Hermitian eigenvalue problems An a posteriori error estimator based on shifts for positive C.R. Acad. Sci. Paris, Ser.I

Contents lists available atScienceDirect

C. R. Acad. Sci. Paris, Ser. I

www.sciencedirect.com

Numerical analysis

An a posteriori error estimator based on shifts for positive Hermitian eigenvalue problems

Estimateur a posteriori pour les problèmes aux valeurs propres hermitiens positifs, reposant sur des décalages

Athmane Bakhta

^a

, Damiano Lombardi

^b

aÉcoledespontsParisTech,France bINRIA,France

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received12September2017 Acceptedafterrevision13April2018 Availableonline20April2018 PresentedbyOlivierPironneau

Thiswork dealswith an a posteriori error estimatorfor Hermitian positive eigenvalue problems.Theproposedestimatorisbasedontheresidualandthedefinitionofsuitable shiftsinthematrix spectrum.The mathematicalproperties (certificationandsharpness) areinvestigatedandsomenumericalexperimentsareproposed.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

ré s u m é

L’objet de ce travail est la mise au point et l’étude d’un estimateur aposteriori pour lesproblèmes auxvaleurs propreshermitiens positifs.L’estimateur proposése base sur uneapproximation delarélationentrel’erreuretlerésiduduproblème. Lespropriétés mathématiquesdel’estimateursontétudiées.Desexperiencesnumériquessontproposées afindevaliderl’estimateur.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

This workdeals withan aposteriorierrorestimatorforHermitian positiveeigenvalue problems.Havingasharperror estimatorforeigenvalues is,ingeneral,a difficulttask.Severalworkswereproposed intheliterature:in[2,3],erroresti- matorsare proposedforthefiniteelement discretizationofeigenvaluesofellipticoperators. Inthereducedbasis context for Stokesequations,an a posteriorierrorestimatorbased onBabuška’sstability theory [1] is proposed in[8]. Ageneral formulationofaposteriorierrorboundsthatcanbe appliedtoseveralsituationsisgivenin[5]. Wealsoreferthereader tothesismanuscript[7],whereerroranalysisiscarriedonfornumerousproblems:coercive,noncoercive,parabolic,Stokes andeigenvalue.Inthiswork,whichwasoriginallymotivatedbyproblemsinvolvingclassicalperiodicSchrödingeroperators,

E-mailaddresses:athmane.bakhta@cermics.enpc.fr(A. Bakhta),damiano.lombardi@inria.fr(D. Lombardi).

https://doi.org/10.1016/j.crma.2018.04.017

1631-073X/©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

an errorestimatorforthe spectrumofpositiveself-adjoint operatorsisproposed, basedontheproblemresidualandthe definitionofshifts.Inparticular,theaimistoestimatetheerrordonewhentheproblemfinelydiscretizedisprojectedon alow-dimensionalbasis,givingrisetoacoarselydiscretizedproblem.

Thedocumentisstructuredasfollows:afterhavingintroducedthenotation,theexpressionoftheestimatorisderived insection2.1.Themainresultsontheanalysisoftheestimatorarepresentedinsection3.Theimplementationandsome numericalexperimentsarepresentedinsection4.

2. Notationandproblemsetting

LetU beaHilbertspaceandA^{bealinearpositive}self-adjointoperatorA∈L

(

U

,

U

)

.Considertheassociatedeigenvalue problem:

A^uexact

= λ

exactu_exact

,

(2.1)

where

λ

exact∈R^∗₊^{istherealpositive}eigenvalue,anduexact∈^{U istheassociated}eigenfunction.

AdiscretefineapproximationofProblem (2.1) isintroducedasfollows:letN∈N^∗,andA∈C^{N×N beamatrix,obtained} bydiscretizingtheoperatorAinEq. (2.1).

Priortodetailingthediscretisedversionsoftheproblem,theRayleighquotientisintroduced:

R

(

v

) =

^{vA v}

vv

.

(2.2)

Themin–maxCourant–Fishertheoremallowsustorecovereigenvaluesandeigenvectorsasaresultoftheoptimisationof thequotient.LetVjbethesetofthevectorsubspacesofdimension j inR^N.Then,

λ

j

=

^max

V∈Vj

minu∈^VR

(

u

)

(2.3)

Thefinelydiscretizedproblemreads:

Au

= λ

u

,

(2.4)

whereu∈C^{N denotesthefinely}discretizedeigenvectorand

λ

thecorrespondingeigenvalue.

