SOLUTION OF A GROUNDWATER CONTROL PROBLEM WITH IMPLICITFILTERING

�A. BATTERMANN

�, J. M. GABLONSKY

�, A. PATRICK

�, C. T. KELLEY

�, K. R. KAVANAGH

�,

T. COFFEY�, AND C. T. MILLER

�Abstract. In thispaperwedescribetheapplicationof a parallelimplementationof theimplicit filtering algorithm

to acontrolproblemfrom hydrology. Weseekto controlthetemperatureatagroupof drinkingwaterwellsby placingbarrierwellsbetweenthedrinkingwaterwellsandawell thatinjectsheatedwaterfrom anindustrialsite.

Keywords. Implicit filtering, Groundwaterflow andtransport,Optimalcontrol,Parallelalgorithms

1. Intr oduction. The objective of this paperis to show how the implicit filtering algorithm[11,15] for noisy optimizationproblemscanbe appliedto optimizationproblemsin hydrology.We focuson a groundwatertemperaturecontrol problem. This problemhassomeof the impor-tantdifficulties,suchasnonconvexity andnonsmoothness,thatonewouldexpectin moredifficultcases,but canuseflow andtransportmodelsandformulationsof theoptimizationproblemthataresufficiently simpleto allow for a completedescriptionin a singlepaper. More difficult problems,with coupledflow and transport,temperaturedependentdensitiesand viscosities,threedimen-sionalgeometries,andmorecomplex flow andtransportequations,will be consideredin futurework.

In thispaper, wesolvethesubsurfaceflow controlproblemwith aparallelimplementation[3]of theimplicit filtering algorithm[10,11,15]. Implicit filtering is asamplingmethodfor optimiza-tion of noisyfunctions.Theproblemhassimpleboundconstraintsandfour optimizationvariables.Theobjective functionis nonconvex, nonsmooth,andhasseverallocal minima. Theoptimizationlandscapein Figure1.1 is aplot of theobjective functionwith two of thevariablessetto zero.

We begin in�

2 by briefly discussingthegroundwaterflow andtransportmodelsusedin thiswork andby formulatingthecontrolproblem.

In�

3 we review the implicit filtering algorithmandits implementationin parallel. Thenin�4 wereporton theresultsof theoptimizationandtheparallelperformance.

2. Groundwater Temperature Control. The problemwe considerin this paperwasgivento usby TGU (TechnologieberatungGrundwasserundUmwelt) GmbH,a consultingengineeringcompany for groundwaterandwater resources.We wish to control the temperaturein a setofdrinkingwaterwells. Thesiteshown in Figure2.1 is in therechargeregion for thesewells. Thereis anindustrialzoneon theright of theshadedregion which injectsheatedwaterin a singlewell,�

Versionof December15,2000.�UniversitatTrier, FachbereichIV, AbteilungMathematik,54286Trier, Germany (batt@uni-trier.de).This author

wassupportedby thefoundationStiftungRheinland–Pfalzfur Innovation.�NorthCarolinaStateUniversity, Departmentof MathematicsandCenterfor Researchin ScientificComputation,

Box 8205,Raleigh,N. C. 27695-8205(tscoffe2@eos.ncsu.edu,jmgablon@unity.ncsu.edu,krkavana@unity.ncsu.edu,Tim Kelley@ncsu.edu,hapatric@unity.ncsu.edu).This researchwaspartially supportedby NationalScienceFoun-dationgrants#DMS-0070641and#DMS-9714811,Army ResearchOfficegrant#DAAD19-99-1-0186,aUSDepart-mentof EducationGAANN fellowship. Computingactivity waspartially supportedby anallocationfrom theNorthCarolinaSupercomputingCenter.�

Departmentof EnvironmentalSciencesandEngineering,104RosenauHall, Universityof NorthCarolina,ChapelHill, NC 27599-7400(casey miller@unc.edu).

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Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

FIG. 1.1. OptimizationLandscape

−0.04

−0.02

0

0.02

0.04

−0.04−0.03−0.02−0.0100.010.020.030.040

100

200

300

400

500

600

700

800

x2

x3

J(0,

x 2,x3,0

)

theinfiltration well. Germanlaw (theWasserhaushaltsgesetz)requiresthatanthropogenicchangesof groundwaterpropertiesbeminimized. In this regulationis therequirementthatdrinking waterbe providedat the lowesttemperaturethat is possibleunderundisturbedconditions.We seektoreducethe temperatureat thedrinking waterwells by minimizing a quadraticfunction involvingpumpingratesat a set of barrier wells, which is an approximatemeasureof cost, anda linearcombinationof pumpingrateandtemperatureatasetof drinkingwaterwells.

