几种常见的优化方法课件

几种常见的优化方法电子结构几何机构函数稳定点最小点Taylor展开：V(x)=V(xk)+(x-xk)V’(xk)+1/2(x-xk)2V’’(xk)+…..当x是3N个变量的时候，V’(xk)成为3Nx1的向量，而V’’(xk)成为3Nx3N的矩阵，矩阵元如：Hessian1几种常见的优化方法电子结构函数稳定点Taylor展开：当x一阶梯度法a.SteepestdescendentSk=-gk/|gk|directiongradient知道了方向，如何确定步长呢？最常用的是先选择任意步长l，然后在计算中调节用体系的能量作为外界衡量标准，能量升高了则逐步减小步长。robust,butslow最速下降法2一阶梯度法Sk=-gk/|gk|directi最陡下降法（SD）

3最陡下降法（SD）3b.ConjugateGradient（CG)共轭梯度第k步的方向标量UsuallymoreefficientthanSD,alsorobust不需要外界能量等作为衡量量利用了上一步的信息4b.ConjugateGradient（CG)共轭2。二阶梯度方法这类方法很多，最简单的称为Newton-Raphson方法，而最常用的是Quasi-Newton方法。Quasi-Newton方法：useanapproximationoftheinverseHessian.Formofapproximationdiffersamongmethods牛顿-拉夫逊法BFGSmethodBroyden-Fletcher-Golfarb-ShannoDFPmethodDavidon-Fletcher-Powell52。二阶梯度方法这类方法很多，最简单的称为Newton-RaMoleculardynamics分子动力学HistoryItwasnotuntil1964thatMDwasusedtostudyarealisticmolecularsystem,inwhichtheatomsinteractedviaaLennard-Jonespotential.Afterthispoint,MDtechniquesdevelopedrapidlytoencompassdiatomicspecies,water(whichisstillthesubjectofcurrentresearchtoday!),smallrigidmolecules,flexiblehydrocarbonsandnowevenmacromoleculessuchasproteinsandDNA.Theseareallexamplesofcontinuousdynamicalsimulations,andthewayinwhichtheatomicmotioniscalculatedisquitedifferentfromthatinimpulsivesimulationscontaininghard-corerepulsions.6Moleculardynamics分子动力学HistWhatcanwedowithMD–CalculateequilibriumconfigurationalpropertiesinasimilarfashiontoMC.–Studytransportproperties(e.g.mean-squareddisplacementanddiffusioncoefficients).–MDintheNVT,NpTandNpHensembles–Theunitedatomapproximation–ConstraintdynamicsandSHAKE–Rigidbodydynamics–MultipletimestepalgorithmsExtendthebasicMDalgorithm7WhatcanwedowithMD–Calcul‘Impulsive’moleculardynamics

1.Dynamicsofperfectly‘hard’particlescanbesolvedexactly,butprocessbecomesinvolvedformanypart(N-bodyproblem).2.Canuseanumericalschemethatadvancesthesystemforwardintimeuntilacollisionoccurs. 3.Velocitiesofcollidingparticles(usuallyapair!)thenrecalculatedandsystemputintomotionagain.4.Simulationproceedsbyfitsandstarts,withameantimebetweencollisionsrelatedtotheaveragekineticenergyoftheparticles.5.Potentiallyveryefficientalgorithm,butcollisionsbetweenparticlesofcomplexshapearenoteasytosolve,andcannotbegeneralisedtocontinuouspotentials. 88Continuoustimemoleculardynamics1.Bycalculatingthederivativeofamacromolecularforcefield,wecanfindtheforcesoneachatomasafunctionofitsposition.2.Requireamethodofevolvingthepositionsoftheparticlesinspaceandtimetoproducea‘true’dynamicaltrajectory.3.StandardtechniqueistosolveNewton’sequationsofmotionnumerically,usingsomefinitedifferencescheme,whichisknownasintegration.4.ThismeansthatweadvancethesystembysomesmalltimestepΔt,recalculatetheforcesandvelocities,andthenrepeattheprocessiteratively.5.ProvidedΔtissmallenough,thisproducesanacceptableapproximatesolutiontothecontinuousequationsofmotion.9ContinuoustimemoleculardynaExampleofintegratorforMDsimulationOneofthemostpopularandwidelyusedintegratorsistheVerletleapfrogmethod:positionsandvelocitiesofparticlesaresuccessively‘leap-frogged’overeachotherusingaccelerationscalculatedfromforcefield.TheVerletschemehastheadvantageofhighprecision(oforderΔt4),whichmeansthatalongertimestepcanbeusedforagivenleveloffluctuations.Themethodalsoenjoysverylowdrift,providedanappropriatetimestepandforcecut-offareused.r(t+Dt)=r(t)+v(t+Dt/2)Dtv(t+Dt/2)=v(t-Dt/2)+a(t+Dt/2)Dt10ExampleofintegratorforMDsOtherintegratorsforMDsimulationsAlthoughtheVerletleapfrogmethodisnotparticularlyfast,thisisrelativelyunimportantbecausethetimerequiredforintegrationisusuallytrivialincomparisontothetimerequiredfortheforcecalculations.Themostimportantconcernforanintegratoristhatitexhibitslowdrift,i.e.thatthetotalenergyfluctuatesaboutsomeconstantvalue.Anecessary(butnotsufficient)conditionforthisisthatitissymplectic.Crudelyspeaking,thismeansthatitshouldbetimereversible(likeNewton’sequations),i.e.ifwereversethemomentaofallparticlesatagiveninstant,thesystemshouldtracebackalongitsprevioustrajectory.11OtherintegratorsforMDsimulOtherintegratorsforMDsimulationsTheVerletmethodissymplectic,butmethodssuchaspredictor-correctorschemesarenot.Non-symplecticmethodsgenerallyhaveproblemswithlongtermenergyconservation.Havingachievedlowdrift,wouldalsoliketheenergyfluctuationsforagiventimesteptobeaslowaspossible.Alwaysdesirabletousethelargesttimesteppossible.Ingeneral,thetrajectoriesproducedbyintegrationwilldivergeexponentiallyfromtheirtruecontinuouspathsduetotheLyapunovinstability.However,thisdoesnotconcernusgreatly,asthethermalsamplingisunaffected⇒expectationvaluesunchanged.12OtherintegratorsforMDsimulChoosingthecorrecttimestep…1.