Acoarseapproximationissolvedbyprojectingthefinelydiscretizedproblemonalow-dimensionalbasis.Moreprecisely, let N∈N^{∗ such that N}N ^{and denote by W}∈C^N×N

,

W =^[w1

, . . . ,

wN] thematrix whose columns are the finely discretizedbasis functions.Let A_N denotethecoarsematrixdefinedas A_N:=^{WHAW.Thus,thecoarse}approximation of theeigenpairsisobtainedbysolving:

AN

φ = λ

N

φ,

(2.5)

whichprovides

λ

N

> λ

,anapproximationfromaboveof

λ

andwhereu_N∈C^N

,

u_N:=^W

φ

,anapproximationoftheassoci- atedeigenvectorreconstructedatthefinelevelofdiscretization.Theinequalitybetweenthecoarselyandfinelydiscretised problemsderivesfromtheoptimisationofthequotient:indeed,sinceinacoarselydiscretisedproblemthespaceonwhich theminimisationiscarriedoutisofsmallerdimension(NN^{),thenthevalueoftheeigenvalueislarger.}

Throughoutthewholedocument,thescalarproductinC^{N willbedenotedbyu}

,

v=^uHv,foru

,

v∈C^N.

In the sequel, we forget aboutthe exact eigenvalues

λ

exact, andour aim is to estimate the error between thefinely approximatedvalue

λ

andthecoarselyapproximatedone

λ

N.

2.1. Derivationoftheestimator

Asformostofthemethodsofaposterioriestimation,arelationshipbetweentheresidualoftheproblemandtheerror onthesolutionissought.Theresidualreads:rN∈C^N

,

rN:=^AuN−

λ

NuN.SincethecoarseproblemisobtainedbyGalerkin projection, itfollows that^rN

,

uN=^{0. Lete}:=^uN−^{u denotethe errorinthe} eigenvectorapproximation. Thefollowing equationfortheresidualisobtained:

r_N

= (

A

− λ)

e

− (λ

N

− λ)

u_N

.

(2.6)

ProjectingEq. (2.6) onuN leadsto:

0

=

uN

, (

A

− λ)

uN

−

u

− (λ

N

− λ) ⇒ ε := λ

N

− λ =

uN

, (

A

− λ)

uN

≥

0

.

(2.7) Theobjectiveistoexpressthisquantityas^rN

,

BrN^,whereB∈C^{N×N isaHermitianmatrix(whichisthe}discretization, atfinelevel,ofaself-adjointoperatorthatwillbemadepreciselater).Thisisdoneintwosteps.

First,anexpressionforthematrixB isderived.Todoso,letusassumethat

λ

N =

λ

⁽ⁱ⁾

,

∀ⁱ

,

1≤ⁱ≤N^{,sothatthematrix}

(

A−

λ

N

)

isinvertible.Here

λ

⁽ⁱ⁾ denotesthei-theigenvalueof A.Thus,reintroducingtheresidualrN intoEq. (2.7) gives:

λ

_N

− λ =

^uN

, (

A

− λ)

u_N

=

^rN

, (

A

− λ

_N

)

⁻¹

(

A

− λ)(

A

− λ

_N

)

⁻¹r_N

.

(2.8) Bydirectidentification,thematrixBreads:

B

= (

A

− λ

N

)

⁻¹

(

A

− λ)(

A

− λ

N

)

⁻¹

.

(2.9)

Note that,inthisform, thematrix B cannot be directlyused,sincetheeigenvalue

λ

isinvolvedinits definition.Second, anapproximationoftheoperatorBisderivedinordertoobtainacomputableandcertifiedapproximationoftheerror.Let a

,

b∈R^{betwoscalars(whoseoptimalvalueswillbeprecisedlater).Then,thematrix Bis}approximatedasfollows

B

˜ := (

A

−

^b

)

⁻¹

(

A

−

^a

)(

A

−

^b

)

⁻¹

,

(2.10)

so that a

,

b can be seenastwo approximationsoftheeigenvalues

λ

and

λ

N andthey aretwo shiftsforthe spectrumof A−

λ

andA−

λ

N respectively.

2.2. Aprioriestimationforeigenvalues

The proposeda posteriorierrorestimatorisdefinedbyexploitingan availablea priorierrorestimator.Severalofsuch estimatorsexistintheliterature,andcanbeadaptedtotheproblemofinterest.Forthepresentwork,anaprioriestimator based ontraces isused, presentedin[9].Since inthe presentwork,positive matricesare considered, thea priorilower valuefor

λ

isdefinedas:

λ ˜ =

max

{

0

, λ ˜

W S

},

(2.11)

where

λ

˜W Sdenotestheresultoftheestimationproposedin[9],whichisnotalwaysguaranteedtobepositive.

3. Analysisoftheestimator:mainresults

Anerrorestimatorissaidtobecertifiediftheestimatederrorisalwayslargerthantheactualerror,anditissaidtobe sharp iftheestimatederroris ascloseaspossible(in some sense dependingon theproblem)to theactual error.Inthis section,we giveaprecisedefinitionoftheproposed aposteriorierrorestimatorandinvestigateunderwhichconditionsit iscertifiedandsharp.

