the theoryfor estimating errors for the Hamilton–Jacobi–Bellmanequation. To obtain the upper error bound, we consider the ‘Krylovregularization’ of the Isaacs equation to build an approximatesub-solution of the scheme. To get the lower error bound, weextend the method of Barles & Jakobsen (2005, SIAM J. Numer.Anal.) which consists in introducing a switching system whosesolutions are local super-solutions of the Isaacs equation.     

Electrical coupling in proton exchange membrane fuel cell stacks: mathematical and computational modelling

^** Corresponding author. Email: wetton{at}math.ubc.ca^*** Email: Peter.Berg{at}uoit.ca^**** Email: caglara{at}uwgb.edu^***** Email: kpromisl{at}math.msu.edu^****** Email: jean.st-pierre{at}ballard.com A mathematical model describing the effects of electrical couplingof proton exchange membrane unit fuel cells through shared bipolarplates is developed. Here, the unit cells are described by simple,steady-state, 1D models appropriate for straight reactant gaschannel designs. A linear asymptotic version of the model isused to give analytic insight into the effect of the coupling,including estimates of the extent of the coupling in terms ofthe number of adjacent cells affected. An efficient numericalmethod is developed to solve the non-linear coupled system.Numerical results showing the effects on stack voltage due toa single cell with anomalous oxidant flow rate are given. Theeffects on stack performance due to end plate effects are alsogiven. It is shown that electrical coupling has a significanteffect on fuel cell performance.     

On a Monte Carlo method for neutron transport criticality computations

^** Email: maire{at}univ-tln.fr^*** Email: denis.talay{at}sophia.inria.fr We give a stochastic representation of the principal eigenvalueof some homogeneous neutron transport operators. Our constructionis based upon the Feynman–Kac formula for integral transportequations, and uses probabilistic techniques only. We developa Monte Carlo method for criticality computations. We numericallytest this method on various homogeneous and inhomogeneous problems,and compare our results with those obtained by standard methods.     

Bayesian networks applied to credit scoring

Email: kchang{at}gmu.eduEmail: RobertFung{at}Fairlsaac.comEmail: alan.lucas{at}hotmail.com^¶Email: BobOliver{at}Fairlsaac.com^||Email: NShikaloff{at}Fairlsaac.com The objectives of this paper are to apply the theory and numericalalgorithms of Bayesian networks to risk scoring, and comparethe results with traditional methods for computing scores andposterior predictions of performance variables. Model identification,inference, and prediction of random variables using Bayesiannetworks have been successfully applied in a number of areas,including medical diagnosis, equipment failure, informationretrieval, rare-event prediction, and pattern recognition. Theability to graphically represent conditional dependencies andindependencies among random variables may also be useful incredit scoring. Although several papers have already appearedin the literature which use graphical models for model identification,as far as we know there have been no explicit experimental resultsthat compare a traditionally computed risk score with predictionsbased on Bayesian learning algorithms. In this paper, we examine a database of credit-card applicantsand attempt to ‘learn’ the graphical structure ofthe characteristics or variables that make up the database.We identify representative Bayesian networks in a developmentsample as well as the associated Markov blankets and cliquestructures within the Markov blanket. Once we obtain the structureof the underlying conditional independencies, we are able toestimate the probabilities of each node conditional on its directpredecessor node(s). We then calculate the posterior probabilitiesand scores of a performance variable for the development sample.Finally, we calculate the receiver operating characteristic(ROC) curves and relative profitability of scorecards basedon these identifications. The results of the different modelsand methods are compared with both development and validationsamples. Finally, we report on a statistical entropy calculationthat measures the degree to which cliques identified in theBayesian network are independent of one another.     

Solving parallel machines scheduling problems with sequence-dependent setup times using variable neighbourhood search

^** Corresponding author. Email: mahdi{at}dcc.ufmg.br^*** Email: martin{at}dcc.ufmg.br^**** Email: mateus{at}dcc.ufmg.br^***** Email: pardalos{at}ufl.edu Variable neighbourhood search (VNS) is a modern metaheuristicbased on systematic changes of the neighbourhood structure withina search to solve optimization problems. The aim of this paperis to propose and analyse a VNS algorithm to solve schedulingproblems with parallel machines and sequence-dependent setuptimes, which are of great importance on the industrial context.Three versions of a greedy randomized adaptive search procedurealgorithm are used to compare with the proposed VNS algorithmto highlight its advantages in terms of generality, qualityand speed for large instances.