Thechoiceoftimestepiscrucial:tooshortandphasespaceissampledinefficiently,toolongandtheenergywillfluctuatewildlyandthesimulationmaybecomecatastrophicallyunstable(“blowup”).2.Theinstabilitiesarecausedbythemotionofatomsbeingextrapolatedintoregionswherethepotentialenergyisprohibitivelyhigh(e.g.atomsoverlapping).3.Agoodruleofthumbisthatwhensimulatinganatomicfluid,thetimestepshouldbecomparabletothemeantimebetweencollisions(about5fsforArat298K).4.Forflexiblemolecules,thetimestepshouldbeanorderofmagnitudelessthantheperiodofthefastestmotion(usuallybondstretching:C—Haround10fssouse1fs).13ChoosingthecorrecttimestepForclassicMD,therecouldbemanytrickstospeedupcalculations,allcenteringaroundreducingtheeffortinvolvedinthecalculationoftheinteratomicforces,asthisisgenerallymuchmoretime-consumingthanintegration.ForexampleTruncatethelong-rangeforces:charge-charge,charge-dipoleLook-uptablesForfirstprinciplesMD,asforcesareevaluatedfromquantummechanics,weareonlyconcernedwiththetime-step.14ForclassicMD,therecouldbeBecausetheinteractionsarecompletelyelasticandpairwiseacting,bothenergyandmomentumareconserved.Therefore,MDnaturallysamplesfromthemicrocanonicalorNVEensemble.Asmentionedpreviously,theNVEensembleisnotveryusefulforstudyingrealsystems.Wewouldliketobeabletosimulatesystemsatconstanttemperatureorconstantpressure.ThesimplestMD,likeverletmethod,isadeterministicsimulationtechniqueforevolvingsystemstoequilibriumbysolvingNewton’slawsnumerically.15BecausetheinteractionsarecMDindifferentthermodynamicensemblesInthislecture,wewilldiscusswaysofusingMDtosamplefromdifferentthermodynamicensembles,whichareidentifiedbytheirconservedquantities.

Canonical(NVT)–Fixednumberofparticles,totalvolumeandtemperature.Requirestheparticlestointeractwithathermostat.

Isobaric-isothermal(NpT)–Fixednumberofparticles,pressureandtemperature.Requiresparticlestointeractwithathermostatandbarostat.