Definition3.1.Theactualerror

ε

:=

λ

−

λ

N isestimatedbythequantity

μ (

a

,

b

) :=

^rN

,

BrN

,

whereBisdefinedin (2.10) andthescalarsa

,

b∈R^dependon

λ

N andonapriorilowerandupperbounds

λ

˜≤

λ

≤ ˜

λ

₊. Wedescribe,inthesequel,howtochoosea

,

b inordertoensurethat

μ (

a

,

b

)

iscertifiedandsharp.

3.1. Certification

Thegoalofthissectionistodeterminethevaluesofaandb suchthat

:= μ (

a

,

b

) − ε =

rN

, (

_B

˜ −

B

)

rN

≥

0

.

(3.1)

Since A isHermitian and B

,

B˜ areobtainedby shifts of A, they sharethesame eigenbasisandthey commute. Hence, it followsthat:

rN

, (

B

˜ −

B

)

rN

=

^N

i=1

rN

,

u⁽ⁱ⁾

²

λ

⁽ⁱ⁾

−

^a

(

b

− λ

⁽ⁱ⁾

)

²

− λ

⁽ⁱ⁾

− λ (λ

N

− λ

⁽ⁱ⁾

)

²

,

(3.2)

where

λ

⁽ⁱ⁾andu⁽ⁱ⁾denoterespectivelythei-theigenvalueandeigenvectorofA.Thus,asufficientconditionfortheestima- tor

μ (

a

,

b

)

tobecertifiedis:

λ

⁽ⁱ⁾

−

^a

(

b

− λ

⁽ⁱ⁾

)

²

− λ

⁽ⁱ⁾

− λ

(λ

N

− λ

⁽ⁱ⁾

)

²

≥

⁰

, ∀

¹

≤

ⁱ

≤

N

.

(3.3)

We shallnowdeterminethevaluesofa andb sothat(3.3) issatisfied.Tothisaim,let

λ

˜≤

λ

be thea priorilower bound definedinsection2.2andlet:

x_i

:= λ

_N

− λ

⁽ⁱ⁾

,

(3.4)

α _:=

a

− λ

_N

, β :=

^b

− λ

_N

, ε _:= λ

N

− λ,

˜

ε _:= λ

N

− ˜ λ.

AftersubstitutionintoEq. (3.3),thefollowingisobtainedforevery1≤ⁱ≤N Q

(

x_i

; α , β) = − α −

^xi

(β +

^xi

)

²

− ε −

^xi

x²_i

≥

⁰

.

(3.5)

We show that there existvalues of

α , β, ε

˜ andconsequently values ofa and b such that the estimator iscertified. We introducethefunction

( α , β) ∈ R

²

→

J

( α , β) := β

²

[ β

²

+

⁴

ε ˜ (β − α ) ]

[

2

β − α − ˜ ε ]

^{2 (3.6)}

andtheset

T

:= {( α , β) ∈ R

²

|

J

( α , β) ≤

^x2_i

, ∀

¹

≤

ⁱ

≤

N

}.

Thus,thefollowingholds.

Proposition3.2.If

( α , β)

∈T ^{thentheestimator}

μ ( α

₊

λ

N

, β

+

λ

N

)

iscertified.

TheproofofthepropositionispresentedinAppendixA.Letusremarkthat,intheexpressionofJ^{,onlyquantitiesthat} canactuallybecomputedappear.Moreover,itholdsthatJ →^{0 as}

β

→^{0.Thismeansthat,forvaluesofb thatarenottoo} farfrom

λ

N,theestimatoriscertified.

3.2. Sharpness

Tostudythesharpnessoftheestimator,anupperaprioribound

λ

˜⁺≥

λ

isintroducedandassumedtosatisfy

λ

˜⁺

> λ

N≥

λ

. Letusrecallthatthedifferencebetweentheestimatederror

μ (

a

,

b

)

andtheactualerror

ε

is

=

μ

−

ε

(definedinEq. (3.1)).

Itholds:

=

^N

i=¹

^rN

,

u⁽ⁱ⁾

^2Q

(

x_i

; α , β),

(3.7)

where theform Q is definedin (3.5). Sincewe donot control the term^rN

,

u^{(i),we consider the} minimization ofthe formQ,inaworst-casesense.Letx₀

,

x₁∈R^suchthatx0

<

x₁andx₀

<

−β ^{andconsidertheintervalI}=R\ [^x0

,

x₁]^.Thus, thefollowingholds:

≤

sup

x∈I

Q

(

x

; α , β)

^rN

²2

.

