Time-local discretization of fractional and related diffusive operators using Gaussian quadrature with applications

an author's

https://oatao.univ-toulouse.fr/21827

https://doi.org/10.1016/j.apnum.2018.12.003

Monteghetti, Florian and Matignon, Denis and Piot, Estelle Time-local discretization of fractional and related

diffusive operators using Gaussian quadrature with applications. (2020) Applied Numerical Mathematics, 155. 73-92.

ISSN 0168-9274

Time-local

discretization

of

fractional

and

related

diffusive

operators

using

Gaussian

quadrature

with

applications

Florian Monteghetti

a

,

∗

,

Denis Matignon

b

,

Estelle Piot

a a_{ONERA/DMPE,UniversitédeToulouse,31055Toulouse,France}

b_{ISAE-SUPAERO,UniversitédeToulouse,31055Toulouse,France}

a

b

s

t

r

a

c

t

Keywords: Fractionalderivative Fractionalcalculus Diffusiverepresentation Eigenvalueproblems Non-classicalmethod Completelymonotonekernel

This paper investigates the time-local discretization, using Gaussian quadrature, of a classofdiffusiveoperatorsthatincludesfractionaloperators,forapplication infractional differential equations and related eigenvalue problems. A discretization based on the Gauss–Legendrequadratureruleisanalyzedboththeoreticallyandnumerically.Numerical comparisons with both optimization-based and quadrature-based methods highlight its applicability. In addition, it is shown, on the example of a fractional delaydifferential equation,thatquadrature-baseddiscretizationmethodsarespectrallycorrect,i.e.thatthey yieldanunpollutedandconvergentapproximationoftheessentialspectrumlinkedtothe fractionalderivative,bycontrastwithoptimization-basedmethodsthatcanyieldpolluted spectrawhoseconvergenceisdifficulttoassess.

1. Introduction

Thebroadfocusofthisarticleisthediscretizationoffractionaloperators usingtheir so-calleddiffusive representation, forapplicationintime-domaincomputationsoreigenvalueproblems.

The diffusiverepresentation offractional operators enablesto recastthem intoan observerofan infinite-dimensional ODE:thelongmemoryoftheoperatorisreflectedintheinfinitedimensionofthecorresponding statespace.Convolution operators that admit such a representation,known asdiffusive operators, havea locallyintegrablecompletely monotone kernel.See[41,40] fordefinitionsoffractionaloperators,[39,6] foran introductiontotheclassofdiffusiveoperators, [21] forexamplesofdiffusiveoperators,and[11,44] forasemigroupformulationofthestate-spacerepresentationinthecontext ofVolterraequations.

Providedthat thediffusive representationissuitablydiscretized,it constitutesatime-local alternativeto,forinstance, fractional linear multistep methods [28] or methods based on the Grünwald–Letnikov approximation [42]. Existing dis-cretizationmethodsforthediffusiverepresentationcanbesplitintotwocategories:methodsthat relyonanoptimization (hereinafter“optimization-based” methods)andpurely analytical methods based onknown quadraturerules (hereinafter “quadrature-based” methods). Note that methods based on discrete diffusive representations are also known as “non-classical”methods[12,4].

In [20], which deals witha fractional monodimensional wave equation, thefractional integral is splitinto two parts, namelyalocalandahistoricalone:whiletheformerisapproximatedadhoc,aGauss–Legendrequadratureruleisemployed

*

Correspondingauthor.

E-mailaddresses:fl[email protected](F. Monteghetti),[email protected](D. Matignon),[email protected](E. Piot).

forthe later,see [25] foran analysis. Anotherapproachconsistsindirectlyusinga quadraturerule,withoutanysplit.To get back toa finiteinterval, one caneither truncatethe semi-infiniteintegration domain[2] oruse achange ofvariable [46,12,4].

In [2], Gauss–LegendreandCurtis–Clenshawquadraturerules are usedon atruncateddomain.A methodproposed in [46],based on aGauss–Laguerre quadrature rulewith achange ofvariable, hasbeen widelyinvestigated andled tothe definitions ofmethods basedinstead ontheGauss–Jacobi quadraturerule [12,4],see[4] foracomparisonthat favors [4, Eq. (23)].

Optimization-based methods have also received scrutiny and enjoyeda wide range of applications, notably in wave propagationproblems.Amethodbasedonalinearleastsquaresoptimizationwherethepoledistributionischosenapriori hasbeenintroducedin[16] fortheidentificationofaleadacidbatteryimpedancemodelusingtime-domainmeasurements. Furtherrefinementshavebeenproposedin[21],withapplicationtoawiderangeofdiffusiveoperators,andin[26],where anonlinearleastsquaresiscomparedwiththemethodproposedin[4],mentionedabove.

TheobjectiveofthispaperistoinvestigatethediscretizationofdiffusiverepresentationsusingGaussianquadrature,for application inthe numerical solutionof fractional differential equationsaswell asrelatedeigenvalue problems.Inspired by classical works on numerical integration [10,1], a family of discretization methods that rely on the Gauss–Legendre quadratureruleisintroducedandanalyzedboththeoreticallyandnumerically.Theanalysisenablestopindownthemost suitable method forapplications.Inparticular,it emphasizes thatthe methodmustbe tailoredto thekernelathand,by contrast withaone-size-fits-allapproach.Numerical comparisonswithexisting discretizationmethods,both optimization andquadraturebased,shedlightonthepracticalinterestoftheproposedmethod.Additionally,itisshownonanumerical example that quadrature-baseddiscretization methods are spectrally correct, i.e.that they yield an unpolluted and con-vergent approximationoftheessential spectrum(linkedto thefractional derivative),bycontrastwithoptimization-based methods.

Thispaperisorganizedasfollows.Section2recallselementaryfactsaboutdiffusiverepresentationsandintroduces the proposed Qβ,N discretizationmethod,where

β

isascalarparametertobesuitablychosenandN isthenumberof

quadra-turenodes.Section3presentsananalysisofthemethodinthecaseoffractionaloperators,whichhighlightsthedependency of

β

uponthe orderofthefractional operator. Numericalapplications andcomparisonsare gatheredinSection 4,where the Qβ,N methodiscomparedagainsttwoexisting methods,one optimization-basedandonequadrature-based.Section 5

investigatestheuseofanonlinearleastsquaresminimizationtorefinethepolesandweightsgivenbythe Qβ,N method.

2. Definitionoftheproposedquadrature-baseddiscretizationmethod

The purposeof thissection isto introduce the proposed Qβ,N discretization method, where

β

is a scalar parameter

to be suitably chosen and N is the number ofquadrature nodes.Aftersome backgroundon diffusive representations in Section2.1,themethodisdefinedinSection2.2,namelyinDefinition4.

