Post-processing with computable error bounds for the finite element approximation of a nonlinear heat conduction problem

Email: ain{at}mcs.le.ac.uk Email: D.Kelly{at}unsw.edu.au^* Email: I.Sloan{at}unsw.edu.au^** Email: swang{at}cs.curtin.edu.au It is shown how the finite element approximation of a nonlinearheat conduction problem may be post-processed to yield enhancedapproximations to the solution and the flux at any point inthe domain. Sharp computable bounds on the accuracy of the post-processedapproximations are derived. A criterion is identified for guidingadaptive refinements of the finite element discretization. Anumerical example is given illustrating the theoretical results.     

A two-time-scale model for hospital patient flow

^** Email: m.ramakrishnan{at}ms.unimelb.edu.au^*** Email: david.sier{at}csiro.au^**** Email: p.taylor{at}ms.unimelb.edu.au Hospitals are currently facing high incidences of ambulancebypass, long waiting lists for elective procedures and a significantproportion of patients experiencing long stays in the emergencydepartment before moving to a bed in a ward. We observe thatthe system of wards and the emergency department in essenceoperates on two time-scales: the wards operate on a time-scaleof days, whereas the state of the emergency department evolveson a much faster time-scale. This observation has motivatedus to construct a two-time-scale model for a hospital system.The faster time-scale is modelled by a continuous time Markovchain, whereas the slower time-scale (ward occupancy) is modelledby a discrete time Markov chain. Given the nature of the system,the state of the wards has a direct impact on the emergencydepartment.     

The Z-transform and linear multistep stability

^** Email: mapjjc{at}maths.bath.ac.uk^*** Corresponding author. Email: ath{at}maths.bath.ac.uk^**** Email: hl{at}maths.bath.ac.uk This paper makes systematic use of control-theoretic methodssuch as the -transform, small-gain theorems and frequency-domainstability criteria in the analysis of the stability behaviourof linear multistep methods. Some of the results in Nevanlinna'swork are recovered and a number of new boundedness and asymptoticproperties of solutions of numerical schemes are obtained. Inparticular, we give a careful and detailed analysis of the nonlinearstability properties of strictly zero-stable methods.     

Combining player statistics to predict outcomes of tennis matches

^** Email: tbarnett{at}swin.edu.au^*** Email: sclarke{at}swin.edu.au With the growth in sports betting, it is possible to bet, bothbefore and during a match, on a range of outcomes in tennis.This paper shows how the standard statistics published by theATP can be combined to predict the serving statistics to beobtained when two given players meet. These statistics are thenused in a spreadsheet model to predict further match outcomes,such as the length of the match and chance of either playerwinning. These calculations can be updated as the match progresses.The method is demonstrated by focusing on a very long men'ssingles match between Roddick and El Aynaoui played at the 2003Australian Open.     

On the solvability for the mixed-type Lyapunov equation

^** Email: xsf{at}math.pku.edu.cn^*** Email: mscheng{at}math.pku.edu.cn In this paper, the linear matrix equation X = AXB^* + BXA^* +Q is considered, which is called the mixed-type Lyapunov equation.Some necessary and sufficient conditions for the existence ofa unique solution are presented. Since a Hermitian positivesemidefinite solution is important from the application pointof view, some sufficient conditions for the existence of a Hermitianpositive semidefinite solution are derived.     

Variable neighbourhood search for redundancy allocation problems

^** Email: ycliang{at}saturn.yzu.edu.tw^*** Email: s929512{at}mail.yzu.edu.tw^**** Email: s927522{at}mail.yzu.edu.tw A variable neighbourhood search (VNS) algorithm has been developedto solve the redundancy allocation problem (RAP). The VNS methodis perfectly suited to those combinatorial problems with potentialneighbourhood structures, as in the case of the RAP. The moststudied configuration of the RAP is a series system of s-independentk-out-of-n:G subsystems the so-called series–parallelsystem. The RAP is to select the optimal combination and redundancylevels of components to meet system-level constraints. Two typesof objectives are considered in this study—system reliabilitymaximization and system cost minimization. The VNS algorithmis tested on sets of benchmark problems and compared to thebest heuristics in the literature such as tabu search, multipleweighted objective heuristic, ant colony optimization and geneticalgorithm. Computational results show the advantages and benefitsof VNS for solving both types of RAP while considering bothsolution quality and computational efficiency.     

