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Underground water quality model inversion of genetic algorithm

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Underground water quality model inversion of genetic algorithmof,water,model,Water

    Underground water quality model inversion

    of genetic algorithm

    GlobalGeology,12(3):164167(2009)

    doi:10.3969/j.issn.1673-9736.2009.03.07

    ArticleID:16739736(2009)03016404

    Undergroundwaterqualitymodelinversion

    ofgeneticalgorithlotgeneticalgorithm

    MARuijieandLIXin

    .

    CollegeofMathematics,JilinUniversity,Changchun130026,China

    2.CollegeoJConstructionEngineering,JilinUniversity,Changchun130026,China Abstract:Theundergroundwaterqualitymodelwithnonlinearinversionproblemisill

    posed,andboilsdown

    tosolvingtheminimumofnonlinearfunction.Geneticalgorithmsareadoptedinanumberofindividualsof

    groupsbyiterativesearchtofindtheoptimalsolutionoftheproblem,theencodingstringsasitsoperationalob

    jeetive,andachievingtheiterativecalculationsbythegeneticoperators.Itisaneffectivemethodofinverse

    problemsofgroundwater,withincomparableadvantagesandpracticalsignificances. Keywords:undergroundwater;qualitymodel;inversion;geneticalgorithm;geneticoperator

    Introduction

    Groundwaterqualitymodelisbasedonthe

    groundwaterresourcesmodelandtheproblemsofthe

    groundwaterquality,Thus,itisinevitabletoinvert

    thehydrogeologicalparameter.Andthefactisthat

    differentgeohydrologicconditionmayproducesimilar orthesamewaterhead,andthegeohydrologiccondi

    tioncannotbealwaysdeterminedmerelybyobserved hydrologicalmaterials.Takethedynamicobservation ofthenaturalgroundwaterortheobservationaldataof thepumpingtestintoconsiderationandinturnunder- standtheproblemsofthegeologicalconditions,which iscalledinverseproblemorindirectproblem. Fromtheperspectiveofmath,inverseproblemis modelidentificationproblem-optimalobjectivefunc..

    tion.Inversionfromobservedvaluestotheparameters ofmediadestinedtothisinversionproblemisnon..fin.. earandillposed.

    Geneticalgorithmwasfoundedin1962byProf. HollandfromMichiganUniversity.Onthebasisofge

    Received24April,2009;accepted10June,2009 neticalgorithminthe1970sDeJongca~iedoutnH- mericalpurenumericalfunctionoptimizationcomputa

    tionalexperimentsonthecomputer,andonthebasis ofaseriesofstudies,summarizedbyGoldberginthe 1980s,thebasicframeworkofgeneticalgorithmwas formed.Itmimicsthetheoryofsurvivalofthefittest andgeneticvariation.ThroughsimulatingDarwin's principle(survivalofthefittest)incentgoodstrut

    ture;andbysimulatingMendel'stheoryofthegenetic variationintheiterativeprocesstomaintaintheexist

    ingstructure,aswellastoseekabetterstructure. GeneticAlgorithmhasaverystrongimplicitparallel- ismandglobalsearchcapability.Whendealingwith

    somelargescaleandhighlynonlinearproblem,ithas uniqueadvantage.Geneticalgorithmisakindofnu- mericalalgorithm,whichisofwideapplicationrange, highefficiencyandwiththecapabilityofglobalopti

    mizationnumericalalgorithm.Itcaneffectivelydeal withmultivariableandcomplexfunctionoptimization

     problemsolving,akindofBionicoptimizationnumer

    icalmethodforinversionwiththeadvantagesofro

    Undergroundwaterqualitymodelinversionofgeneticalgorithm165

    bustness,parallelandhighefficiency

    Mathematicmodel

    Let'ssupposethatthereareNwellsintheregion G(Zhoueta1.,2000)madebytheclosedCHileF, thedistributionasshowninFig.1,andamongthem mwellsareextractionwells,

    Fig.1Distributionofwells

    Newellsareobservationwells.m+Ne=N:

    wellcentroidcoordinatesare(,Y),WellRadiusis rci,WellPerimeterareF;To(Y)asthecenterof acircle,torcifortheradius,thiscirculardomainre

    cordedasKci(i=1,2,,N);alsoassumedthat

    theN.extractionwellshasthreewellsofgivenproduc

    tionvolumesandheadsha(a=1,2,,N)onthe

    wall;has?explorationwellsonthewallonlygiven headha(a=N+1,N+2,,?+),hereN+?,

    :m.threeobservationwellscentroidgiventhewell

    headvalueh(,Y)(i=r+1,,m+Ne),andset

    valueh(x,Y,t)totheborder,intheregionG

    giventheinitialheadh1(,Y).