2.1. Diffusiverepresentation

Inthispaper,weconsiderthediscretizationofso-calleddiffusivekernels,expressedas

h

(

t

)

:=

∞



0

e−ξtH

(

t

)

μ(ξ )

d

ξ

(

t

∈ R),

(1)

where H istheHeavisideorunitstepfunction(H(t

)

=

1 fort

>

0,nullelsewhere)and

μ

∈

C((

0,

∞))

isthediffusiveweight. Bydefinition,diffusivekernelsarelocallyintegrableon

[

0,

∞)

,i.e.h

∈

L1

loc

(

[

0,

∞))

,sothatthediffusiveweightsatisfies

∞



0

μ(ξ )

1

+ ξ

d

ξ <

∞.

Notethat,ingeneral, h isnotintegrableover

(0,

∞)

.Thisclassofkernelsisphysicallylinkedtonon-propagating diffusion phenomena, encounteredinviscoelasticity[11,44,29],electromagnetics[18],andacoustics[38] [35,Chap. 2].See[39,21,6,

26] andreferencesthereinforfurtherbackgroundondiffusiverepresentationsandtheirapplications.BydefiningtheLaplace transformas

ˆ

h

(

s

)

:=

∞



0 h

(

t

)

e−stds

(

(

s

) >

0

),

theidentity(1) reads

ˆ

h

(

s

)

=

∞



0

μ(ξ )

s

+ ξ

d

ξ.

Remark1.As definedherein,a diffusive kernelisa locallyintegrablecompletely monotone kernelon

(0,

∞)

.A diffusive kernelh isintegrableon

(0,

∞)

ifandonlyif[19,Thm. 5.2.5]

∞



0

μ(ξ )

ξ

d

ξ <

∞,

whichisnotthecaseforthekernelsconsideredinthispaper,seethethreeexamplesbelow.

Remark2(Terminology).Inthispaper,weusethefollowingterminology:thediffusiverepresentation ofh istheidentity(1), whilethefunction

μ

iscalledthediffusiveweight.Thisslightlydiffersfrom[39] where

μ

iscalledthediffusive representa-tionofh.Thequantity

μ

isalsoknownunderothernamessuchasspectralfunction [18] orrelaxationspectrum [29].

Thecomputationalinterestofdiffusivekernelsisthat,formally,theconvolutionoperatoru

→

h



u admitsthefollowing infinite-dimensionaltime-localrealization

⎧

⎪

⎪

⎨

⎪

⎪

⎩

∂

t

ϕ(

t

, ξ )

= −ξ

ϕ(

t

, ξ )

+

u

(

t

),

ϕ(

0

, ξ )

=

0

(ξ

∈ (

0

,

∞)),

h



u

(

t

)

=

∞



0

ϕ(

t

, ξ )μ(ξ )

d

ξ,

(2)

whereu isacausalinput. Afunctionalframeworkforthisrealizationhasbeenproposed in[11,44]. Letusnowlistthree examplesofdiffusiveoperatorscoveredbythediscretizationmethodintroducedinSection2.2.

1. TheRiemann–Liouvillefractionalintegral,definedas[41,§ 2.3] [32]

Iαu

:=

Yα



u

,

where

α

∈ (

0

,

1

)

andthefractionalkernelis

Yα

(

t

)

:=

H

(

t

)

(α)

t1−α

, ˆ

Yα

(

s

)

=

1

sα

.

(3)

Theassociateddiffusiveweightis

μ

α

(ξ )

:=

sin

(απ

)

πξ

α

.

(4)

2. Anotherdiffusivekernelisthezeroth-orderBesselfunctionofthefirstkind[31,§ 3.3]

J0

(

t

)

H

(

t

)

= +

eit ∞



0

μ

1_/2

(ξ )

√

−ξ +

2ie −ξt_d

_ξ

₊

_e−it ∞



0

μ

1_/2

(ξ )

√

−ξ −

2ie −ξt_d

_ξ

₍₅₎

= +

2



⎡

⎣

eit ∞



0

μ

1/2

(ξ )

√

−ξ +

2ie −ξt_d

_ξ

⎤

⎦ ,

where

μ

1/2 isgivenby(4) andi istheunitimaginarynumber.

3. ThefractionalCaputoderivative,definedas[5] [40,§ 2.4.1] [32]

dαu

:=

I1−αu

˙

,

(6)

whereu is

˙

thestrongderivative.Itformallyadmitstheinfinite-dimensionaltime-domainrealization(contrastwith(2))

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

∂

t

ϕ(

t

, ξ )

= −ξ

ϕ(

t

, ξ )

+

u

(

t

),

ϕ(

0

, ξ )

=

u

(

0

)

ξ

(ξ

∈ (

0

,

∞)),

dαu

(

t

)

=

∞



0

(

−ξ

ϕ(

t

, ξ )

+

u

(

t

))

μ(ξ )

d

ξ,

(7)

whereu isasufficientlyregularcausalinput.Ifu

(0)

=

0,thendα u matchestheRiemann–Liouvillefractionalderivative. Thesethreeconvolutionoperatorscanbediscretizedusingthe Qβ,N method,introducedinSection2.2below.

2.2. Discretizationmethod

Thecausalkernelh givenby(1) isdiscretizedusingN first-orderkernelsas

h

(

t

)

hnum

(

t

)

:=

N



n=1

μ

ne−ξntH

(

t

) (

t

∈ R).

(8)

IntheLaplacedomain,thisreads

ˆ

h

(

s

)

ˆ

hnum

(

s

)

=

N



n=1

μ

n s

+ ξ

n

(

(

s

) >

0

).

In this work, we seekto find an expression for

(ξ

n

,

μ

n

)

that applies atleast to the kernels listed inSection 2.1, whose

diffusiveweights

μ

aremonotoneon

(0,

∞)

withasingularityat

ξ

=

0,whichleadstothefollowingassumption. Assumption3.Thediffusiveweight

μ

∈

C((

0,

∞))

hasapower-lawsingularityat

ξ

=

0,i.e.

μ(ξ )

=

O



1

ξ

α

,

(9) with

α

∈ (

0,1).

Followingclassicalworksonnumericalquadrature[10,Chap. 3] [1,§ 5.6],thefollowingtwomethodscouldbeenvisaged todealwithasingularintegrallike(1).

1. Consider

μ

asaweightfunctionanddefine eitheranewsetofGaussnodes(ifpossible)oranewproductquadrature rulewithequidistantnodes[1,§ 5.6].

2. Recoveracontinuousintegrandusingachangeofvariables.Forexample,forthisintegral,MATLAB®

_integral

_function

usesthechangeofvariable

ξ

=



v 1−v



2

,see[43,§ 4.2].