Transformations between isospectral membranes yield conformal maps

^** Email: H.Gottlieb{at}griffith.edu.au In two dimensions, contrary to the known situation that a conformaltransformation yields isospectral density functions for vibratingmembranes, it is shown that a coordinate transformation leadingto isospectral densities must be conformal.     

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 the numerical solution of involutive ordinary differential systems: enhanced linear algebra

^** Email: jukka.tuomela{at}joensuu.fi^*** Corresponding author. Email: arponen{at}maths.warwick.ac.uk^**** Email: villesamuli.normi{at}joensuu.fi We analyse some Runge–Kutta type methods for computing1D integral manifolds, i.e. solutions to ordin-ary differentialequations and differential-algebraic equations. We show thatwe can compute the solutions which respect all the constraintsof the problem reliably and reasonably quickly. Moreover, weshow that the so-called impasse points are regular points inour approach and hence require no special attention.     

The Helmholtz equation in a locally perturbed half-plane with passive boundary

^** Email: mduran{at}ing.puc.cl^*** Email: ignacio.muga{at}ucv.cl^**** Email: nedelec{at}cmapx.polytechnique.fr In this article, we study the existence and uniqueness of outgoingsolutions for the Helmholtz equation in locally perturbed half-planeswith passive boundary. We establish an explicit outgoing radiationcondition which is somewhat different from the usual Sommerfeld'sone due to the appearance of surface waves. We work with thehelp of Fourier analysis and a half-plane Green's function framework.This is an extended and detailed version of the previous articleDurán et al. (2005, The Helmholtz equation with impedancein a half-plane. C. R. Acad. Sci. Paris, Ser. I, 340, 483–488).     

The group and Riesz basis properties of string systems with time delay and exact controllability with boundary control

^** Email: gqxu{at}tju.edu.cn^*** Email: Jiajg{at}mailst.xjtu.edu.cn The group property of a string system with time delay in boundaryand the Riesz basis property of eigenvectors of the system arediscussed in the present paper. It is proved that, when thefeedback with delay > 0, the system also associates a C₀group, and its eigenvectors (generalized eigenvectors) forma Riesz basis in Hilbert space . This result shows that timedelay may destroy the stability of the system, but the groupand Riesz basis properties are kept. As a consequence, the exactcontrollability of the system with boundary control is given.     

Newton's method on Riemannian manifolds: Smale's point estimate theory under the {gamma}-condition

^** Email: cli{at}zju.edu.cn^*** Email: wjh{at}zjut.edu.cn The -conditions for vector fields on Riemannian manifolds areintroduced. The -theory and the -theory for Newton's methodon Riemannian manifolds are established under the -conditions.Applications to analytic vector fields are provided and theresults due to Dedieu et al. (2003, IMA J. Numer. Anal., 23,395–419) are improved.     

An a posteriori error indicator for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations

^** Email: paul.houston{at}nottingham.ac.uk^*** Corresponding author. Email: ilaria.perugia{at}unipv.it^**** Email: schoetzau{at}math.ubc.ca We introduce a residual-based a posteriori error indicator fordiscontinuous Galerkin discretizations of H(curl; )-ellipticboundary value problems that arise in eddy current models. Weshow that the indicator is both reliable and efficient withrespect to the approximation error measured in terms of a naturalenergy norm. We validate the performance of the indicator withinan adaptive mesh refinement procedure and show its asymptoticexactness for a range of test problems.     

Supraconvergence of a finite difference scheme for solutions in Hs(0, L)

^** Email: silvia{at}mat.uc.pt^*** Email: ferreira{at}mat.uc.pt^**** Email: grigo{at}math.tu-berlin.de In this paper we study the convergence of a centred finite differencescheme on a non-uniform mesh for a 1D elliptic problem subjectto general boundary conditions. On a non-uniform mesh, the schemeis, in general, only first-order consistent. Nevertheless, weprove for s (1/2, 2] order O(h^s)-convergence of solution andgradient if the exact solution is in the Sobolev space H^1+s(0,L), i.e. the so-called supraconvergence of the method. It isshown that the scheme is equivalent to a fully discrete linearfinite-element method and the obtained convergence order isthen a superconvergence result for the gradient. Numerical examplesillustrate the performance of the method and support the convergenceresult.     

