A unified framework for the numerical solution of general quadratic matrix equations

^** Email: vassilios.tsachouridis{at}ieee.org^*** Email: N.karcanias{at}city.ac.uk^**** Email: ixp{at}le.ac.uk Algebraic quadratic equations are special cases of a singlegeneralized algebraic quadratic matrix equation (GQME). Thispaper focuses on the numerical solution of the GQME using probability-1homotopy methods. A synoptic review of these methods and theirapplication to algebraic matrix equations is provided as background.A large variety of analysis and design problems in systems andcontrol are reported as special cases of the presented frameworkand some of them are illustrated via numerical examples fromthe literature.     

Explicit stability conditions for time-discrete vector Lur'e-type systems

^** Email: gilmi{at}cs.bgu.ac.il^*** Email: rmedina{at}ulagos.cl Non-linear discrete-time Lur'e-type systems are considered.Explicit stability conditions are suggested. They are formulatedin terms of the determinants of characteristic polynomials.Moreover, an estimate for the region of attraction of the zerosolution is derived.     

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.     

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.     

Boundary feedback stabilization of coupled sine-Gordon equations

^** Email: koba{at}cntl.kyutech.ac.jp^*** Email: sakamoto{at}cntl.kyutech.ac.jp This paper is concerned with global stabilization of the systemgoverned by coupled sine-Gordon equations without damping. Astabilizer is constructed by boundary velocity feedback. Theclosed-loop system is shown to be well posed by the non-linearsemigroup approach. Moreover, using a multiplier method, globalexponential stabilization of the closed-loop system is proved.     

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.     

A posteriori error estimates of discontinuous Galerkin methods for non-standard Volterra integro-differential equations

^** Email: jingtang{at}lsec.cc.ac.cn^*** Email: hermann{at}math.mun.ca In this paper we establish a posteriori error estimates forthe discontinuous Galerkin (DG) method applied to linear, semilinearand non-standard (non-linear) Volterra integro-differentialequations. We also present an analysis of the DG method withquadrature for the memory term. Numerical experiments basedon three integro-differential equations are used to illustratevarious aspects of the error analysis.     

Optimal allocation of reliability tasks to mitigate faults during system development

^** Email: walter.johnston{at}baesystems.com^*** Email: j.quigley{at}strath.ac.uk^**** Email: lesley.walls{at}strath.ac.uk This paper considers reliability planning for a concept designfor a new system where a portfolio of possible reliability developmenttasks exists; the goal is to find a selection and sequence oftasks to achieve reliability targets subject to time constraintsat minimal cost. This is non-trivial given that each task potentiallycan expose several different weaknesses and each weakness potentiallycan be exposed by several different tasks. We use a Bayesianpoint process model to estimate the system reliability. Theprior distribution maps to a fault register and relates directlyto a set of potential engineering modifications.The likely impactof each task can be assessed using the point process model.An integer programming approach is used to sequence and scheduletasks under the constraint that contractual reliability requirementsmust be met. An illustrative example is provided and an extensionto system availability is proposed.     

On a modified version of ILDM approach: asymptotic analysis based on integral manifolds

^** Email: vbykov{at}cs.bgu.ac.il^*** Email: goldfarb{at}cs.bgu.ac.il^**** Email: vladimir{at}bgumail.bgu.ac.il^***** Email: umaas{at}itt.mach.uni-karlsruhe.de Using the method of integral (invariant) manifolds, the intrinsiclow-dimensional manifolds (ILDM) method is analysed. This isa method for identifying invariant manifolds of a system's slowdynamics and has proven to be an efficient tool in modellingof laminar and turbulent combustion. It allows treating multi-scalesystems by revealing their hidden hierarchy and decomposingthe system dynamics into fast and slow motions. The performedanalysis shows that the original ILDM technique can be interpretedas one of the many possible realizations of the general framework,which is based on a special transformation of the original coordinatesin the state space. A modification of the ILDM is proposed basedon a new definition of the transformation matrix. The proposednumerical procedure is demonstrated on linear examples and highlynon-linear test problems of mathematical theory of combustionand demonstrates in some cases better performance with respectto the existing one.     

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.     

An explicit expression for the surface-impedance tensor of a compressible monoclinic material in a state of plane strain

^** Email: y.fu{at}keele.ac.uk^*** Email: d.w.brookes{at}keele.ac.uk We present an explicit expression for the surface-impedancetensor associated with a compressible monoclinic elastic materialin a state of plane strain. Among a wide range of applications,such an explicit expression can, for instance, be used to writedown an explicit secular equation for the surface or interfacialwave speed, and an explicit bifurcation condition for surfaceor interfacial wrinkling. A previously known quartic secularequation for the surface-wave speed is recovered as a by-product.     

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.     

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.     

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.     

Robust constrained linear regulator problem

^** Email: mesquine{at}ucam.ac.ma^*** Email: a.benlamkadem{at}ucam.ac.ma The robust constrained state and control regulator problem isconsidered. Necessary and sufficient conditions of positiveinvariance are established. A linear programming approach ispresented in order to construct, for an uncertain constrainedlinear system, a stabilizing linear state feedback control.The control law transfers asymptotically to the origin any initialstate belonging to a given set, while constraints on the stateand the control vectors are respected.     

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.     

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.     

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.     

Regular linear systems governed by systems with state, input and output delays

^** Email: hadd{at}ucam.ac.ma^*** Email: idrissi{at}ucam.ac.ma In this paper, we give a new reformulation of linear systemswith delays in input, state and output. We show that these systemscan be written as a regular linear system without delays. Thetechnique used here is essentially based on the theory recentlydeveloped by Salamon and Weiss and the shift in semigroup properties.Our framework can be applied, in particular, when the delayoperators are given by Riemann–Stieltjes integrals.     

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.