Figure2.2shows therelative locationsof thewells. Theinjectionwell is thesquareat thefarright, thebarrierwells theverticalrow in themiddle,andthedrinkingwaterwellsarethearrayontheleft.

Numericalexperimentsshow that a steady-statesolutionis obtainedafter eight to ten yearsof real time. Becauseof this we mayusethefour steadystatepumpingratesascontrolvariables.For the work reportedherewe neglect the vertical dimensionand the dependenceof viscosityanddensityon temperature.Theseassumptionsenableusto decoupletheequationsfor flow andtemperatureandto usea two-dimensionalsimulatorfor each.Giventhecontrols,wecansolvefortheflow andusetheresultsfrom theflow codeto computethetemperaturedistribution.

To determinetheflow wecomputethepiezometrichead� from

�� ��� � ����������� �������(2.1)

andappropriateinitial/boundaryconditions. In (2.1),�

is the storagecoefficient,��� �"!$# � is the

thicknessof the aquifer,��� �"!$# � is the hydraulicconductivity, and � is a sourceterm. Fromthe

headwecomputethemeanmacroscopicporevelocityvectorvia

% �'&��� �( !

(2.2)

where( is theeffectiveporosityof theporousmedium.

2

FIG. 2.1. Mapof theSite

FIG. 2.2. Well Locations

Infiltration well

Drinking water well

Barrier well

After solving theflow equation,we modeltemperaturein a way thata solutetransportcodecanbeusedto solve therelevantequations[7].

3

Thewatertemperature) satisfies

*�+,)+�-/. 0�1�2�34150 )7698;: 150 )=<(2.3)

wherethethermalretardationfactoris

* . >@?BA$CEDFCEGHCA$IJDKILGMION(2.4)

In (2.3), A$I . P N Q is thevolumefractionof theaqueousphase,ARC . P N S is thevolumefractionofthesolid phase,DFC$GJC . > N TKU'V,WYX�Z\[

is theheatcapacityof thesoil, and DKIHGJI . ] N >^TK_`U'V,WYX�Z\[is theheatcapacityof thefluid. For saturatedflow, A . A$I .

Thethermaldispersiontensoris

acbed . f�g\h : h i bjd ? 2 f,k 8 f�g 6ml b l dh : h <(2.5)

wherei bed

is theKroneckeri, and

f,k . >nPoXand

f�g . >^Xarelongitudinalandtransversaldisper-

sivity valuesthatarecharacteristicof the porousmedium.3

is a nonsmoothfunctionof : , andhenceof p . Thisaccountsfor thenonsmoothnessthatis clearlyvisible in Figure1.1.

We formulatetheoptimizationproblemas

qsrutvnw^x V 2 py6 . p�z�p ?�{ z,) N(2.6)

Here p}| *7~is the vector of steady-statepumpingratesat the control wells, ) | *7~

is thetemperatureat thedrinkingwaterwells,and

{ . 2 N PoQK�`_ < N P�>`>^� < N P ]`] P < N P ]`� > 6�z�| * ~is avectorof therelativepumpingrates(in

X�ZJW`�n� G ) at thesewells. Thetruncationerrorin theflowandtransportcodescontributelow-amplitudenoiseto

V.

Theboundconstraintswereimposedto accountfor limits in thepumpingrates.Thesecon-straintswere not active at the solution, and the optimizationwas essentiallyan unconstrainedproblem.

3. Implicit Filtering . Implicit filtering [11,15] is a projectedquasi-Newton iterationwhichusesdifferencegradients,reducingthe differenceincrementasthe optimizationprogresses.Themethodwasdesignedfor problemswith objectivefunctionsthataresmallperturbationsof smoothfunctions.Ourparadigmis �

.� C ?��

(3.1)

where� C is smooth,and

h � 2�� 6 h is small. In practice�

is usuallynonsmoothandsometimesdiscon-tinuous.

Implicit filtering is a samplingmethod. This meansthat the optimizationis directedonlyby informationon function values,with no gradientinformation. Implicit filtering differs fromclassicalsamplingmethodssuchastheNelder-Mead[17] or Hooke-Jeeves[12] algorithmsin thatit is readilyimplementedin parallel[3,4,6] by simplyperformingthefunctionevaluationsneededfor the differencegradientin parallel. The potentialfor quasi-Newton acceleration[5, 11,15]

4

is a featurethat other parallelizablesamplingmethods,suchas the PDS method[8, 19,20] orDIRECT [9,13,14], cannotexploit. Theresultsreportedin this paperwereobtainedwith IFFCO,aFORTRAN implementationof implicit filtering [3].

