Monte Carlo method.docx

Monte Carlo method.docx

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MonteCarlomethod

MonteCarlomethod

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Computationalphysics

Numericalanalysis ·Simulation

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MonteCarlomethods（orMonteCarloexperiments）areaclassofcomputationalalgorithmsthatrelyonrepeatedrandomsamplingtocomputetheirresults.MonteCarlomethodsareoftenusedincomputersimulationsofphysicalandmathematicalsystems.Thesemethodsaremostsuitedtocalculationbyacomputerandtendtobeusedwhenitisinfeasibletocomputeanexactresultwithadeterministicalgorithm.[1]Thismethodisalsousedtocomplementtheoreticalderivations.

MonteCarlomethodsareespeciallyusefulforsimulatingsystemswithmanycoupleddegreesoffreedom,suchasfluids,disorderedmaterials,stronglycoupledsolids,andcellularstructures（seecellularPottsmodel）.Theyareusedtomodelphenomenawithsignificantuncertaintyininputs,suchasthecalculationofriskinbusiness.Theyarewidelyusedinmathematics,forexampletoevaluatemultidimensionaldefiniteintegralswithcomplicatedboundaryconditions.WhenMonteCarlosimulationshavebeenappliedinspaceexplorationandoilexploration,theirpredictionsoffailures,costoverrunsandscheduleoverrunsareroutinelybetterthanhumanintuitionoralternative"soft"methods.[2]

TheMonteCarlomethodwascoinedinthe1940sbyJohnvonNeumann,StanislawUlamandNicholasMetropolis,whiletheywereworkingonnuclearweaponprojects（ManhattanProject）intheLosAlamosNationalLaboratory.ItwasnamedaftertheMonteCarloCasino,afamouscasinowhereUlam'suncleoftengambledawayhismoney.[3]

Contents

∙1Introduction

∙2History

∙3Definitions

o3.1MonteCarloandrandomnumbers

o3.2MonteCarlosimulationversus"whatif"scenarios

∙4Applications

o4.1Physicalsciences

o4.2Engineering

o4.3Computationalbiology

o4.4ComputerGraphics

o4.5Appliedstatistics

o4.6Games

o4.7Designandvisuals

o4.8Financeandbusiness

o4.9Telecommunications

∙5Useinmathematics

o5.1Integration

o5.2Simulation-Optimization

o5.3Inverseproblems

o5.4Computationalmathematics

∙6Seealso

∙7Notes

∙8References

∙9Externallinks

Introduction

MonteCarlomethodappliedtoapproximatingthevalueofπ.Afterplacing30000randompoints,theestimateforπiswithin0.07%oftheactualvalue.Thishappenswithanapproximateprobabilityof20%.

MonteCarlomethodsvary,buttendtofollowaparticularpattern:

1.Defineadomainofpossibleinputs.

2.Generateinputsrandomlyfromaprobabilitydistributionoverthedomain.

3.Performadeterministiccomputationontheinputs.

4.Aggregatetheresults.

Forexample,consideracircleinscribedinaunitsquare.Giventhatthecircleandthesquarehavearatioofareasthatisπ/4,thevalueofπcanbeapproximatedusingaMonteCarlomethod:

[4]

1.Drawasquareontheground,theninscribeacirclewithinit.

2.Uniformlyscattersomeobjectsofuniformsize（grainsofriceorsand）overthesquare.

3.Countthenumberofobjectsinsidethecircleandthetotalnumberofobjects.

4.Theratioofthetwocountsisanestimateoftheratioofthetwoareas,whichisπ/4.Multiplytheresultby4toestimateπ.

Inthisprocedurethedomainofinputsisthesquarethatcircumscribesourcircle.Wegeneraterandominputsbyscatteringgrainsoverthesquarethenperformacomputationoneachinput（testwhetheritfallswithinthecircle）.Finally,weaggregatetheresultstoobtainourfinalresult,theapproximationofπ.

Ifgrainsarepurposefullydroppedintoonlythecenterofthecircle,theyarenotuniformlydistributed,soourapproximationispoor.Second,thereshouldbealargenumberofinputs.Theapproximationisgenerallypoorifonlyafewgrainsarerandomlydroppedintothewholesquare.Onaverage,theapproximationimprovesasmoregrainsaredropped.

History

BeforetheMonteCarlomethodwasdeveloped,simulationstestedapreviouslyunderstooddeterministicproblemandstatisticalsamplingwasusedtoestimateuncertaintiesinthesimulations.MonteCarlosimulationsinvertthisapproach,solvingdeterministicproblemsusingaprobabilisticanalog（seeSimulatedannealing）.

