Suggestions for F urther Investigations

Theuser friendliness ouldbeimproved byinluding more optional

param-eters to the problem setup-le. The setup-le should dene the following

optionalparameters:

∆

^{Initialtrust region radius for Algorithm1.}

max _f1

^{Maximal numberof funtion}evaluations inAlgorithm1.

max _f2

^{Maximal numberof funtion}evaluations inAlgorithm2.

max _f3

^{Maximal numberof funtion}evaluations inAlgorithm3.

η x

^{Steplength in}diffjaobianwhenalulatingJaobian wrt.

x

^.

η p

^{Steplength in}diffjaobianwhenalulatingJaobian wrt.

p

^.

ε _F

^{Usedinthe stopping riterion for thene modelobjetive funtion.}

ε K

^{Usedinthe stopping riterion for thegradient mathing.}

ε hx

^{Usedinthe stopping riterion for thestep lengthfor}

x

^-iterates.

ε _hp

^{Usedinthe stopping riterion for thestep lengthfor}

p

^-iterates.

opts Options formarquardt ([1e-8 1e-4 1e-4 200 1e-12℄).

diagA Parameter dening thenumberof mapping parameters

(0: fullmatrix

A

^{,1: diagonal matrix}

A

⁾

Someof the above options arealready used in theSMIS framework, ie.

∆

^,

max _f1

^,

max _f2

^{,anddiagA,andalsotwo toleraneparameters} orresponding to

ε F

^,

ε K

^,

ε hx

^and

ε hp

^.

We also list some possible hanges inorder to inrease the auray of the

omputations:

•

^{Introduingthestep lengthparameter}

η

^{asaninput parameterto the}

funtion diffjaobian.

•

^{Making itpossible to exploitexat gradients, ifthey areavailable.}

•

^{Providinga setof} reommendabletolerane parameters.

•

^{Testingdierent strategies for updating thepenaltyfator.}

•

^{Testingother weighting fator} strategies.

•

^Further investigations of how the number of mapping parameters in-uenethe results.

•

^{Further analysison theoptimalmapping} parameters.

•

^{Testingdierent strategiesonerningthenumberof mapping}

param-eters,eg. releasing aparameter inevery main iteration.

•

^{Further analysisof thetest results,eg. looking at the}termination ri-teria for the sub-algorithms.

•

^{Further analysisof theiteration sequenes.}

•

^{Deningdierentresidual}formulations,eg.lettingtheformulation de-pend onthe numberofiterates available. Possiblyomittingthe

gradi-ent residual inthe rstiterations in orderto minimize thenumber of

ne modelevaluations.

Chapter 6

Conlusion

Inthis thesis aSpae Mappingalgorithm hasbeen presented and tested on

some test problems. We have made a number of investigations in order to

make a robust implementation and analyze the harater of the Parameter

Extration probleminregard to thesolutions,wean expet tond.

Thesteplengthusedinthenitediereneapproximationsisfoundtobeof

greatimportane,inasethe exatgradientsofthemodelarenot available.

The step length inuenes both the trunation errors and the rounding

er-rors,and basedonresults fromtheTLT2problem andthetheoretial error

funtions, we nd a suitable value for the step length to be approximately

10 ^{− 5}

^.

The number of equations in the Parameter Extration problems depends

on, how many main iterations we have made. A large part of the

Param-eter Extration problems are therefore underdetermined, and onsequently

thereareinnitelymanysolutions. Thereisalsothepossibilityofhaving an

overdeterminedproblem,inwhihasewe maybeabletoeetthesolution

bymeans of the residual formulation.

The formulation of the residual inluding the normalization fators, the

weighting fators and the penalty fator makes it possible to give eah of

the residual elements an individual priority. In this way we an more or

less hoose whih order of loal and global agreement the surrogate model

shouldsatisfy. Thesolutionwendisinuenedbythehoieoftheresidual

formulation.

Itmustbelaried,thatwearenotalwayssure,thatthevariousfatorseven

inuenethe results. Thisdependsentirely ontheharater oftheproblem,

WehavehosentodenetheParameterExtrationproblemsasleastsquares

problems, and solve them by the Marquardt algorithm. The Marquardt

methodiswell-suitedfortheproblem,sine thedampingparameterontrols

the step length and ensures, that the Marquardt equations have a unique

solution.

Ifthe rankof

J _r

^{isnot full,the solution depends onthedamping parameter}

µ

^{. For small}

µ

^{-valuestheMarquardtsolutionisnearlyidentialtothe}

mini-mumnormsolution, whihisthesolutionlosesttothepreviousone. Inthis

asetheMarquardtequationsgivethesameeetasaformofregularization

wrt.the previous solution.

If we regularize the problems, we are guaranteed, that the Parameter

Ex-tration problems are overdetermined in all iterations. The regularization

termensures,thatwe ndasolutionlosetotheprevioussolution. We have

shown, that solving the Parameter Extration problem for the regularized

residual formulation orresponds to a speial ase of the Marquardt

equa-tionswiththedampingparameter

1 + µ

^{andahangedrighthandside. The}

regularizationmeans,thatwe annot besureto satisfythegradient math,

ifthe solutionis farfrom theprevious.