Supposeweseekto solve �s�u��^�L�s�����y�(3.2)

where ������������ �E¡�¢9£'���y�¤¢¥£§¦¥¢©¨Fª(3.3)

Here,�¡�¢©¨ �¢¬«y­ and

�¦¥¢®¨ �¢¯«y­ aresequencesof realnumberssuchthat

°7± ² ¡³¢ ² ¦¥¢ ²µ´± ª(3.4)

Herewedenotethe ¶ th componentof thevector � by � �y��¢ to distinguishthecomponentindex fromtheiterationindex. Wedenoteby · the ¸�¹ projectiononto

�. For �º�»�7¼

· � �y��¢� ½¾¿

¾À¡³¢ if � �y��¢¥£µ¡³¢� �y�¤¢ if ¡�¢ ² � �y��¢ ² ¦¥¢¦9¢ if � �y��¢¥Á§¦9¢

(3.5)

Implicit filtering asimplementedin IFFCObeginsby scaling�

to theunit cube( ¡�¢�ÃÂ

and

¦¥¢�ÅÄ

for all ¶ ). For ²BÆ £

ªÈÇ , let ÉËÊ � denotethe finite differenceapproximationof É �

with stepsize Æ thatusescentraldifferencesif all pointsof thecentraldifferencestencilarein�

andone-sideddifferencesin thosedirectionsin which onepoint in the stencil is not in�

. Therestriction Æ £ÃªÌÇ impliesthatat leasttwo pointswill bein thestencilin any coordinatedirection(thecenterandat leastoneof �ÎÍ Æ,Ï , where Ï is theunit vectorin thatdirection). Thestencilisusedbothto approximatethegradientandto provide oneof theterminationcriteria. Let Ð ���"Ñ Æ �bethedifferencestencilabout� in

�with stepsize Æ . We call thecondition

��� �y�ң�s�u�

ÓR�^ÔKÕÖ�^× ÊJØ ���ÚÙK�(3.6)

stencilfailure. In theunconstrainedcase[2,15] stencilfailureimpliesthat É �YÛ�§Ü

� Æ � . A similarresultalsoholdsin theboundconstrainedcase,wherestencilfailureimpliesthat

� ° · � � ° É �YÛJ� �y�Ý��§Ü

� Æ �MªWeterminatethequasi-Newtoniterationfor agivenvalueof Æ afterastencilfailurefor thisreason.

IFFCOoffersachoiceof SR1andBFGSquasi-Newtonupdates.For boundconstrainedprob-lemswe recommendthe SR1update. We will formally describethe algorithm. We begin withAlgorithm fdquasi, which is afinite differenceprojectedquasi-Newton iterationfor (3.2).

Implicit filtering is a sequenceof calls to fdquasi with the differenceincrementsor scalesreducedaftereachreturnfrom fdquasi.

Thereareseveral convergencetheoremsfor implicit filtering [5, 11,15]. We statea typicalresultfrom [15] for completeness.

THEOREM 3.1. Let � satisfy(3.1)andlet É �YÛ beLipschitzcontinuous.Let Æ�Þàß Â,�� Þ ¨ be

theimplicit filtering sequence, and Ð Þ � Ð ���"Ñ Æ�Þ � . Assumethat fewer than áKâãá � backtracksaretakenfor all but finitelymany ä . Thenifå �u�Þ\æ=ç � Æ�Þè´éÆëê ­Þ �Îì5í

ÓR�nÔ5î �eïð�ÚÙK�m�e��ñÂ

(3.7)

5

Algorithm 1 fdquasi ò�ó"ôMõ�ô�ö,÷ãø`ó"ô$ù�ôMúûô\øK÷ãø`óyüöþýÃÿwhile ö��;ö�÷ãøKó and �\ó���� ò ó����mõ�ò�óyü$ü��� ùOú do

computeõ and ��5õif (3.6)holdsthen

terminateandreportstencil failur eend ifupdatethemodelHessian� if appropriate;solve ��� ý�����Yõ�ò óyüuseabacktrackingline search,with atmost øF÷ãø`ó backtracks,to find asteplength �if øK÷ãø`ó backtrackshavebeentakenthen

terminateandreportline search failur eend ifó�� � ò ó�������üö�� ö��µÿ

endwhileif ö��/ö,÷ãø`ó reportiteration count failur e

Algorithm 2 imfilter ò ó"ôMõOô®ö,÷ãø`ó"ô$ù�ô��Yú �"!Fô\øK÷ãøKóyüfor #sý%$�ô'&(&'& dofdquasi ò�ó"ôMõ�ô®ö�÷ãøKó"ôÝù�ôMú)�5ô\øK÷ãøKóyü

end for

thenanylimit pointof thesequence�nó*�"! is a critical pointof õ"+ .The implicit filtering methodhasmany parameters,the sequenceof scales,the termination

parameterù , andthelimits øK÷ãø`ó andö,÷ãø`ó on theinnerandouteriterations.Wewill discussoursettingsof thoseparametersin , 4.

