The ADI Model Problem

174 Pages·2013·1.23 MB·English

by  Eugene Wachspress (auth.)

Eugene Wachspress
The ADI
Model
Problem
The ADI Model Problem

Eugene Wachspress
The ADI Model Problem
123
EugeneWachspress
EastWindsor,NewJersey
USA
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DOI10.1007/978-1-4614-5122-8
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In lovingmemoryof mywifeof53 years
Natalie

Preface
Thisworkisanupdatededitionofmyself-publishedmonographonTheADIModel
Problem [Wachspress, 1995]. Minor typographic corrections have been made in
Chaps.1–4. A few innovations have been added such as non-iterative alignment
ofcomplexspectrainSect.4.6.In1995thetheorywaspleasingbutapplicationwas
limited. With regardto discretized elliptic equations,the restriction to rectangular
gridsforwhichreasonableseparableapproximationscouldbegeneratedwaspartic-
ularlylimiting.LyapunovandSylvestermatrixequationsopenednewapplications.
Although the commutationproperty was no longer a problem there were limiting
factorsheretoo.Accurateestimatesforcrucialeigenvalueswererequiredinorder
to find effective iteration parameters. This led to a study described in Sect.3.8 in
Chap.3ofsimilarityreductiontobandedupperHessenbergformwhichendedwith
“Intime,anefficientandrobustschemewillemergeforapplicationtotheLyapunov
problem.”SuchaschemeispresentedinChap.5ofthisedition.Crucialeigenvalues
includethosewithsmallrealpart,thosewhichsubtendlargeanglesattheoriginand
thatoflargestmagnitude.Whenthereducedmatrixisexpressedinsparseformthe
MATLABprogramEIGSprovidespreciselythetoolneededforthisdetermination.
This left only the limiting fact that ADI, although competitive with methods
like Bartels–Stewart [Bartels, 1972], did not offer enough advantage to supplant
standard methods. Lack of familiarity of practitioners with the new theory was
also detrimental. The saving innovation was that of Penzl [Penzl, 1999]. He
demonstratedthattheADIapproachallowedonetoreducearithmeticsignificantly
forproblemswith low-rankright-handsides. Thiswas notpossible with the other
schemes. Further analysis by Li and White [Li and White, 2002] reduced the
arithmetic even further. Now the ADI approach was worthy of consideration by
practitioners. It is my hope that it will become the method of choice not only for
low-rankproblemsbutalsoformostLyapunovandSylvesterproblems.
My programs are all written for serial computation. The ADI approach par-
allelizes readily. I have described in Sect.5.7 a scheme for approximating a full
symmetricmatrixbyasumoflow-rankmatrices.Eachofthelow-rankmatricesmay
be used as a right-hand side in parallel solution of low-rank Lyapunov equations.
I only discuss symmetric right-hand sides in Sect.5.7. Sylvester equations have
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nonsymmetricright-handsides.BiorthogonalLanczosschemesextendtononsym-
metricmatrices.Lanczosdifficultiesassociatedwithnear-zeroinnerproductsmay
bepreemptedinthisapproachbyrestarttofindthenextlow-rankcomponent.
In 1995 I was still writing FORTRAN programs. I now work exclusively with
MATLAB.Chapter6nowincludesMATLABimplementationoftheorydescribed
in Chaps.1–5. These programs have treated successfully all my test problems.
Theorywas supportedin thatwhencrucial eigenvalueswere computedaccurately
theobservederrorreductionagreedwithprediction.
This work describes my effort over the past 50 odd years related to theory
and application of the ADI iterative method. Although ADI methods enjoyed
widespreaduse initially,modelproblemconditionswere notpresentandlittle use
was made of the theory related to elliptic functions.The more recentrelevanceto
LyapunovandSylvesterequationsstimulatedthefirstsignificantapplication.Iam
hopeful that this will lead to further analysis and that other areas of application
willbeuncovered.ThatthesimplystatedminimaxproblemdefinedinEqs.8–10in
Chap.1couldleadtothetheoryexposedinthisworkhasneverceasedtoamazeme.
EastWindsor,NJ EugeneWachspress
Contents
1 ThePeaceman–RachfordModelProblem................................. 1
1.1 Introduction............................................................ 1
1.2 ThePeaceman–RachfordMinimaxProblem......................... 3
1.3 EarlyParameterSelection............................................. 8
1.4 OptimumParametersWhenJ D2n.................................. 9
1.5 OptimumParametersfromChebyshevAlternance .................. 11
1.6 EvaluatingtheErrorReduction....................................... 16
1.7 ApproximateParameters.............................................. 20
1.7.1 ErrorReduction................................................ 20
1.7.2 IterationParameters........................................... 21
1.8 UseofOtherDataonSpectraandInitialError....................... 22
2 TheTwo-VariableADIProblem............................................ 25
2.1 RectangularSpectra ................................................... 25
2.2 W.B.Jordan’sTransformation ........................................ 26
2.3 TheThree-VariableADIProblem .................................... 31
2.4 AnalysisoftheTwo-VariableMinimaxProblem .................... 33
2.5 GeneralizedADIIteration............................................. 43
3 ModelProblemsandPreconditioning..................................... 47
3.1 Five-PointLaplacians ................................................. 47
3.2 TheNeutronGroup-DiffusionEquation.............................. 49
3.3 Nine-Point(FEM)Equations.......................................... 51
3.4 ADIModel-ProblemLimitations..................................... 54
3.5 Model-ProblemPreconditioners...................................... 55
3.5.1 PreconditionedIteration....................................... 55
3.5.2 Compound Iteration (Quotations from
Wachspress1963) ............................................. 56
3.5.3 UpdatedAnalysisofCompoundIteration.................... 57
3.6 InteractionofInnerandOuterIteration .............................. 59
3.7 Cell-CenteredNodes .................................................. 64
3.8 TheLyapunovMatrixEquation....................................... 64
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