AnearlyvariantoftheMonteCarlomethodcanbeseenintheBuffon'sneedleexperiment,inwhichπcanbeestimatedbydroppingneedlesonafloormadeofparallelstripsofwood.Inthe1930s,EnricoFermifirstexperimentedwiththeMonteCarlomethodwhilestudyingneutrondiffusion,butdidnotpublishanythingonit.[3]

In1946,physicistsatLosAlamosScientificLaboratorywereinvestigatingradiationshieldingandthedistancethatneutronswouldlikelytravelthroughvariousmaterials.Despitehavingmostofthenecessarydata,suchastheaveragedistanceaneutronwouldtravelinasubstancebeforeitcollidedwithanatomicnucleus,andhowmuchenergytheneutronwaslikelytogiveofffollowingacollision,theLosAlamosphysicistswereunabletosolvetheproblemusingconventional,deterministicmathematicalmethods.StanisławUlamhadtheideaofusingrandomexperiments.Herecountshisinspirationasfollows:

ThefirstthoughtsandattemptsImadetopractice[theMonteCarloMethod]weresuggestedbyaquestionwhichoccurredtomein1946asIwasconvalescingfromanillnessandplayingsolitaires.ThequestionwaswhatarethechancesthataCanfieldsolitairelaidoutwith52cardswillcomeoutsuccessfully?

Afterspendingalotoftimetryingtoestimatethembypurecombinatorialcalculations,Iwonderedwhetheramorepracticalmethodthan"abstractthinking"mightnotbetolayitoutsayonehundredtimesandsimplyobserveandcountthenumberofsuccessfulplays.Thiswasalreadypossibletoenvisagewiththebeginningoftheneweraoffastcomputers,andIimmediatelythoughtofproblemsofneutrondiffusionandotherquestionsofmathematicalphysics,andmoregenerallyhowtochangeprocessesdescribedbycertaindifferentialequationsintoanequivalentforminterpretableasasuccessionofrandomoperations.Later[in1946],IdescribedtheideatoJohnvonNeumann,andwebegantoplanactualcalculations.

–StanisławUlam[5]

Beingsecret,theworkofvonNeumannandUlamrequiredacodename.VonNeumannchosethenameMonteCarlo.ThenamereferstotheMonteCarloCasinoinMonacowhereUlam'sunclewouldborrowmoneytogamble.[1][6][7]Usinglistsof"truly"randomrandomnumberswasextremelyslow,butvonNeumanndevelopedawaytocalculatepseudorandomnumbers,usingthemiddle-squaremethod.Thoughthismethodhasbeencriticizedascrude,vonNeumannwasawareofthis:

hejustifieditasbeingfasterthananyothermethodathisdisposal,andalsonotedthatwhenitwentawryitdidsoobviously,unlikemethodsthatcouldbesubtlyincorrect.

MonteCarlomethodswerecentraltothesimulationsrequiredfortheManhattanProject,thoughseverelylimitedbythecomputationaltoolsatthetime.Inthe1950stheywereusedatLosAlamosforearlyworkrelatingtothedevelopmentofthehydrogenbomb,andbecamepopularizedinthefieldsofphysics,physicalchemistry,andoperationsresearch.TheRandCorporationandtheU.S.AirForceweretwoofthemajororganizationsresponsibleforfundinganddisseminatinginformationonMonteCarlomethodsduringthistime,andtheybegantofindawideapplicationinmanydifferentfields.

UsesofMonteCarlomethodsrequirelargeamountsofrandomnumbers,anditwastheirusethatspurredthedevelopmentofpseudorandomnumbergenerators,whichwerefarquickertousethanthetablesofrandomnumbersthathadbeenpreviouslyusedforstatisticalsampling.

Definitions

ThereisnoconsensusonhowMonteCarloshouldbedefined.Forexample,Ripley[8]definesmostprobabilisticmodelingasstochasticsimulation,withMonteCarlobeingreservedforMonteCarlointegrationandMonteCarlostatisticaltests.Sawilowsky[9]distinguishesbetweenasimulation,aMonteCarlomethod,andaMonteCarlosimulation:

asimulationisafictitiousrepresentationofreality,aMonteCarlomethodisatechniquethatcanbeusedtosolveamathematicalorstatisticalproblem,andaMonteCarlosimulationusesrepeatedsamplingtodeterminethepropertiesofsomephenomenon（orbehavior）.Examples:

∙Simulation:

Drawingonepseudo-randomuniformvariablefromtheinterval（0,1]canbeusedtosimulatethetossingofacoin:

Ifthevalueislessthanorequalto0.50designatetheoutcomeasheads,butifthevalueisgreaterthan0.50designatetheoutcomeastails.Thisisasimulation,butnotaMonteCarlosimulation.

∙MonteCarlomethod:

Theareaofanirregularfigureinscribedinaunitsquarecanbedeterminedbythrowingdartsatthesquareandcomputingtheratioofhitswithintheirregularfiguretothetotalnumberofdartsthrown.ThisisaMonteCarlomethodofdeterminingarea,butnotasimulation.

∙MonteCarlosimulation:

Drawingalargenumberofpseudo-randomuniformvariablesfromtheinterval（0,1],andassigningvalueslessthanorequalto0.50asheadsandgreaterthan0.50astails,isaMonteCarlosimulationofthebehaviorofrepeatedlytossingacoin.

KalosandWhitlock[4]pointoutthatsuchdistinctionsarenotalwayseasytomaintain.Forexample,theemissionofradiationfromatomsisanaturalstochasticprocess.Itcanbesimulateddirectly,oritsaveragebehaviorcanbedescribedbystochasticequationsthatcanthemselvesbesolvedusingMonteCarlomethods."Indeed,thesamecomputercodecanbeviewedsimultaneouslyasa'naturalsimulation'ora