Themostsignificantopportunityfor parallelismis in thecomputationof ��5õ , whereall thefunctionevaluationsfor ó.-0/=ò ó"ôMúOü areindependent.Onecanalsoperformtheline searchfunc-tion evaluationsin parallel. In , 4.2we show how theparallelismcanbeeffectively exploitedbyIFFCO.

4. Computational Results. Thecomputationsreportedin thissectionweredoneon theIBMSP/2supercomputerlocatedat theNorth CarolinaSupercomputerCenterrunningIBM AIX 4.3.This IBM SP/2consistsof 180 nodes,whereeachnodeconsistsof four 375 MHz Power3-IIprocessors.Eachnodehas2 GB of memory. WeusedtheIBM xlf 7.1FORTRAN compiler.

The parametersin the implicit filtering algorithmwere ù�ý ÿ , øK÷ãø`ó ý 1 , ö,÷ãø`ó ý ÿ'$ ,243 ý5&6$"7 for 89ý'ÿ`ô'&(&'&Lô97 , and : 3 ý��;&6$<7 for 89ý�ÿ`ô'&'&'&Hô97 . WeusedtheSR1quasi-Newtonmethodandimposeda limit of 50 function evaluationson the optimization. The scaleswere ú 3 ý>=@? 3for ÿ��A8��AB . We terminatedthe optimizationafter we expendedthe budgetof 50 functionevaluations.

The parallelismwas in the simultaneousevaluationsof the objective function to form thedifferencegradients.We discretizedtheflow equationson a 7C=�DE7@= meshandusedMODFLOW[16] to computethe piezometrichead.Fromthe headwe extractedthe velocity vectorandusedMT3D [18], a transportsimulator, to computethe temperaturedistribution on the mesh.See[1]

6

for a morecompleteaccountof themodel,theboundaryconditions,andtheunderlyingphysicalassumptions.Wecomputedthesteady-statesolutionsusingaccuratetemporalintegrationoutto tenyears.For theflow simulation120time stepsof 30 daysaretaken. Thetransportintegrationwasexplicit, andwetook150transportstepsfor eachflow step.MODFLOW andMT3D communicatevia disk I/O.

4.1. Effectivenessof the Control. In Figures4.1 and4.2 we plot contoursof temperature.We normalizethe FHG<I9J temperatureof thegroundwaterto zeroandthe F�K<I9J temperatureof thewaterfrom the injectionwell to one. The injectionwell is locatedat thebox on theright sideofthe plume,the control wells at the vertical row of diamondsin the centerof the plume,andthedrinking waterwells at thecirclesto theleft of thecontrolwells. Thetemperatureof theinjectedwateris K I J warmerthattheambientgroundwatertemperatureof F'G I J . This leadsto anincreaseof F I J at thedrinkingwaterwells for theuncontrolledflow, to highsatisfytheregulations.

Thefiguresclearlyshow thattheoptimizedpumpingratesreducethetemperatureatthedrink-ing wells andthat thesizeof thehigh temperatureplumehasbeenreduded.Themaximumtem-peratureat thedrinking waterwells is F'GMLNF'I9J for thecontrolledflow, which is within regulatorylimits.

FIG. 4.1. TemperatureDistribution: UncontrolledFlow

0

0.1

0.2

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0.4

0.5

0.6

0.7

0.8

0.9

1

5 10 15 20 25 30 35 40

5

10

15

20

25

30

35

40

4.2. Parallel Performance. As wedescribedearlierin O 3, therearetwo opportunitiesfor par-allelismin IFFCO,theevaluationof thegradientandtheline search.Weexploit thesepossibilitiesin our implementationby usingthePVM parallelprogramminglibrary.

Theprocessorson eachnodeshare2 gigabytesof memory(which did not affect our compu-tations)anda local, temporarydirectory. We usedthis temporarydirectoryfor thedatafiles andtemporaryfiles we neededin our simulation. Sincefour processorssharedthe samelocal direc-tory, we addeda uniquetaskidentificationnumber(TID) to eacheachtemporaryfile to preventthedifferentprocessorsfrom writing to thesamefile.