A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension

^** Email: eymard{at}math.univ-mlv.fr^*** Email: gallouet{at}cmi.univ-mrs.fr^**** Corresponding author. Email: herbin{at}cmi.univ-mrs.fr Finite-volume methods for problems involving second-order operatorswith full diffusion matrix can be used thanks to the definitionof a discrete gradient for piecewise constant functions on unstructuredmeshes satisfying an orthogonality condition. This discretegradient is shown to satisfy a strong convergence property forthe interpolation of regular functions, and a weak one for functionsbounded in a discrete H¹-norm. To highlight the importance ofboth properties, the convergence of the finite-volume schemefor a homogeneous Dirichlet problem with full diffusion matrixis proven, and an error estimate is provided. Numerical testsshow the actual accuracy of the method.     

Determining the optimal constrained multi-item (Q, r) inventory policy by maximising risk-adjusted profit

^** Email: john.betts{at}infotech.monash.edu.au^*** Email: robertj{at}unimelb.edu.au The predominant approach to determining replenishment batchsizes for capital constrained multi-item inventories is to assumethat at some point in time the replenishment of all items willcoincide, and that batch sizes are small enough that the constraintis not violated when this event occurs. However, when an inventoryconsists of a large number of independently replenished components,the probability that all replenishments coincide is very small.The standard approach thus results in unnecessarily conservativebatch sizes that under-utilise the available resource, resultingin lower profit than would be the case if a small risk of violatingthe constraint was tolerated. In this paper, a new approachto determining constrained batch sizes is presented where, fora certain average investment, the probability of exceeding abinding, or fixed, constraint on capital is determined. Thisprobability is used to define an adjustment factor to be appliedto expressions for company profit so that an optimal trade-offbetween maximising profit and reducing risk of failure is obtainedsimply by optimising this adjusted profit. By optimising profitadjusted for the risk of exceeding the constraint, the new modelyields batch sizes that are larger, and result in greater profitabilitythan those recommended under traditional models.     

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.     

Solving the continuous space p-centre problem: planning application issues

^** Email: wei.97{at}osu.edu^*** Email: murray.308{at}osu.edu^**** Email: xiao.37{at}osu.edu The Voronoi diagram heuristic has been proposed for solvingthe p-centre problem in continuous space. However, importantassumptions underlie this heuristic and may be problematic forpractical applications. These simplifying assumptions includeuniformly distributed demand, representing a region as a rectangle;analysis of a simple Voronoi polygon in solving associated one-centreproblems and no restrictions on potential facility locations.In this paper, we explore the complexity of solving the continuousspace p-centre problem in location planning. Considering theissue of solution space feasibility, we present a spatiallyrestricted version of this problem and propose methods for solvingit heuristically. Theoretical and empirical results are provided.     

Radially isospectral annular membranes

^** Email: h.gottlieb{at}griffith.edu.au By utilizing transformations of both the radial coordinate andthe radial wave function, densities of annular membranes whichare radially isospectral to any given radial density are produced.In particular, new families of annular membrane densities arefound which are radially isospectral to a uniform membrane.Some new generalizations for completely isospectral annularmembranes (including angle-dependent densities) are also given.     

Improving the accuracy of the mini-element approximation to Navier-Stokes equations

^** Email: blanca{at}imati.cnr.it^*** Email: frutos{at}mac.cie.uva.es^**** Corresponding author. Email: julia.novo{at}uam.es A technique to improve the accuracy of the mini-element approximationto incompressible the Navier–Stokes equations is introduced.Once the mini-element approximation has been computed at a fixedtime, the linear part of this approximation is postprocessedby solving a discrete Stokes problem. The bubble functions neededto stabilize the approximation to the Navier–Stokes equationsare not used at the postprocessing step. This postprocessingprocedure allows us to increase by one unit (up to a logarithmicterm) the H¹ norm rate of convergence of the velocity and correspondinglythe L² norm of the pressure. An error analysis of the algorithmis performed.