Tosimplifytheimplementation,wechoosethesecondmethod,i.e.weseekasuitablechangeofvariables

: (−

1

,

1

)

→ (

0

,

∞), (−

1

)

=

0

,

(

1

)

= ∞,

sothattheright-handsideoftheidentity

h

(

t

)

=

1



−1

μ((

v

))

e−(v)t

˙(

v

)

dv (10)

canbeaccuratelydiscretizedusingtheGauss–Legendrequadraturerule

(

vn

,

wn

),

thusyielding

ξ

n

:= (

vn

),

μ

n

:=

wn

˙(

vn

)

μ(ξ

n

),

(11)

where

˙

denotesthederivativeof

.

Giventhesingularitycondition(9),anaturalchoiceis[10,§ 3.1] [1,§ 5.6]

β

(

v

)

:=



1

+

v 1

−

v

1 β

, β >

0

.

(12)

This change of variables results from the composition of v

→

1₁₋+_vv, which maps

(

−

1,1) to

(0,

∞)

, and the power law v

→

v1β_.Using

β,therepresentation(10) reads

h

(

t

)

=

2

β

1



−1 e−t  1+v 1−v 1 β

(

1

−

v

)

−1−1β

₍

₁

+

_v

₎

β1−1

_μ



₁

₊

_v 1

−

v

1 β



dv

,

(13)

whichleadstothedefinitionofthe Qβ,N discretizationmethodgivenbelow.

Definition4.The Qβ,N discretizationof(1) is(8) with

ξ

n

:=



1

+

vn 1

−

vn

1 β

,

μ

n

:=

wn 2

β

(

1

+

vn

)

1 β−1

₍

₁

−

_v n

)

−1− 1 β

_μ

_(ξ

n

) ,

(14)

Intuitively,one mayexpectthebest valuefor

β

to bedependent onpropertiesofthediffusive weight

μ

,suchasthe valueof

α

in(9).Section3investigatesthisforthecaseoffractionaloperators.

3. Analysisforfractionaloperators

Thepurposeofthissectionistoshowthat,forthefractionalkernel(3),thebestpracticalvaluefor

β

isgivenby(22). ThetheoreticalanalysisispresentedinSection3.1andexamplesofapproximationerrorsareprovidedinSection3.2.

3.1. Theoreticalanalysis

Letusrecallthefollowingstandardtheorem.

Theorem5(Convergencerate).Let

(

vn

,

wn

)

betheGauss–Legendrequadratureruleandp anonnegativeinteger.Iff

∈

C

p

(

[−

1,1

])

,

then lim N→∞N p







1



−1 f

(

v

)

dv

−

N



n=1 wnf

(

vn

)





 =

0

.

Inparticular,iff

∈

C

∞

(

[−

1,1

])

thenspectralconvergenceisachieved.

Proof. Since f isatleastcontinuouson

[−

1,1

]

,wehavetheestimate[1,Thm. 5.4]







1



−1 f

(

v

)

dv

−

N



n=1 wnf

(

vn

)





 ≤

4 inf deg q≤2N−1



f

−

q



L∞([−1,1])

.

Theconclusionfollowsfromapolynomialapproximationresult[13,Thm. I.VIII].

Giventheaboveresult,tofindtheoptimalvaluefor

β

inthe Qβ,N discretization,itissufficienttostudytheregularity

oftheintegrandin(13).Letusnowfocusonthefractionalkernel(3),whichisthesimplestkernelthatsatisfies(9).Afirst convergenceresultissummarizedinthepropositionbelow.

2

Proposition6.Let

β >

0,N

∈ N

∗,andYα,numbetheQβ,NdiscretizationofYα with

α

∈ (

0,1).If

β

≤

1

−

α,

(15)

thenYα,num

(

t

)

→

Yα

(

t

)

asN

→ ∞

foranyt

>

0.If,additionally,

1

β

∈ N,

α

β

∈ N,

(16)

then



Yα

(

t

)

−

Yα,num

(

t

)

 =

k→∞

O



n−k



_{foreverypositiveintegerk andt}

_>

_0.

Proof. Thediffusiverepresentation(13) ofthefractionalkernel(3) reads

Y_α

(

t

)

=

1



−1



β

(

t

,

v

)

dv

,

(17) with



β

(

t

,

v

)

:=

2 sin

(απ)

π

β

e −t1+v 1−v 1 β

(

1

−

v

)

−1+α−β1

₍

₁

+

_v

₎

−1+1−βα

_.

Since



β

(

t

,

·)

∈

C

∞

((

−

1,1)),theonlytaskistoinvestigatethesingularitiesat

−

1 and1.Thereisnosingularityatv

=

1 as

longast

>

0,since x

→

e−t  1_{+ 2x}  1 β x1+ 1−βα

isinfinitelydifferentiableat0+,withoutassumptionon

α

and

β

.Sincev

→

e−t 

1+v 1−v

1 β

hasalimitasv

→ −

1+,theintegrand



β

(

t

,

·)

iscontinuous ifandonlyif(15) holds.Furthermore,



β

(

t

,

·)

∈

C

∞

(

[−

1,1

])

FromProposition6,aconvergenceresulton



Yα

− ˜

Yα



L1_(,T)forany



>

0 andT

>



canbereadilydeduced,although

thisisnotsufficientfortime-domaincomputations.Indeed,foraninputu

∈

L2

(0,

T

)

∩

L∞

(0,

T

),

wehavethestraightforward estimate

|

h



u

(

T

)

|



u



L∞(0,T)

≤ 

h



_L1_(0,T)

,

(18)

whichjustifiesaninterestinapproximatingtheL1 _{normofh.Thisrequiresanadditionalconstrainton}

_β

_{,seeProposition7.}

Proposition7.Let

β >

0,N

∈ N

∗,andYα,numbetheQβ,NdiscretizationofYα with

α

∈ (

0,1).Ifboth(15) and

β

≤

α

(19)

hold,thenlimN→∞



Yα,num



L1_(0,T)

= 

Yα



_L1_(0,T)foranyT

>

0.If,additionally,(16) holds,thenthisconvergenceisspectral. Proof. Let T

>

0.Since

α

∈ (

0,1),wehave



β

∈

L1

((0,

T

)

× (−

1,1)).TheFubinitheoremyields



Yα



L1_(0,T)

=

1



−1

β

(

T

,

v

)

dv

,

(20) where

β

(

T

,

v

)

=

2 sin

(απ)

π

β



1

−

e−T  1+v 1−v 1 β



(

1

−

v

)

−1+βα

₍

₁

+

_v

₎

−1− α β

_.

It issufficient to investigatethe regularity of

β

(

T

,

·)

at v

= ±

1.

β

(

T

,

·)

is continuous at 1 ifand only if(19) holds.

Expandingaround v

= −

1 yields



1

−

e−t  1+v 1−v 1 β



(

1

+

v

)

−1−βα

=

_t

₍

₁

+

_v

₎

1− α β −1

₍

₁

−

_v

₎

−β1

+ · · · ,

hencecontinuityat

−

1 isachievedifandonlyif(15) holds.If,additionally,(16) isassumed,then

β

(

T

,

·)

∈

C

∞

(

[−

1,1

])

.