(3.8)

Thetermsup_x∈IQ

(

x;

α , β)

canbecomputedexplicitly.Itisreducedtofindthezerosofacubicpolynomial(detailsinthe proof).However,togetsomeinsightintotheestimation,someapproximationsareintroduced.Letusconsiderthefunction

K

: R

^y

→

^K

(

y

; α , β) =

²

β − α

(

y

+ β)

²

+ β

² y

(

y

+ β)

²

,

wheretheparameters

α

and

β

aredefinedin (3.4).Thesecondmainresultreadsasfollows.

Proposition3.3.Letx0=^β(β_(2β−^−˜_α^ε)₎^andx1

>

x0andconsidertheintervalI=R

/

[^x0

,

x1]^.Thus,

β +

2

β − α ₊

2

ε ˜

K

(

x0

; α , β)

1/2

≤ ˜ λ

⁺

− λ

N

⇒

^sup

x∈I

Q

(

x

; α , β) ≤

^K

(

x0

; α , β).

Moreover,theoptimalvalues

α

^∗and

β

^∗arethesolutiontotheconstrainedminimizationproblem

( α

^∗

, β

^∗

) ∈

^argmin

(α,β)∈R²

K

(

x₀

; α , β)

(3.9)

β +

(

2

β − α +

²

ε ˜ )

K

(

x₀

; α , β)

1/2

≤ ˜ λ

⁺

− λ

_N

.

(3.10)

Theproof ofthepropositionispresentedinAppendixB.Wepointoutthatall thequantitiesappearingcanbeactually computed.

3.3. Asimplifiedversion

Inthissection,a simplifiedversionispresented,forwhichsomeestimationcan beperformedanalytically. Letk∈R^+. Considerthefollowingshifts:

b

= (

1

+

^k

)λ

N

,

(3.11)

a

= ˜ λ +

^k

λ

N

.

Thisimplies:

β =

^k

λ

N

,

(3.12)

α = β − ˜ ε .

Hence,Eq. (3.9) canbeexpressedasafunctionof

β

only.Thefollowingpropositionholds.

Proposition3.4.When

α , β

aredefinedasinEq.(3.12),

≤

⁶

^rN

²₂

λ

N

+ ˜ λ .

TheproofofthepropositionispresentedinAppendixC.Remarkthat,forasufficientlysmallvalueofk,theestimatoris certified,sincek→^{0 implies}

β

→^0.

4. Implementationandnumericalexperiments

Inthissection, anefficientimplementationoftheproposedestimatorisdescribedandsomenumericalexperimentsare proposedtoassessitsproperties.

4.1. Efficientimplementation

The numerical implementation of the estimator is discussed in this section. One of the desirable properties of an a posterioriestimatoristobecheapfromacomputationalpointofview.Theexpressionofthepresentestimatoris:

μ =

^rN

, (

A

−

^b

)

⁻¹

(

A

−

^a

)(

A

−

^b

)

⁻¹r_N

.

^(4.1)

Thisexpressioninvolvestheinverseofthematrix A−^{b.Evenifthematrix A issparse(whichisthecaseformostofthe} applications),thecomputationoftheinversewouldbeprohibitive.Instead,thefollowingcomputationisperformed:

(

A

−

^b

)

yN

=

^rN

,

(4.2)

μ ₌

yN

, (

A

−

^a

)

yN

,

(4.3)

hintingthatalinearsystemfor y_N hastobesolved.Thisischeaperthancomputingtheinverse.Inorderforittobeeven cheaper,an iterativemethodisused.Since A isHermitian,itseigenvalues arereal,sothat,sinceb induces asimpleshift in theeigenvalues, their imaginarypart remains zero.Thus, iterations ofthebi-conjugate gradient stabilizedmethodare effective[4].Asan initialguessfortheiteration,we take y⁽_N⁰⁾=^uN.Thisisaparticularlygoodguess,whichwouldbe the exact solution ifb=

λ

. Bydoing so, onlyfew matrix vector products are actually neededto havea goodapproximation of yN.Thecostofthisoperationislow,especiallywhen A issparse.Whensolvingtheproblem,itisimportanttoreachan accurateapproximationofthequantity

μ

ratherthan yN perse.Tothisend,let

μ

^(k)bethevalueof

μ

obtainedatthek-th iterationofthebi-conjugategradientstabilizedmethod;anappropriatestoppingcriterionis|

μ

^(k+1)−

μ

^(k)|

< δ

,where

δ

is auser-definedtolerancethatcanbechosentofixacertainnumberofdigitsintheapproximationof

μ

.

Table 1

Errorestimatorforthees- timator derived from the resultsofProposition3.3.