    KnownleakagerechargeintensityE(Xu,2003)

    rechargeq,initialvalue0,exteriorboundary'shead hl,PartialwallboundaryheadhandflowrateQ(u =1,2,,,,),partoftheobseilationheadh.(a

    =m+1,,m+Are),toreversehydrogeologyparam

    etersofthemathematicsmode1.

    2Geneticalgorithmsteps

    Geneticalgorithmisbasedontheprincipleofbi

    ologicalevolution,aglobaloptimizationalgorithm, drawingontheBiologicalnaturalselectionandGenet

    icevolutionmechanismtodevelopaglobaloptimiza

    tionofadaptiveprobabilisticsearchalgorithm,the productofcombinationofbiologicalgenetictechnology andcomputertechnology.

    Geneticalgorithmhasthreebasicelements:eneo- dedmode,geneticoperatorsandfitnessfunction. Encodingofwhichistoconver[thesolutionsof theproblemintocodedstringstosimulatethewaythe Biologicalchromosomesgeneticcomputingforgenetic operations;operatorisaseriesofoperations,thesim

    ulationofindividuaIsurvivalenvironment;fitness functionismeasuringfunction,thecriteriaforevalua

    tionoftheviabilityoftheindividua1.Apartfromthe threefundamentalelements,thegeneticalgorithmalso involvessomeparametersofcontrollingfactors,name

    lygroupsize,crossoverprobability,mutationproba

    bility,geneticalgebraandsoon.Geneticalgorithm basicideaistoabstractthebiologicalevolution processandthendescribedasreproduction,crossover andmutationoperatorthreeoperators.Geneticalgo

    rithmcaneffectivelydealwithmultivariateandthe

solvingofcomplexfunctionrelationoptimizationprob

    lems,duetogeneticalgorithmnotrequireddifferenti

    able,parallelandwidelyapplicable.Thegeneral structureofgeneticalgorithmmodelanditsapplica

    tionsareasfollows:

    (1)Geneticencoding:Willsolvetheproblemin whicheachvariableisseenasagene,accordingto thetypeofvariablesandrange,andselecttheappro

    priatedigitbinarycodetobeencodedseparately, calledthegeneticcode,suchas:=[aI,a2,a3, (4j?

    (2)fitnessfunction:Geneticalgorithmwhenin search,substantiallywithoutusingexternalinforma

    tion,onlyappropriatevaluefunctionasthebasis, Zhu(1997)usedspeciesofeachindividual'sfitnessto carryoutthesearch.Accordingtothefunctionalrela

    tionshipbetweenproblemsolvingandgeneencoding rulesca1cI】】ataljndividua1s'thefitnessfunctionthat I:

    F=l(,Y,z)i=1,2,,n

    Intheapplicationofthemodel,theobjective functionoftheoptimizingmodelwastakenaspartof thegeneticalgorithmindividualfitnessfunction,em beddedinthesolvingprocessoffitnessfunction,real

    166MaR.J.andLiX

    izingthecouplingofgeneticalgorithmandthenumeri

    calmethod,thecouplingmodeis

    Intheformulas:Msizeofagroup,thatis,the

    nutuberofchromosomes:肋一thenumberofcalculat-

    ingpoints;Fitness(i)-iindividualfitness

Theestablishmentofspecies:biologyliveinna

    tureintheformofspeciesgroup.AspeciesP(t)has Nindividuals:P(t)=(A1,A2,,An),(Ortega&

    Rheinboldt,1970;Allgower&Georg,1983).Asa startingpointfortheinitialevolutionspeciesP(0) canbegeneratedrandomlyorbyothermeans.Gener

    allyadoptrandomlygeneratedapproachtoproduce populationscalepopsizechromosomesindividuals.