2

Remark 8.Proposition 7 states convergence of the L1 _{norm, but not convergence in the L}1 _{norm, i.e. lim}

N→∞



Yα

−

Yα,num



L1_(0,T)

=

0 (which impliesconvergenceofthe L1 norm).This propositioncan thereforebe deemed insufficientin

viewoftheestimate(18);however,itgivesasecondconstrainton

β,

namely(19),whichispracticallyuseful.

Remark9(Frequencydomain).Asimilar studyin the frequencydomain reachesthe same conclusion.For anys

=

0 with

(

s

)

≥

0,wehave

ˆ

Yα

(

s

)

=

2

β

sin

(απ)

π

1



−1

(

1

−

v

)

αβ−1

₍

₁

+

_v

₎

1−α β −1 s

(

1

−

v

)

β1

+ (

₁

+

_v

₎

1β dv

,

andthe integrand iscontinuous on

[−

1,1

]

ifandonlyif(15, 19) hold. If, furthermore,(16) holds,then theintegrand is infinitelysmoothandwehavespectralconvergencefor







Y

ˆ

α

(

s

)

−

N



n=1

μ

n s

+ ξ

n







.

Forany

ω

m

>

0,wereadilydeducethat



i

ω

Yα

ˆ

(

i

ω

)

−

i

ω



nN=1

μn

iω+ξn



L2(−ωm,ωm) hasthesameconvergenceproperties.

Practicalchoiceof

β

Basedontheaboveresults,thefollowingrulescanbefollowedtochoose

β

inpractice. 1. If

α

∈ (

0,1)

∩ Q

suchthat

α

=

n0 n1 withni

∈ N

∗_,then

β

1

:=

1 n1 (21) satisfiesthecondition (15),(16),and(19) sothatthe Qβ1,N methodyieldsaspectrallyconvergentapproximation.This

valueisalsosuitedfor

α

∈ (

0,1)

∩ (R\Q)

with

α

n0 n1.

Fig. 1. Errors(23,24,25)andmaximumpole(26) forh=Yα withα=58.( )Qβ,N withβ= β1=18.( )Qβ,N withβ= β2.( )Qβ,N with

β= β2×0.99.( )Qβ,Nwithβ= β2×1.01.( )Qβ,Nwithβ= β3.

2. Theconditions(15) and(19) suggestusingalargervalueof

β,

namely

β

2

:=

min

(α,

1

−

α),

(22)

whichyields atleasta convergent approximationfromPropositions6 and7.Section 3.2belowshowsthat

β

2 isthe

mostinterestingchoiceformoderatevaluesofN. 3.2. Numericalillustrations

Toinvestigatenumericallytheinfluenceof

β

ontheconvergenceofthe Qβ,N method,wedefine threeerrors.Thefirst

oneisinthefrequencydomain

ε

_∞,ωm

:=







1

−

ˆ

hnum

ˆ

h

(

i

ω)







L∞(−ωm,ωm)

,

(23)

with

ω

m

>

0 agivenangularfrequency.Thesecondandthirdonesareinthetimedomain,namely

ε

T

:=





1

−

hnum h



(

T

),

ε

1,T

:=



h



_L1_(0,T)

− 

hnum



L1_(0,T)





h



_L1_(0,T)

,

(24)

withT

>

0.Fromnowon,weset

ω

m

=

T

=

104

,

(25)

so that we considerbroadband approximationsof thekernel h.We first considerh

=

Yα, covered by theresults of Sec-tion3.1,withfourvaluesof

α

andthenconcludewithh

=

J0.Computationsaredonewithdoubleprecisionfloatingpoint.

The case h

=

Yα with

α

=

5

8

0.62

≥

1

2 is shown in Fig. 1. The choice

β

1

=

18 achieves spectral convergence, with

saturation at double precision, asexpected from Section 3.1. The value

β

2

=

1

−

α

does not converge spectrally, butit

providesabetterapproximationformoderatevaluesofN.Thevalue

β

3

:=

max

(α,

1

−

α),

whichdoesnotsatisfy(15),istheleastinterestingoption.Thesensitivityoftheerrorsobtainedwith

β

= β

2 ishighlighted

bythecurvescorrespondingto

β

=

0.99

× β

2 and

β

=

1.01

× β

2,whicharesignificantlyworseinthetimedomain.These

error plotshighlight that the time-domain norms do add information: here, the sensitivity to

β

2 cannot be seen inthe

frequencydomainforinstance,whileitistheoppositeforothervaluesof

α

coveredbelow. TheupperrightplotofFig.1givesthemaximumpole

ξ

max

:=

max

n

ξ

n

.

(26)

Thisquantityisespeciallyimportantwhenusinganexplicitschemetoadvancetherealization(2) intime,sincethetime steptypicallyscalesas

O(ξ

max−1

).

Theplotshowsthat

ξ

max

=

O

(

N

2

β

_).

Giventhatthehigher

ξ

max,themorecostlythetimeintegration,onemayexpectthatahighervalueof

ξ

maxsystematically

yields amoreaccurate discretization.However,thisneednot bethecase:althoughthisisindeedverifiedfor

β

1,

β

2,and

Fig. 2. Errors(23,24,25)andmaximumpole(26) forh=Yαwithα=12.( )Qβ,N withβ=12.( )Qβ,Nwithβ= β2×0.99.( )Qβ,Nwith

β= β2×1.01.

Fig. 3. Errors(23,24,25)andmaximumpole(26) forh=Yα withα=27.( )Qβ,N withβ= β1=1₇.( )Qβ,N withβ= β2.( )Qβ,Nwith

β= β2×0.99.( )Qβ,Nwithβ= β2×1.01.( )Qβ,Nwithβ= β3.

Fig. 4. Errors(23,24,25)andmaximumpole(26) forh=Yαwithα=

√ 2−1 √ 2 .( )Qβ,Nwithβ= β1= 1 7.( )Qβ,Nwithβ= β2.( )Qβ,Nwith β= β2×0.99.( )Qβ,Nwithβ= β2×1.01.( )Qβ,Nwithβ= β3.

Fig. 2 plots theerror graphs for

α

=

1

2. Here, the values

β

1,

β

2, and

β

3 are identicalso that the Qβ,N discretization

enjoys spectral convergence,withdoubleprecisionon

ε

T reachedforaround 100variables.The sensitivitytoachangein

β

around

β

2 canbeseeninbothfrequency-domainandtime-domainerrors:althoughthevaluesof

ξ

max remainclose,the

approximationsaresignificantlyworsefor0.99

× β

2 and1.01

× β

2.