N ε μ

200 6.95 7.74 250 5.44 6.01 300 3.82 4.14 350 2.94 3.14 400 1.89 1.92

Table 2

ErroranderrorestimationfordifferentvaluesoftheparameterkandbasissizeN.

k=0.01 k=0.05 k=0.1 k=0.2

N ε μ N ε μ N ε μ N ε μ

200 6.72 7.71 200 6.81 7.89 200 6.84 8.26 200 6.74 9.51

250 5.36 6.31 250 5.38 6.18 250 5.44 6.26 250 5.28 6.28

300 4.14 5.08 300 4.12 4.81 300 4.16 4.68 300 4.05 4.47

350 2.98 3.91 350 2.99 3.64 350 2.99 3.40 350 2.97 3.09

400 1.93 2.85 400 1.93 2.58 400 1.94 2.33 400 1.95 2.08

4.2. Randommatrixsmallesteigenvalue

Forthisfirstsynthetictest,wecreatearandommatrix A˜ ∈R^{N×N with}N=^{500,andconsiderthepositivesymmetric} matrix Adefinedas:

A

=

^{c I}

+

¹ 2

A

˜ + ˜

^{A with c}

=

¹

− λ

min

,

(4.4)

where

λ

_min is the smallesteigenvalue of thematrix A˜ + ˜^{A. As a} consequence,the matrix A is Hermitian andpositive definite,anditssmallesteigenvalue,whoseestimationerrorisinvestigated,isequalto1.Then,theeigenvalueproblemis projected inthecoarse basis consistingofthefirst N vectorsofthe canonicalbasis with N

<

N^{,sothat projecting A is} equivalenttotakethefirstN rowsandcolumnsofA.

Since the matrix is random, 32 samples were taken, and foreach of them the test was performed. The values N= [²⁰⁰

,

250

,

300

,

350

,

400]^{wereusedtocomputean}approximationofthespectrum.

TheestimatorobtainedasaresultofProposition3.3wasimplementedandtested.Theoptimisationproblemwassolved byusingastandardglobaloptimisationalgorithm.TheresultispresentedinTable1.

The estimatordescribedin Section3.3was used fordifferentvaluesofk= [⁰

.

01

,

0

.

05

,

0

.

1

,

0

.

2]^{.The meanofthetrue} errorandthe meanof its aposteriori estimationare reportedin Table2.The estimator israthersharp, andthisistrue in generalforall the valuesofk and N. Aninteresting trend can be observed.For betterdiscretizations (higher N), the parameterk that allowsforbetter estimationsis larger.Thisis inaccordancewiththetheoretical estimationprovidedin Eq. (5.18).Indeed,forbetterdiscretization,

λ

N iscloserto

λ

,meaningthat,thelarger N,thelower

λ

N.Consequently,the ratio ^λ_λ^˜+

N becomeslargerandsodoestheoptimalvalueofk.

The results ofthe simplified estimator are, ingeneral, lesssharp than theestimator obtained by solving the optimi- sationproblem. However, thelatter,despite thefact thatonly twoscalars haveto bedetermined, mightbe cumbersome to besolved for,because ofthelarge numberof inequalityconstraints involved.Thus, thesimplified estimatorisa good trade-offbetweensimplicityandsharpness.IntheapplicationtotheSchrödingeroperator,onlytheresultsforthesimpli- fiedestimatorarereported.Thestudyofthedefinitionofestimatorsbasedonoptimisationproblemssimilartotheonein Proposition3.3willbetheobjectoffurtherinvestigations.

4.3. LowestenergybandsofaperiodicSchrödingeroperator

Thesecondtestcaseisrelatedtotheapplicationthatfirstmotivatedthiswork,theapproximationofthelowerenergy statesoftheSchrödingeroperator.Forall j∈Z^,letej

(

x

)

:=^√1_2π^eijx.Foralls∈N^{∗,letusdefine}

X_s

:=

^Span

e_j

|

^j

∈ Z, |

^j

| ≤

^s

(4.5) anddenoteby Ns:=^2s+^{1 thedimensionof X}s andby Xs:^L2per→^Xs the L²_per

(

0

,

2π

)

orthogonal projectoronto Xs.We considerthe2π^{-periodicrealvaluedpotential V} ∈^L2per

(

0

,

2π)^{(plottedinFig.1a)anddefinedbytheFourierexpansion}

V

(

x

) =

3 j=−³

V

ˆ

_je_j

,

V

ˆ

₀

=

^{2 and} V

ˆ

_j

=

¹

+

⁰

.

5 i

,

V

ˆ

_−j

= ˆ

^Vj

, ∀

^j

>

0

.

Fig. 1.A posteriori error estimator for the three lowest energy bands of a one-dimensional periodic Schrödinger operator.