Isobaric-isenthalpic(NpH)–Fixednumberofparticles,pressureandenthalpy.Unusual,butrequiresparticlestointeractwithabarostatonly.16MDindifferentthermodynamicAdvancedapplicationsofMDWewillthenstudysomemoreadvancedMDmethodsthataredesignedspecificallytospeedup,ormakepossible,thesimulationoflargescalemacromolecularsystems.Allthesemethodsshareacommonprinciple:theyfreezeout,ordecouple,thehighfrequencydegreesoffreedom.Thisenablestheuseofalargertimestepwithoutnumericalinstability.Thesemethodsinclude:–Unitedatomapproximation–ConstraintdynamicsandSHAKE–Rigidbodydynamics–Multipletimestepalgorithms17AdvancedapplicationsofMD17RevisionofNVEMDLet’sstartbyrevisinghowtodoNVEMD.Recallthatwecalculatedtheforcesonallatomsfromthederivativeoftheforcefield,thenintegratedthee.o.m.usingafinitedifferenceschemewithsometimestepΔt.Wethenrecalculatedtheforcesontheatoms,andrepeatedtheprocesstogenerateadynamicaltrajectoryintheNVEensemble.Becausethemeankineticenergyisconstant,theaveragekinetictemperatureTKisalsoconstant.However,inthermalequilibrium,weknowthatinstantaneousTKwillfluctuate.IfwewanttosamplefromtheNVTensemble,weshouldkeepthestatisticaltemperatureconstant.18RevisionofNVEMD18ExtendedLagrangiansThereareessentiallytwowaystokeepthestatisticaltemperatureconstant,andthereforesamplefromthetrueNVTensemble.–Stochastically,usinghybridMC/MDmethods–Dynamically,viaanextendedLagrangianWewilldescribethelattermethodinthislectureAnextendedLagrangianissimplyawayofincludingadegreeoffreedomwhichrepresentsthereservoir,andthencarryingoutasimulationonthisextendedsystem.Energycanflowdynamicallybackandforthfromthereservoir,whichhasacertainthermal‘inertia’associatedwithit.AllwehavetodoisaddsometermstoNewton’sequationsofmotionforthesystem.19ExtendedLagrangians19ExtendedLagrangiansThestandardLagrangianLiswrittenasthedifferenceofthekineticandpotentialenergies:Newton’slawsthenfollowbysubstitutingthisintotheEuler-Lagrangeequation:Newton’sequationsandLagrangianformalismareequivalent,butthelatterusesgeneralisedcoordinates...20ExtendedLagrangians..20CanonicalMDSo,ourextendedLagrangianincludesanextracoordinateζ,whichisafrictionalcoefficientthatevolvesintimesoastominimisethedifferencebetweentheinstantaneouskineticandstatisticaltemperatures.Themodifiedequationsofmotionare:TheconservedquantityistheHelmholtzfreeenergy.(modifiedformofNewtonII)21CanonicalMD(modifiedformofCanonicalMDByadjustingthethermostatrelaxationtimetT

(usuallyintherange0.5to2ps)thesimulationwillreachanequilibriumstatewithconstantstatisticaltemperatureTS.TSisnowaparameterofoursystem,asopposedtothemeasuredinstantaneousvalueofTKwhichfluctuatesaccordingtotheamountofthermalenergyinthesystematanyparticulartime.ToohighavalueoftTandenergywillflowveryslowlybetweenthesystemandthereservoir(overdamped).ToolowavalueoftTandtemperaturewilloscillateaboutitsequilibriumvalue(underdamped).ThisistheNosé-Hooverthermostatmethod.22CanonicalMD22CanonicalMDTherearemanyothermethodsforachievingconstanttemperature,butnotallofthemsamplefromthetrueNVTensembleduetoalackofmicroscopicreversibility.Wecallthesepseudo-NVTmethods,andtheyinclude:–BerendsenmethodVelocitiesarerescaleddeterministicallyaftereachstepsothatthesystemisforcedtowardsthedesiredtemperature–GaussianconstraintsMakesthekineticenergyaconstantofthemotionbyminimisingtheleastsquaresdifferencebetweentheNewtonianandconstrainedtrajectoriesThesemethodsareoftenfaster,butonlyconvergeonthetruecanonicalaveragepropertiesasO(1/N).23CanonicalMD23Isothermal-isobaricMDWecanapplytheextendedLagrangianapproachtosimulationsatconstantpressurebysimplyaddingyetanothercoordinatetooursystem.Weuseη,whichisafrictionalcoefficientthatevolvesintimetominimisethedifferencebetweentheinstantaneouspressurep(t),measuredbyavirialexpression,andthepressureofanexternalreservoirpext.TheequationsofmotionforthesystemcanthenbeobtainedbysubstitutingthemodifiedLagrangianintotheEuler-Lagrangeequations.Thesenowincludetworelaxationtimes:oneforthethermostattT,andoneforthebarostattp.24Isothermal-isobaricMD24Isothermal-isobaricMDTheisknownastheNosé-Hoovermethod(Melchionnatype)andtheequationsofmotionare:25Isothermal-isobaricMD25Isothermal-isobaricMD26Isothermal-isobaricMD26Constraintdynamicsfreezethebondstretchingmotionsofthehydrogens(oranyotherbond,inprinciple).Weapplyasetofholonomicconstraintstothesystem,whichar