Theconclusionsfor

α

=

2

7

0.28

≤

1

2,showninFig.3,areidenticalto

α

=

5

8.Theonlydifferenceisthatthesensitivity

to

β

2isonlyseeninthefrequency-domainnorm

ε

∞,ωm.Fig.4showstheerrorsobtainedfor

α

=

√

2−1

√

2

0.29,avalueclose

to 2₇ butirrational. Themain differenceis theerrorobtainedfor

β

= β

1,whichis lessaccurate inthe frequencydomain

compared to

α

=

2

7.However,thehierarchybetweenthe Qβ,N methodsisidentical,andthe othererrorsare similar.The

choices

β

=

₄₁1 (justified by

α

12

41)and

β

=

1

3 (justified by

α

1

3),notshownhere,deliverpoorerresults.Overall,Fig.4

illustratesthattheirrationalityof

α

isnotamajorconcerninpractice.

Insummary,Figs.1–4show that,formoderatevaluesofN,the Qβ,N methodwith

β

= β

2 deliverssatisfactory

conver-genceresultsforany

α

∈ (

0,1),rationalorirrational.Inaddition,thefactthat

β

2

≥ β

1 impliesthattheQβ2,N methodyields

a lowermaximumpole(26) than Qβ1,N,whichisofparticularinterestfortime-domainsimulations.Thesetwoproperties

Fig. 5. Errors (23,24,25) and maximum pole (26) for h=J0. ( ) Qβ,Nwithβ=12. ( ) Qβ,Nwithβ= β2×0.99. ( ) Qβ,Nwithβ= β2×1.01.

Fig.5givestheerrorsobtainedinapproximatingtheBesselfunction J0,whosediffusiverepresentationisgivenby(5):

the results are similar to that shown inFig. 2 for the fractional kernel oforder 1

_/

2, since the diffusive weights of both kernelshaveasimilarbehavior.

Thecomputationalmeritsofthe Qβ,N-methodwith

β

= β

2 arefurtherinvestigatedinSection4,wherenumerical

appli-cationsaregathered.

4. Numericalapplicationsandcomparisons

Thepurposeofthissectionistoinvestigatethecomputationalpropertiesofthe Qβ,N methodaswellascomparethem

to those oftwo existing methods: one quadrature-based, recalledin Section 4.1,andone optimization-based, recalledin Section4.2.Thecomparisoniscarriedoutintheotherthreesections:Section4.3gathersapproximationerrors,Section4.4

focusesonthesimulationofafractionaldifferentialequation,andSection4.5investigatesspectralcorrectness,whichturns outtobeanimportantfeatureofthe Qβ,N method,and,moregenerally,ofquadrature-basedmethods.

4.1. Birk–Songquadraturemethod

Afterreviewingexistingmethods,notably[46] and[12],BirkandSongproposedthechangeofvariable

ξ

=

β

(

v

)

with

β

=

1₄.However,theyproposetouseaGauss–JacobiquadratureruleinsteadofaGauss–Legendreone(therebyintroducing asingularityatv

= −

1 intheintegrand),whichleadstothediscreterepresentation[4,Eq. (23)]

ξ

n

:=



1

− ˜

vn 1

+ ˜

vn

4

,

μ

n

:=

8 sin

(απ

)

π

˜

wn

(

1

+ ˜

vn

)

4

,

(27)

where

(

v

˜

n

,

w

˜

n

)

istheGauss–Jacobiquadraturerulefortheweightfunctionv

→ (

1

−

v

)

2α+1

(1

+

v

)

−(2α−1)with

α

:=

1

−

2

α

.

(Bewarethat,in[4,Eq. (23)],“

α

”denotestheorderoftheCaputoderivative,whereasherein,

α

istheorderofthefractional integral.)

4.2. Optimizationmethod

Webrieflyrecallheretheoptimizationmethoddefinedin[21,§ 4.3],whichconsistsinaleastsquaresoptimization.The mainchallengeofsuchanoptimizationisthath

ˆ

num isnonlinearwithrespecttothepoles

(ξ

n

)

n,whichfurthermorehavea

widevariationsincetheoretically

ξ

∈ (

0,

∞)

.Toavoidthiscomputationaldifficultythemethodproceedsasfollows. 1. Thethreeinputparameters,namelyN

∈ J

2,

∞J

,

ξ

min

>

0,and

ξ

max

> ξ

minarechosen.

2. TheN poles

ξ

n arelogarithmicallyspacedin

[ξ

min

,

ξ

max

]

:

ξ

n

= ξ

min



ξ

max

ξ

min

n−1 N−1

(

n

∈ J

1

,

N

K).

3. Let A

:=



(

i

ω

k

+ ξ

n

)

−1



k,n

∈ C

K×N _andb

_:=



_h

ˆ

₍

_i

_ω

k

)



k

∈ C

K_{,wheretheK angularfrequencies}

_ω

karealsologarithmically

spacedin

[ξ

min

,

ξ

max

]

.TheN weights

μ

n arecomputedwithalinearleastsquaresminimizationof

J

(μ)

:= 

C

μ

−

d



2₂

=

K



k=1







N



n=1

μ

n i

ω

k

+ ξ

n

− ˆ

h

(

i

ω

k

)







2

,

(28)

Fig. 6. Errors(23,24,25)andmaximumpole(26) forh=Yαwithα=58.( )Qβ,N withβ= β1=18.( )Qβ,N withβ= β2.( )Birk–Song method(27).( )Qβ,Nwithβ=14.

Fig. 7. Errors(23,24,25)andmaximumpole(26) forh=Yα withα=12.( )Qβ,N withβ=12.( )Birk–Songmethod(27).( )Qβ,N with

β=14. C

:=



(

A

)

(

A

)



∈ R

2K×N

_,

_d

_:=



(

b

)

(

b

)



∈ R

2K

_.

Providedthat2K

>

N theproblemisoverdeterminedandcanbedirectlysolvedbyapseudo-inverse.Therealityofthe weights

μ

nisenforcedthroughthedefinitionofC andd,whichseparatesrealandimaginaryparts.However,notethat

thesignofeach

μ

n isunconstrained.

Thistechnique isparticularlysuitedfortime-domainsimulations,where

ξ

max isnaturally known(frome.g. theminimum

acceptabletimesteporthemaximumfrequencyofinterestinwavepropagationproblems);itcanalsohandlemorecomplex representations that involve additional poles. For a given N and

ξ

max, there is usually an optimal range for the lower

bound

ξ

min,whichgovernsthelong-timebehaviorofhnum,whichmustnotbechosen toosmall.Forthediffusive kernels

considered herein, a logarithmic spacing ofthe poles

ξ

n is satisfactory (alinear spacingyields poorerresults). In all the

applicationspresentedinthissection,weset

K

=

104

.