Theabsolutelycontinuousspectrum(see[6] forthedetails)oftheperiodicShcrödingeroperator A= −_dx^d2+^{V isobtained} astheunionofthediscretespectraoftheBloch operators Aq:=

(

−ⁱ_dx^d +^q

)

²+^V,whereq ∈ [−¹

/

2

,

1

/

2[^{.Thus,forevery} q∈ [−¹

/

2

,

1

/

2[^{,aneigenvalueproblem}

A_qu

= λ

u (4.6)

issolvedintheHilbertspaceU=^H1per

(

0

,

2π

)

.Thesolutionstotheeigenvalueproblem(4.6) arenumericallyapproximated using a Galerkin methodin Fourierspace Xs. Forall s∈N^{∗,we denote by}

λ

_q^s_,1≤ · · · ≤

λ

_q^s_,Ns the eigenvalues (rankedin increasing order,countingmultiplicity)oftheoperator A^s_q:= XsA_q ^∗_X

s.ForQ ∈N^{∗,weintroducearegular}discretization gridoftheBrillouinzone[−¹

/

2

,

1

/

2]^anddenoteby

^∗_Q := {^q0= −¹₂

,

q1

,

q2

,

· · ·

,

qQ =¹₂}^.

In the presenttest, we investigate the aposteriori errorestimator presented inSection 3.3 on thefirst three energy bandsq∈

^∗_Q →

λ

^s_q,mform=¹

,

2

,

3,wherethefinediscretizationoftheoperatorisobtainedwithN=^{501 andthecoarse} discretization with N=13 corresponding respectively to sref=^{250 and s}=^{6. The a posterioriestimation of the error}

ε

q,m=

λ

^s_q,m−

λ

q^sref,m forthepointsq intheBrillouinzoneandthefirstthreebands(m=¹

,

2

,

3)isdenotedby

μ

^s_m,q.InFig.1 areplottedthetrueerror

ε

q,m=

λ

_q^s_,m−

λ

^sq^ref,mandtheaposteriorierrorestimator

μ

_m^s_,q computedasexplainedpreviouslyfor differentvaluesofthecoefficientk

>

0,namelyk=⁰

.

1

,

0

.

5

,

1.

5. Conclusionsandperspectives

Anaposteriorierrorestimatorhasbeenproposed,basedontheresidualandonthedefinitionofshiftsofthespectrum ofthematrix.Itisshowntobeconditionallycertified,anditprovidesasharpestimation.Futuredirectionsofinvestigation consistinextendingthistypeofestimatortonon-positiveandpotentiallynon-Hermitianeigenvalueproblems,andtoapply thistorealisticcases.

Acknowledgments

ThepresentworkwouldnothavebeenpossiblewithoutfruitfuldiscussionswithVirginieEhrlacherandDavidGontier.

Appendix A. ProofofProposition3.2

The aimisto provethat if

( α , β)

belongsto theset T^{,then theestimator}

μ (

a

,

b

)

iscertified wherea=

α

+

λ

N and

β

=^b+

λ

N. As we already mentioned in Section 3.1, a sufficient condition for the estimatorto be certified is that the functionR^x→^Q

(

x;

α , β)

definedin (3.5) innonnegative.

Letassumethatx =^{0 and}

β

= −^{x.Thefirstcondition}correspondstothefactthattheestimatedeigenvalue

λ

N isexactly equaltooneofthetrueeigenvaluesof A.Thisisrelatedtotheapproximationoftheproblem,nottotheestimator.Itcan happeneitherif N istoosmallandtheapproximation ispoor,orif

λ

N=

λ

.Thislattercase, whichcorrespondstoa zero error,issolvable.Indeed,thedenominatorinQ vanishes,buttheresidualisorthogonaltotheeigenvector,thusmakingthe wholetermvanishingintheseriesinEq. (3.2).Thecondition

β

= −^{xisanactualcondition forb,whichwillbeanalyzed} lateron.Thisconditionimpactsthesharpnessmorethanthefactthatthemethodiscertified.

Theexpressionof Q isdeveloped,leadingto

Q

(

x

; α , β) = (

2

β − α ₋ ε )

x²

− β(

2

ε ₋ β)

x

− ε β

²

x²

(β +

^x

)

²

.

(5.1)

ThesignofQ isdeterminedbythesignofthenumerator.Thediscriminantofthenumeratoris

( α , β) = β

²

(

2

ε ₋ β)

²

+

⁴

ε β

²

(

2

β − α ₋ ε ) = β

²

[ β

²

+

⁴

ε (β − α ) ] .

(5.2) TwocasesarepossibleforQ tobenonnegative.Thefirstcaseiswhen

( α , β) <

0 and

(

2

β

−

α

₋

ε )

≥^{0.Thevaluesof}

α

and

β

givenbythissituationarenotusefulinourcontext.Thesecondcase,whichismoreinteresting,iswhen

( α , β)

≥⁰ and

(

2

β

−

α

₋

ε )

≥^0.