    Thatcanavoidsearchsomepointsnotnecessary,so actuallyequaltosearchformorepoints,peculiarto thegeneticalgorithmofimplicitparallelism. Reproduction:SelectindividualsfromspeciesP (t)copytospeciesP(t+1).Eachindividualrepro

    ductionprobabilityofthebreedingopportunityiscon

    trolledbythefertilityprobablityPs.ThevalueofPs dependsoneachindividual'sfitnessfunctionFi;In theprocessofcopyingwilloperatecrossoverandmu tation.(He,2003,Garcia&Gould,1980).The choiceofcrossiscontrolledbycrossoverprobability Pm.Usingthemutationprobability,theformulais: Pm(+1)=Pm(k)-E0.3Pm(1)]/k,;

    intheformula,kstandsforgeneticalgebra,kis onbehalfofthelargestgeneticalgebra,P(1)repre

    sentsthefirstgenerationmutationprobabilityandPm (k)the七一thgenerationmutationprobability.Assoon asthereproduction,crossoverandmutationofspecies P(t+1)iscompleted,speciesP(t+1)toreplace speciesP(t),whichachievesagenerationofbreed

    ing.

    Strategyselection:EvaluatethespeciesP(+

    1),testingthespeedofevolutionandconvergence degrees,todeterminewhethertheevolutionsophisti- cated.Ifimmature.thencontinuewitheachgenera

    tionreproductionandevolution,sothatthequalityof individualspecieshasbeenoptimized-Maintainthe optimalstrategy.retentionofmoptimalindividualsin thepreviousgeneration.andtherestselectpopsizem

    optimalindividualsinthenextgeneration,suchasso' phisticated,thentheendofthesolutionprocess, whenreceivedbytheindividualandthespeciesisto solvetheproblemoftheoptimalsolution(Georg, 1990:Allgower&Georg,1990).Ifmature,then endthesolvingprocessofthepopulation.Thenthe individualsandspeciesobtainedistheoptimalsolu

    tionofproblemsolving.

    3Application

    Theinitialpopulationsareselectedfromabout 1000randomlysolutionindividualsaccordingtotheir fitnessfunctionvaluesquality,anditisbettertoset thefollowingparametersthroughthecalculationval

    ue:populationsizem=100,theprobabilityofcross

    computingP=0.7.theprobabilityofmutationcalcu

    lationpc=0.033.alengthofstringF=4.

    Table1Someknownpriorconditionsfortheparameters /m.d

    Table2OptimalresultsofGAinversion

    /m?d

    After100timesofiteration,thesolutionremains stable,increasingthenumberofiterationshasno

effectontheresults,andatthistimeithasbeensimi

    lartotheoptimalsolution,thetimespentatotalof4 seconds.

    3Conclusions

    Althoughthegeneticalgorithmtoachieveoptimal Undergroundwaterqualitymodelinversionofgeneticalgorithm167 over100timesiteration,butonlyfoursecondstorun, indicatingthatgeneticalgorithmisaoptimization methodofahighdegreeofrobustnessandaneffective waytosolvethegroundwaterinverseproblem,with theadvantagesothertraditionalinversionmethodscan notcomparewithandpracticalsignificance. Rererences

    AllgowerEL,GeorgK.1983.PredictorCon'ectorandsimpli

    cialmethodsforapproximatingfixedpointsandzeropoints ofnonlinearmappingsinmathematicalprogramming.Ber

    lin:Spinger,163184.

    GarciaCB.GouldFJ.1980.Relationsbetweenseveralpath followingalgorithmsandlocalglobalNewtonmethods.SI

    AMReview,22(3):263274.

    GeorgK.1990.Anoteonstepsizecontrolfornumericalcurve following.ProceedingoftheNATOadvancedresearchin

    stituteonhomotopymethodsandglobalconvergence.New York:PleniumPress.472-498.

    HeDK,WangFL,ZhangCM.2003.EstablishmentofPa

    rametersofgeneticalgorithmBasedonUniformDesign. Journalol'NortheasternUniversity,24(5):409.411.(in ChinesewithEnglishabstract)

    OmegaJM,RheinboldtWC.1970.Iterativesolutionofnon

1inearequationsinSeveralVariables.NewYork:Aca

    demicPress,1572.

    XuQ,ChenRQ,GuanYL,eta1.Theshortestpathanalysis basedongeneticalgorithms.JournalofEastChinaC.eo

    logicalInstitute,2003,26(2):168172.(inChinese

    withEnglishabstract)

    ZhouzH.ChenzQ.ChenSF.2000.Researchoffieldthee

    rybasedadaptiveresonanceneuralnetwork.Journalof NanjingUniversity:NaturalSciences,36(2)140147.

    (inChinesewithEnglishabstract)

    ZhuHS.1997.Featuresandapplicationsofmultigroupge

    neticalgorithms.SysterasEngineeringTheory&Practice. 7885.finChinesewithEnglishabstract)

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