(29)

Remark10.Thereisan inherentdifficulty whencomparing theaboveoptimizationmethodwiththe Qβ,N method,since

both do not havethesame numberof parameters:1forthe Qβ,N method(namely thenumber ofquadraturenodes N,

since

β

hasbeenchosentobe

β

2 followingtheanalysisofSection3), 3fortheoptimizationmethod(namely N and the

minimum and maximum poles

ξ

min and

ξ

max). In all the results presented below, the parameters

ξ

min and

ξ

max of the

optimizationmethodhavebeenempiricallychosentoyieldthebestresults. 4.3. Approximationerrors

InthespiritofSection3.2,herearegatheredcomparisonsoftheapproximationerrors.

ComparisonwiththeBirk–Songmethod. Acomparisonbetweenthe Qβ,N methodandtheBirk–Songmethod(27) isshown

inFig.6for

α

=

5

8 andFig.7for

α

=

1 2.

Let usfirst consider the case

α

=

5

8.The behavior of

ε

∞,ωm highlights the accuracy of the Birk–Songmethod in the

Fig. 8. Errors(23,24,25)andmaximumpole(26) forh=Yαwithα=12.( )Qβ,N withβ= β2.Optimizationwithξmax=104:( )ξmin=10−16, ( )ξmin=10−14,( )ξmin=10−10,( )ξmin=10−6.

Fig. 9. Errors(23,24,25)andmaximumpole(26) forh=Yαwithα=58.( )Qβ,N withβ= β2.Optimizationwithξmax=104:( )ξmin=10−16, ( )ξmin=10−14,( )ξmin=10−10,( )ξmin=10−6.

N

≤

70,asone mayexpect fromthechangeofvariablethatdefinestheBirk–Songmethod.Asimilartrendisseeninthe timedomain,althoughtheretheclosest Qβ,N methodforN

≤

70 isthat obtainedwith

β

= β

2.Allthemethodsarecloser

for

α

=

1

2,atleastinthetime domain,andthe method Qβ,N with

β

=

1₄ hasalmostidenticalconvergencepropertiesto

thatoftheBirk–Songmethod.

Asalreadymentionedwhencomparingthevarious Qβ,N methodsinSection3.2,thegraphsof

ε

∞,ωm,

ε

T,and

ε

1,T alone

arenotsufficienttocomparediscretizationmethods:onemusttakeintoaccountthevalueof

ξ

max,showninthetopright

plotofFigs.6and7.TheseplotsshowthatfortheBirk–Songmethodwehave

ξ

max

=

O(

N 2

β

₎

_with

_β

=

1

4,i.e.

ξ

max

=

O(

N8

),

whichimpliessignificantlylargervaluesthatthe Qβ2,N methodrecommendedfromtheanalysisofSection 3.2.Theimpact

oftheselargevaluesof

ξ

max isofconcernwhenusingexplicittime-marchingscheme,seeSection4.4.

Comparisonwiththeoptimizationmethod. Theerrorsfor

α

=

1

2 and

α

=

5

8 areplottedinFigs.8and9,respectively.Giventhe

resultsofSection 3,onlythe Qβ,N methodwith

β

= β

2 isconsidered.Forthe(three-parameter)optimizationmethod,we

choose

ξ

max

=

ω

m

=

104andplottheerrorsforvariousvaluesof

ξ

min:theresultshowsthattheoptimalvalueof

ξ

mindoes

stronglydependuponN,sothat

ξ

minisnotstraightforwardtochooseapriori.However,providedthatthevalueof

ξ

minis

well-chosen,theoptimizationmethodcanoutperformthe Qβ2,N methodonarangeofN,whichjustifiesitspopularityin

large-scaleapplicationswherethevalueofN iscritical.Notethat,bycontrastwithSection3.2,thecomparisonisrestricted to N

∈ J

1,50

K

,sinceoutsideofthisinterval,themaximumpole

ξ

max ofthe Qβ2,N methodissignificantlylargerthan10

4

sothat thecomparisonwouldnot be fair.Insummary,here, themainadvantage ofthe Qβ2,N-method isthatit hasjust

oneparameter.

Torefinethecomputedpolesandweights,onemayconsidertheuseofanonlinearleastsquaresminimizationbyadding thefollowingfourthsteptothethree-stepoptimizationmethoddescribedinSection 4.2:

4. Compute N newweights

(

μ

n

)

n andpoles

(ξ

n

)

n witha nonlinearleastsquaresminimization oftheright-handsideof

(28),startingfromthepoleschoseninstep2andtheweightsobtainedinstep3,withthefollowinglinearconstraints

μ

n

≥

0

, ξ

n

≥

0

, ξ

n

≤ ξ

max

(

n

∈ N).

Fig.10showstheapproximationerrorsobtainedusingthetrust-regionalgorithmimplementedinMATLAB®

lsqnon-lin

.Both theconvergence speed ofthe nonlinear optimizationstage and thequality of the endresultstrongly depend upontheinitialpolesdistribution,whichmakesthismethodunpractical (forexample,thecase

ξ

min

=

10−14 isdifficultto

Fig. 10. Errors(23,24,25)andmaximumpole(26) forh=Yα withα=21.( )Qβ,N withβ= β2.Four-stageoptimizationwithξmax=104:( )

ξmin=10−16,( )ξmin=10−14,( )ξmin=10−10.(SameparametersasFig.8.)

Fig. 11. FDE(30) fory0=1 andg=1.NumericalsolutionscomputedwithRKF84,t=9×10−3,andN=6.( )Qβ,Nwithβ= β2(ξmin=1.221×10−3,

ξmax=8.189×102).( )Optimization(ξmin=10−3,ξmax=102).( )Optimization(ξmin=10−4,ξmax=102).(Leftonly)( )Exactsolution(31), ( )Exactsolutionforg=0.

convergewhile

ξ

min

=

10−16isalmostinstantaneous.).Furthermore,theapproximationerrorgraphsshow thatitis

unsat-isfactory, sothat it isnot worth considering in practice.In fact,nonlinear optimizationis bestused incombination with quadraturerules: thisisinvestigatedinSection 5.Thisnonlinearfour-stageoptimizationmethodisnotfurtherconsidered intheremainingofthissection:“optimizationmethod”willdenotethethree-stepmethoddescribedinSection 4.2.

4.4. Fractionaldifferentialequation

Letusconsiderthefollowingscalarfractionaldifferentialequation

˙

y

(

t

)

=

ay

(

t

)

−

g d12_y

₍

_t

_),

_y

₍

₀

₎

=

_y0

₍

_t

_>

₀

_),

₍₃₀₎

where y is

˙

thestrongderivativeandd12 _{istheCaputoderivativedefinedin(6).Theexactsolutionof(30) canbeexpressed}

usingtheMittag-LefflerfunctionEα,β as[32,Ex. 1.6]

ye

(

t

)

:=

y0

λ

1

− λ

2



λ

1E1_/2,1



λ

2

√

t



− λ

2E1_/2,1



λ

1

√

t



,

(31)

where

λ

1 and

λ

2 aretherootsofs

→

s2

+

gs

−

a.TheleftplotofFig.11showstheexactsolutionon

[

0,tf

]

withtf

=

100,

y0

=

1,andforboth g

=

0 (i.e. standardODE) andg

=

1 to highlighttheeffectofthefractional derivative.Toaccurately

evaluate ye,werelyonthealgorithmproposedin[17].