Assumethat

(

2

β

−

α

−

ε )

≥^{0,whichtranslatesto}

2

β − α ≥ ε ⇐

2

β − α ≥ λ

N

− ˜ λ,

(5.3)

meaning that, if a

,

b are chosen such that 2

β

−

α

is larger than the difference betweenthe estimated value andthe a priori lowerbound, then thecondition will be automaticallysatisfied.Let x_1,2

,

x₂≤^x1 denotethetwo real zerosofthe numerator:

x1,2

= β(

2

ε ₋ β) ± β [ β

²

+

⁴

ε (β − α ) ]

^1/2

2

[

2

β − α − ε ] .

(5.4)

Thus,Q

(

x;

α , β)

isnonnegativeifx∈R\

(

x2

,

x1

)

, whichautomaticallyimpliesthatx =^{0,sincethezerosareofoppositesign.}

Observethatthecase

β

=⁰ ⇒ ^b=

λ

N leadstoacertifiedestimatorx1,2=^{0,alongwiththecondition x} =^0.Nevertheless, thisestimatorisnotsharp.

Finally,weconsiderthegapbetweenthetwozeros:

(

x₁

−

^x2

)

²

= β

²

[β

²

+

⁴

ε (β − α )]

[

²

β − α ₋ ε _]

²

^,

^(5.5)

whichcannotbecomputed,sinceitdependsupon

ε

.Nevertheless,thefollowingholds

(

x₁

−

^x2

)

²

≤ β

²

[β

²

+

⁴

ε ˜ (β − α )]

[

²

β − α _{− ˜} ε _]

²

⁼

^J

⁽ α , β),

(5.6)

whereJ ^{wasdefinedinEq. (3.6).Toconcludetheproof,ifx2}≥

(

x₁−^x2

)

²thenx∈R\

(

x₂

,

x₁

)

,whichensuresthatQ

(

x;

α , β)

isnonnegativeimplyingthattheestimator

μ ( α

+

λ

N

, β

+

λ

N

)

iscertified.

Fig. 2.Pictureofthefunctionsinvolved:thefunctionQ(x)isinsolidblackline,thefunctionsK,K₋areindashedredline;thepointsx0,x1,2andx_∗are depictedaswell.

Appendix B. ProofofProposition3.3

The objectiveofthisproof isto providea boundforthe infinitynorm of Q

(

x;

α , β)

definedinEq. (3.5). Theproof is dividedintotwostepsconsistinginstudyingthefunction Q (seeFig.2)ontwointervals,namelyx

>

x₁ and−∞

<

x

<

x_∗, wherex_∗ willbedefinedlateron.

First,letx

>

x₁.Letusbound Q fromabovebymeansofamonotonicallynon-increasingfunction.Itholds:

Q

(

x

; α , β) = (

2

β − α ₋ ε )

x²

− β(

2

ε ₋ β)

x

− ε β

²

x²

(β +

x

)

²

≤

²

β − α

(

x

+ β)

²

+ β

²

x

(

x

+ β)

²

:=

K

(

x

).

(5.7)

The maximum of Q

(

x

)

, inthe interval x

>

x1 is reachedfora point x, whichis strictly largerthan x1, since Q

(

x1

)

=^0.

Furthermore,onecan easilycheckthat

∂

xK≤^{0,sothat K is}monotonicallynon-increasing.Thus, ifx0

<

x1,then K

(

x0

) >

K

(

x₁

) >

Q

(

x

),

∀^x

>

x₁.

Apoint x0

<

x1 isprovidedby x0=^β(β_(2β−^−˜_α^ε)₎^{.Thus,theinfinitynormof Q,onthe intervalx}

>

x1,isboundedby K

(

x0

)

, whichisafunctionof

α , β, ε

˜.

The second partof the proof consistsin studying Q forx

<

−β. This interval isrelevant, especially when thelower eigenvalues areestimated. Indeed,x=

λ

N−

λ

⁽ⁱ⁾,sothat mostofthe xarenegative.Let

λ

bethen-theigenvalue,theone wearecurrentlyestimating,andlet

λ

⁺bethesuccessiveeigenvaluedifferentfrom

λ

,i.e.

λ

⁺

> λ

.

First,itcanbecheckedthat:

Q

≤

²

β − α ₊

2

ε ˜ )

(

x

+ β)

²

:=

^K−

(

x

),

(5.8)

byconsideringthatx

<

−

β

and

ε

˜

> ε

. Apointx_∗islookedfor,suchthat:

K₋

(

x^∗

) =

K

(

x0

),

(5.9)

leadingto:

x_∗

= − β ± (

2

β − α ₊

2

ε ˜ )

^1/2

K^1/2

(

x0

) .