The PVM programminglibrary leadsto the useof the master-slave parallel programmingparadigm.In thecomputationamasterprocessordid all thework in IFFCOexceptfor thefunction

7

FIG. 4.2. TemperatureDistribution: ControlledFlow

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5 10 15 20 25 30 35 40

5

10

15

20

25

30

35

40

Numberof Processors Run-time(in sec.) Speedup1 5582.89 1.0000

2+1 2986.31 1.86954+1 1618.64 3.44918+1 1050.15 5.3163

TABLE 4.1Paralell efficiency

evaluations. The time neededto do this was muchsmallerthan the time neededto evaluateafunction. Therefore,we usedthe masterto run IFFCO andusedboth the masterandthe slavesto do the function evaluationsneededduring the evaluationof the gradientandthe line search.We only neededto sendshortmessagesbetweenthemasterandtheslaves,sothecommunicationtimeswereverysmallcomparedto thecomputations.Thismeansweonly neededto usethebasicsendandreceivemechanismsprovidedby PVM.

ThePVM implementationavailableon theIBM SP/2neededadedicatedprocessorto run thePVM server. In Table4.1 we show thetimesneededto solve theproblemwith differentnumbersof processors.We recordthenumberof processorsas,for example, PRQTS to emphasizethatoneprocessorwasneededasthePVM server(acharacteristicof theIBM SP/2PVM). Thelastcolumnof thetableshows thespeedupfactor U

VXWZY\[Y V^]where Y\[ is the time neededwith oneprocessorand Y V is the time neededwith _\Q`S processors.Perfectspeedupfor ourconfigurationwouldbe

UVXW _ .

Notethat it doesnot make sensefor this problemto usemorethannineprocessors.At mosteightprocessorsarerequiredfor evaluatingthegradient,andoneis requiredasthePVM server.Table4.1showsgoodparallelperformance.

8

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[2] D. M. BORTZ AND C. T. KELLEY, Thesimplex gradientandnoisyoptimizationproblems, in ComputationalMethodsin OptimalDesignandControl,J.T. Borggaard,J.Burns,E. Cliff, andS.Schreck,eds.,vol. 24ofProgressin SystemsandControlTheory, Birkhauser, Boston,1998,pp.77–90.

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[5] T. D. CHOI AND C. T. KELLEY, Superlinearconvergenceand implicit filtering, SIAM J. Optim.,10 (2000),pp.1149–1162.

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StateUniversity, Centerfor Researchin ScientificComputation,August1998.[10] P. GILMORE, An Algorithmfor OptimizingFunctionswith Multiple Minima, PhDthesis,North CarolinaState

University, Raleigh,NorthCarolina,1993.[11] P. GILMORE AND C. T. KELLEY, An implicit filtering algorithmfor optimizationof functionswith manylocal

minima, SIAM J.Optim.,5 (1995),pp.269–285.[12] R. HOOKE AND T. A. JEEVES, ‘Dir ectsearch’ solutionof numericalandstatisticalproblems, Journalof the

Associationfor ComputingMachinery, 8 (1961),pp.212–229.[13] D. R. JONES, TheDIRECTglobaloptimizationalgorithm. to appearin theEncylopediaof Optimization,1999.[14] D. R. JONES, C. C. PERTTUNEN, AND B. E. STUCKMAN, Lipschitzian optimizationwithout the Lipschitz

constant, J.Optim.TheoryAppl., 79 (1993),pp.157–181.[15] C. T. KELLEY, Iterative Methodsfor Optimization, no. 18 in Frontiersin Applied Mathematics,SIAM,

Philadelphia,1999.[16] M. G. MCDONALD AND A. W. HARBAUGH, A modularthree-dimensionalfinite-differencegroundwaterflow

model, U.S.GeologicalSurvey Techniquesof WaterResourcesInvestigations,Book6,Ch.A1, Reston,VA,1988.

[17] J. A. NELDER AND R. MEAD, A simplex methodfor functionminimization, Comput.J.,7 (1965),pp.308–313.[18] S. S. PAPADOPULOS, MT3D: A modularthree-dimensionaltransportmodel,Version1.5,Documentationand

User’sGuide, S.S.Papadopulos& Associates,Inc.,Bethesda,Maryland,1992.[19] V. TORCZON, On theconvergenceof themultidimensionaldirectsearch, SIAM J. Optim.,1 (1991),pp.123–

145.[20] , Ontheconvergenceof patternsearch algorithms, SIAM J.Optim.,7 (1997),pp.1–25.

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