Comparisonwiththeoptimizationmethod. Weseektocomputenumericalsolutionsof(30) witharelativeaccuracyof,say, 6%.WiththeQβ,N method,thesoleparameterofwhichis N,thisaccuracytargetisattainedforanyN

≥

6,sothatweset

N

=

6 fortheoptimizationmethodaswell. Thecorresponding numericalsolutions areshowninFig.11,whichalsoplots therelativeerrorforother valuesof

ξ

minand

ξ

max.Time-integrationisperformedusingafourth-ordereight-stageexplicit

Runge–Kutta method,namely the RKF84 from[45, Tab. A.9],with a timestep of



t

=

9

×

10−3, whichis the maximum stabletimestepforallmethods.AsexpectedfromSection3.2,bothmethodsyieldsimilarresults.

ComparisonwiththeBirk–Songmethod. Forthetimestep



t

=

9

×

10−3_{,usedinFig.11,theBirk–Songmethod(27) yields}

a stableresultonlyforN

≤

2.Forinstance,for N

=

3,thestability timestepisfoundtobe



tmax

=

2.37

×

10−3,whichis

a significantreduction.Thiscanbe explainedby thelarge valuesof

ξ

max,alreadyhighlightedinSection4.3.Thistimestep

reductioncouldbebalancedbyanaccuracyincrease.Toinvestigatethis,Fig.12plotsacomparisonbetweentheBirk–Song and Qβ2,N methodsatatimestep well-belowthe stabilitylimit, namely



t

=

10−

3 _{for N}

₌

_{3.Onthisexample,the Q}

β,N

Fig. 12. FDE(30) fory0=1 andg=1.NumericalsolutionscomputedwithRKF84,t=10−3andN=3.( )Qβ,Nwithβ= β2(ξmin=1.613×10−2,

ξmax=6.198×101).( )Birk–Songmethod(27) (ξmin=3.139×10−4,ξmax=3.185×103).(Leftonly)( )Exactsolution(31).

4.5. Eigenvalueapproachtostability

Toconcludethis section onnumerical applications,let usconsidera casewhere the Qβ,N andoptimizationmethods

haveradicallydifferentproperties.Weareinterestedinstudyingthestabilityofthesolutionofthefollowingvector-valued fractionaldelaydifferentialequation

˙

x

(

t

)

=

Ax

(

t

)

+

Bx

(

t

−

τ)

−

g I2d1−αx

(

t

),

(32) with A

=

1 2



−

3 1 1

−

3



,

B

=

1 4



1 1 1 1



,

I2

=



1 0 0 1



,

τ

=

10

,

α

=

5 8

.

(33)

ThematricesA and B arechosensothat(32) isasymptoticallystableforanyg

≥

0,

τ

≥

0,and

α

∈ (

0,1)[36,Thm. 7]. Tostudythestabilityof(32),werecastitintoanabstractCauchyproblem

d X dt

(

t

)

=

A

X

(

t

),

X

(

0

)

∈

H

,

X

:=

⎛

⎝

ψ

x

ϕ

⎞

⎠ ∈

H

,

(34)

whichisknown asan eigenvalueapproach. Thedefinitionof

A

isobtainedby usingthediffusiverepresentationof d1−α andrewritingthetime-delaytermasan observerofatransportequation ontheboundedinterval

(

−

τ

,

0)[14,§ VI.6] [9, § 2.4] [34,Chap. 2],whichleadsto

A

X

:=

⎛

⎜

⎝

Ax

+

B

ψ (

−

τ)

−

g I2

$

_∞ 0 [

−ξ

ϕ(ξ )

+

x]

μ

α

(ξ )

d

ξ

dψ dθ

−ξ

ϕ(ξ )

+

x

⎞

⎟

⎠ ,

withstate-space H anddomain

D

(

A

)

givenby

H

:= C

2

×

L2

(

−

τ

,

0

; C

2

)

×

L2_{ξμα(ξ )}

(

0

,

∞; C

2

),

D

(

A

)

:=

&

(

x

, ψ,

ϕ)

∈

H







ψ

∈

H1

₍

₋

_τ

_,

₀

_{; C}

2

₎

−ξ

ϕ(ξ )

+

x

∈

L2_(1+ξ)μ α(ξ )

(

0

,

∞; C

2

₎

'

,

wheretheweightedL2spaces L2_ξμ

α(ξ )andL 2 (1+ξ)μα(ξ )aredefinedas L2_f

(

0

,

∞) :=

⎧

⎨

⎩

ϕ

: (

0

,

∞) → C

measurable







∞



0

|

ϕ(ξ )

|

2f

(ξ )

d

ξ <

∞

⎫

⎬

⎭

,

with f

(ξ )

= ξ

μ

α

(ξ )

and f

(ξ )

= (

1

+ ξ)

μ

α

(ξ ),

respectively. Foradditionalbackgroundonthissemigroupformulation,see [33,37].

Remark11(Motivation).Theequation(32) isatoymodelmeanttocheckthesuitabilityofagivendiscretizationmethodfor stabilitystudies,thusvalidatingitsuseforequationsthatdonotenjoytheoreticalresults.Practicalstabilitystudiesconsist in computingstability regions, see e.g. [34] fordelay equations. When usingan eigenvalue approach,it is ofparamount importance that the spectrum of

A

be accurately approximated, something which is lessof concern with time-domain simulations.

Fig. 13. Spectrum σ(Ah)for(32,33)obtainedwith the Qβ,N methodwithβ= β2.Transport equationdiscretizedwith Np=80 nodes.( )N=400 (S(Ah)= −8.12×10−11).( )N=200 (S(Ah)= −1.29×10−9).( )N=11 (S(Ah)= −1.00×10−4).

Fig. 14. Spectrumσ(Ah)for(32,33)obtainedwiththeoptimizationmethodwithξmax=104.TransportequationdiscretizedwithNp=80 nodes.N=400 with( )ξmin=10−15(S(Ah)= +3.5×1012);( )ξmin=10−10(S(Ah)= −7.1×10−11).N=20 with( )ξmin=10−15(S(Ah)= +2×10−15);( )ξmin= 10−10_(S(

Ah)= −10−12).

