Optimal pruning in parametric differential equations

Initial value problems for parametric ordinary differential equations (ODEs) arise in many areas of science and engineering. Since some of the data is uncertain, traditional numerical methods do not apply. This paper considers a constraint satisfaction app

OptimalPruninginParametricDifferentialEquations

MichaJanssen

UCL

10,PlaceSainteBarbe1348Louvain-LaNeuve

PascalVanHentenryckBrownUniversity

Box1910

Providence,RI02912

Abstract

Initialvalueproblemsforparametricordinarydifferentialequations(ODEs)ariseinmanyareasofscienceandengineering.Sincesomeofthedataisuncertain,traditionalnumericalmethodsdonotapply.ThispaperconsidersaconstraintsatisfactionapproachthatenhancestraditionalintervalmethodswithapruningcomponentwhichusesarelaxationoftheODEandHermiteinterpolationpolynomials.Itsolvesthemaintheoreticalandpracticalopenissueleftinthisapproach:thechoiceofanoptimalevaluationtimefortherelaxation.Asaconsequence,theconstraintsatisfactionapproachisshowntoprovideaquadratic(asymptotical)improvementinaccuracyoverthebestintervalmethods,whileimprovingtheirrunningtimes.Experimentalresultsonstandardbenchmarkscon rmthetheoreticalresults.

YvesDeville

UCL

10,PlaceSainteBarbe1348Louvain-LaNeuve

1Introduction

Initialvalueproblems(IVPs)forordinarydifferentialequations(ODEs)arisenaturallyinmanyapplicationsinscienceandengineering,includingchemistry,physics,molecularbiology,andmechanicstonameonlyafew.Anordinarydifferentialequationisasystemoftheform

oftendenotedinvectornotationbyor.AninitialvalueproblemisanODE

.Inaddition,inpractice,itisoftenthecasethattheparametersand/orthewithaninitialcondition

initialvaluesarenotknownwithcertaintybutaregivenasintervals.Hencetraditionalmethodsdonotapplytotheresultingparametricordinarydifferentialequationssincetheywouldhavetosolvein nitelymanysystems.Intervalmethods,pioneeredbyMoore[Moo66],provideanapproachtotackleparametricODEs.Thesemethodsreturnenclosuresoftheexactsolutionatdifferentpointsintime,i.e.,theyareguaranteedtoreturnintervalscontainingtheexactsolution.Inaddition,theyaccommodateeasilyuncertaintyintheparametersorinitialvaluesbyusingintervalsinsteadof oating-pointnumbers.Intervalmethodstypicallyapplyaone-stepTaylorintervalmethodandmakeextensiveuseofautomaticdifferentiationtoobtaintheTaylorcoef cients[Eij81,Kru69,Moo66,Moo79].Theirmajorproblemhoweveristheexplosionofthesizeoftheboxesatsuccessivepointsastheyoftenaccumulateerrorsfrompointtopointandloseaccuracybyenclosingthesolutionbyabox(thisiscalledthewrappingeffect).Lohner’sAWAsystem[Loh87]wasanimportantstepinintervalmethodswhichfeaturesef cientcoordinatetransformationstotacklethewrappingeffect.Morerecently,NedialkovandJackson’sIHOmethod[NJ99]improvedonAWAbyextendingaHermite-Obreschkoff’sapproach(whichcanbeviewedasageneralizedTaylormethod)tointervals(seealso

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