Thealgorithmwiththevariousresidualformulationsistestedonthree

prob-lems: The Rosenbrok problem in its original form and in the augmented

version provides a theoretial example, and theTLT2 and TLT7 problems

areexamplesofengineering design problems ofmore pratial harater.

Thetest resultsshow, thatthe algorithm workswell inaseof both solving

theregularized and the unregularized problem. In several ases thelatteris

atually resulting inafaster onvergene to theoptimizer,and thesolution

is found, before there is a possibility of having overdetermined Parameter

Extration problems. Weonlude, thatfor theonsidered test problems it

isno obstale, iftheParameter Extration problems areunderdetermined.

TheRosenbrokproblemshowssomeinteresting featuresonerningthe

op-timalmappingparameters. TheRosenbrokfuntionisspeial,beausethe

seond response funtion is linear and only depends on one of the design

parameters. Thishasonsequenesfor themapping parametersthroughthe

iteration sequene, inthe sense thatseveral of themapping parameters are

not hanged form theinitial values.

For theaugmentedRosenbrokproblemthesamefeaturesarefoundfor the

optimalmapping parameters.

Inthe engineeringdesignproblemTLT2wegetgoodonvergeneresultsfor

But when we onsider the approximation errors from using the surrogate

model dened by the optimal mapping parameters, the two versions give

very dierent results.

The regularized problem provides good surrogate approximations and

gen-erally, the surrogate model is better than a lassial Taylor model

approx-imation - not only over a large region of the design parameter spae, but

alsoneartheexpansionpoint. Forthesenariowithoutregularizationofthe

residualwe getaverypoorsurrogatemodel, even though thealgorithm has

onvergedto the ne modeloptimizer.

It shows,that for this problemthe normalization fatorshave an inuene.

Weahieveaeptableresultsbothregardingonvergeneandthequalityof

thesurrogatemodelapproximationfor salingofonly thegradient residual.

It is not possible to onlude anythingin general on the normalization

fa-tors, but in this ase they must be omitted to ensure a suitable surrogate

model.

The importane of the quality of the surrogate model approximation of

ourse depends on the design problem: If we are only interested in the

optimal design parameters, the surrogate model is perhaps unimportant.

Butinmodellingproblemswe maybeinterested inreplaing thenemodel

withthesurrogatemodelandexaminingthisinsteadatalowomputational

expense. Insuhaaseanunauratesurrogateapproximationisnotuseful.

The last test problem TLT7 is of larger dimension. The results here show

fastonvergeneintheaseoftheunregularized ParameterExtration

prob-lems,andthe redutionofthe mappingparameters havenoinueneonthe

results. Butwhenthe regularizationterm isadded,there isnoonvergene.

It isnot possible to explainthis behaviour.

Itappliesforalltheonsideredtestproblems,thattheSpaeMapping

algo-rithm provides fast onvergene results ompared to lassial optimization

methods.

Appendix A

Short User's Guide for The

SMIS Framework

InthefollowinghapterabriefoverviewoftheSMISframeworkinMatlab

isgiven.

A.1 The Problem Setup-le

Thesetup-le for thegivenproblem isplaed intheorrespondingproblem

diretory. The setup-le ontains information on the ne and oarse model

optimizers and linear equality/inequality onstraints for the optimization

problem.

Thesearedened by:

A _(1:leq,:) x + b _(1:leq,:) = 0

^and

A (leq+1:l,:) x + b (leq+1:l,:) ≥ 0

Thesetup-le mustreturn astruture

S

^{ontaining thefollowing elds:}

n The numberofdesign parameters.

m The numberofn parameters.

fine The m-leimplementing thene model funtion.

oarse The m-leimplementing theoarse modelfuntion.

parf Vetor withthe sampling points forthene model.

par Vetor withthe sampling points fortheoarse model.

paro Struture withinformation usedfor plotting.

xast The nemodeloptimizer.

zast The oarse modeloptimizer.

A

l

^-by-

n

^{matrix deningtheonstraints (}

l =

^size

(

^A

, 1) − leq

⁾

b

l

^{olumn vetor deningthe}onstraints.

leq Numberof linearequality onstraints.

x Initial guessfor oarse modeloptimizer.

delta Initial trustregion radius for surrogate optimizationproblem.

eps1 Used instopping riterion forthe mainproblem.

eps2 Used instopping riterion forthe Parameter Extration problems.

maxfun1Maximal numberof funtion evaluations inAlgorithm1.

maxfun2Maximal numberof funtion evaluations inAlgorithm2

(also usedinAlgorithm 3).

p1 Norm deningtheobjetivefuntion of themain problem.

p2 Norm deningtheobjetivefuntion of theParameter

Extration problems. When usingthealgorithm version

withmarquardt thisoption hasno eet.

iontr Parameter usedforontrolling the omputation:

1: starts algorithm,2: startsalgorithm andprints information.

diagA Parameter deningnumberof mapping parameters

(0: fullmatrix

A

^{,1: diagonal matrix}

A

⁾

Theoptions formarquardt aredened inthele'lmm'aseg.opts

=

^([1e-8

1e-14 1e-14 200 1e-12℄).

In document IMM YGBY2004ESAESRETR.54 ei (Sider 92-100)