(5.10)

Tobesurethattherearenovaluesinproximityoftheasymptotes,thesignminuscanbepicked.

Toconclude,ifx

<

x_∗,then Q

(

x

)

≤^K−

(

x

)

≤^K−

(

x^∗

)

=^K

(

x₀

)

andhence^Q∞≤^K

(

x₀

)

intheinterval

(−∞

≤^x≤^x∗

)

∪

(

x≥^x1

)

.

Lastly, theconditionrelating

( α , β)

to x

<

x^∗ isdetailed. Letusconsiderthat, thexcloserto−

β

is theoneforwhich x=

λ

N−

λ

⁺.Let

λ

˜⁺be alowerbound fortheeigenvalue

λ

⁺,that, werecall,itisthefirstsuccessiveeigenvaluedifferent from

λ

.Then,itholds:

β + (

2

β − α +

²

ε ˜ )

^1/2

K^1/2

(

x₀

) ≤ ˜ λ

⁺

− λ

_N

.

(5.11)

Appendix C. ProofoftheProposition3.4

Thevaluesof

α , β

forthesimplifiedestimatoraresubstitutedintotheSystem(3.9),leadingto:

x0

= β(β − ˜ ε )

(

3

β − ˜ ε ) ,

(5.12)

K

(

x₀

; α , β) =

³

β − ˜ ε

(

x0

+ β)

²

+ β

²

x0

(

x0

+ β)

²

,

(5.13)

β + (

3

β + ˜ ε )

^1/2

K^1/2

(

x0

) ≤ ˜ λ

⁺

− λ

N

.

(5.14)

Attheexpenseofalittlebitofsharpness,thissystemcanbefurthersimplified:

x0

= (β − ˜ ε )

3

,

(5.15)

K

(

x₀

; α , β) ≤

⁶

β − ˜ ε ^,

^(5.16)

β ≤

¹

1

+

²

/ √

6

λ ˜

⁺

− λ

_N

.

(5.17)

Thus,iftheestimatedgapbetween

λ

andthesuccessiveeigenvaluedifferentfrom

λ

is

λ

˜⁺−

λ

N,then,theoptimal

β

is:

β =

¹

1

+

2

/ √

6

λ ˜

⁺

− λ

N

⇒

k

=

¹ 1

+

2

/ √

6

λ ˜

⁺

λ

N

−

1

.

(5.18)

Forthis:

≤

⁶

rN

²₂

(

1

+

²

/ √

6

)˜ λ

⁺

−

²

/ √

6

λ

N

+ ˜ λ ≤

⁶

rN

²₂

λ

N

+ ˜ λ .

(5.19)

Thelastinequalityisderivedbyconsideringthat

λ

˜⁺

> λ

N. References

[1]I.Babuška,W.C.Rheinboldt,A-posteriorierrorestimatesforthefiniteelementmethod,Int.J.Numer.MethodsEng.12 (10)(1978)1597–1615.

[2]R.G.Durán,C.Padra,R.Rodríguez,Aposteriorierrorestimatesforthefiniteelementapproximationofeigenvalueproblems,Math.ModelsMethods Appl.Sci.13 (08)(2003)1219–1229.

[3]V.Heuveline,R.Rannacher,Aposteriorierrorcontrol forfiniteelementapproximationsofellipticeigenvalueproblems,Adv.Comput.Math.15 (1) (2001)107–138.

[4]C.T.Kelley,IterativeMethodsforLinearandNonlinearEquations,SocietyforIndustrialandAppliedMathematics,1995.

[5]Y.Maday,A.T.Patera,J.Peraire,Ageneralformulationforaposterioriboundsforoutputfunctionalsofpartialdifferentialequations;applicationtothe eigenvalueproblem,C.R.Acad.Sci.Paris,Ser.I328 (9)(1999)823–828.

[6]M.Reed,B.Simon,MethodsofModernMathematicalPhysics.IV:AnalysisofOperators,Elsevier,1978.

[7]D.V.Rovas,Reduced-BasisOutputBoundMethodsforParametrizedPartialDifferentialEquations, PhDthesis,MassachusettsInstituteofTechnology, Cambridge,MA,USA,2003.

[8]G.Rozza,D.B.PhuongHuynh,A.Manzoni,Reducedbasisapproximationandaposteriorierrorestimationforstokesflowsinparametrizedgeometries:

rolesoftheinf-supstabilityconstants,Numer.Math.125 (1)(2013)115–152.

[9]H.Wolkowicz,G.P.Styan,Boundsforeigenvaluesusingtraces,LinearAlgebraAppl.29(1980)471–506.