Thestabilityofthefractionaldelaydifferentialequation(32) followsfrompropertiesofthespectrumof

A

.Theoretically, thespectrumof

A

consistsoftwodistinctparts:(a)isolatedeigenvalueswithfinitealgebraicmultiplicity;(b)anessential spectrumon

(

−∞,

0)ifg

=

0.Thisessentialspectrumimpliesthat

S

(

A

)

:=

sup

λ∈σ(A)

(λ) =

0

,

sothat(32) cannotbeexponentiallystable,butisindeedasymptoticallystable.

Let uschose g

=

2 and try to recover this stability result numerically, by computing the spectrum of

A

h, a

finite-dimensionalapproximationof

A

,whichrequirestodiscretizeboththetime-delayandthefractionalderivative.

The monodimensional transport equation on

(

−

τ

,

0) can be discretized using any numerical scheme suited to the transport equation. Herein, we use a discontinuous Galerkin finite element method [22], whose spectral properties are well-known [23],on1elementwithNp nodes(i.e.apolynomialofdegree Np

−

1).Forthelargevalue Np

=

80,the

spec-trumissatisfactoryintheregionofinterest,sothatanywitnessedspectralpollutionstemsfromtheapproximationofthe fractionalderivative.

Thefractional derivativeisapproximatedwithN variables

ϕ

n,sothatthematrix

A

h issquare with

(2

+

Np

+

N

)

lines.

Figs. 13and14plotthespectraobtainedusingboththe Qβ,N andtheoptimizationmethods.Inbothcases,thestructure

ofthespectrumisconsistentwiththetheory;therearehowevermajordifferencesbetweenthetwomethods.

Since the Qβ,N method has one parameter, it is straightforward to assess convergence.In the region of interest, the

spectrum isconvergedforN

≥

11,seeFig.13.TherightplotofFig.13showsthat theessentialspectrumisonlymadeof realeigenvalues.Moreover,wehavethemostimportantpropertythat

S

(

A

h

)

:=

max

λh∈σp(Ah)

(λ

h

)

isnegativeforallvaluesofN.Hence,thestabilityresultisverifiednumerically.

LetusnowturntotheoptimizationmethoddescribedinSection4.2.Letusset

ξ

max

=

104 so thatthetworemaining

freeparametersare N andthelowerbound

ξ

min.Fig.14plotsthespectraobtainedusingtwovaluesforN and

ξ

min,namely

smallones(N

=

20 and

ξ

min

=

10−15)andlargeones(N

=

400 and

ξ

min

=

10−10).Theleftplotshowsthatthestructureof

thespectrumisapparentlyidenticaltothatobtainedwiththe Qβ,N method,withareasonablyconvergedpointspectrum.

However, thezoomgivenintherightplotshowsthattheessentialspectrumispolluted.Significantlyforastabilitystudy, thespectrumcanbecomeslightlyunstable,seethepositivevalueofS

(

A

h

)

for

ξ

min

=

10−15:although S

(

A

h

)

remainsclose

tozero,itssigndependsonthechoiceof

ξ

minandN. Thisimpliesthattheoptimizationmethodisnot suitedtocompute

thespectrumof

A

inthisexample.

TheBirk–Songmethod(27) alsoenjoysspectralaccuracy,seeFig.15.AtN

=

11,thespectrumiswell-convergedandthe essential spectrum isboth non-polluted andstable. Thissuggeststhe conjecturethat spectral correctness isexhibitedby

Fig. 15. Spectrumσ(Ah)for(32,33)obtainedwithtwoquadraturemethodswithN=11.TransportequationdiscretizedwithNp=80 nodes.( )Qβ,N methodwithβ= β2.(S(Ah)= −10−4).( )Birk–Songmethod(27) (S(Ah)= −3.5×10−8).

Fig. 16. Spectrumσ(Ah)for(35,33)obtainedwiththeQβ,Nmethodwithβ= β2andN=400.TransportequationdiscretizedwithNp=80 nodes.( ) τ0=0 (S(Ah)= −6×10−10).( )τ0=2τ(S(Ah)=6.3×10−2).

everyquadrature-basedmethods,sothat theyshouldbepreferredtooptimization-basedonesforanyapplicationwherea correctspectrumisneeded.

Letusconcludethissectionwithtwoadditionalexamples.

ApplicationtoBesselfunction. AsrecalledinSection2.1,diffusiverepresentationsneednotberestrictedtofractional opera-tors.Letusconsideramorecomplexequationthan(32),forinstancethememorydelayequation

˙

x

(

t

)

=

Ax

(

t

)

+

Bx

(

t

−

τ)

−

g I2J0



x

(

t

−

τ

0

),

(35)

where

τ

0

≥

0 and A, B,g,and

τ

aregivenby(33).Similarlyto(32),thediffusiverepresentationof J0givenby(5) enables

toformulatean abstractCauchyproblem(34).However,since theweight

μ

iscomplex-valued,theasymptoticstabilityof (35) cannot be established using the energy methodfollowed in [36, Thm. 7] for(32). Hence the need fora numerical stability study. Fig.16 plots the discrete spectrum obtainedwiththe Qβ,N method fortwo valuesof

τ

0,namely

τ

0

=

0

and

τ

0

=

2

τ

.Thespectrumexhibitstwostraightlinesthatstartfrom

−

i and

+

i,whicharethecutschosentoextendthe

Laplacetransform

ˆ

J0tothelefthalf-plane.Thisdiscretizationenablesustoconcludethatthecase

τ

0

=

0 isasymptotically

stablewhilethecase

τ

0

=

2

τ

isunstable.

Constrainedoptimization. The pollution ofthe essentialspectrum visibleinthe rightplotofFig. 14can be linked tothe unconstrained natureofthelinearleastsquaresoptimizationdescribedinSection4.2.Morespecifically,itiscausedbythe negativityofsomeweights

μ

n (whichisnotnecessarilyapracticalconcernfortime-domaincomputations).Analternative

isthereforetouse,insteadofthepseudo-inverse,an iterativeoptimizationalgorithmthatenforces thenonnegativity con-straint.ThisisillustratedbyFig.17,whichpresentsspectraobtainedbyminimizing(28) with

μ

≥

0 usingthenonnegative leastsquaresalgorithm[24,(23.10)] throughitsimplementationinMATLAB®

lsqnonneg

.Thespectralpollutionis signif-icantlyreduced,buthasnotcompletelydisappeared,sinceslightlyunstablespectra(orevenunconvergedspectra)canstill beobtainedforsomepoorlychosenparameterssuchas

ξ

min

=

10−15,

ξ

max

=

104,andN

=

50.Thecostofthisoptimization

algorithm,aswell asits difficultytoconvergeforsometriplets

(ξ

min

,

ξ

max

,

N

),

isapracticalchallenge tothecomputation

ofstabilityregions.

5. Improvementofthequadraturemethodusinganonlinearleastsquaresoptimization

The discussion of Section 4.3 hashighlighted that thesole use of a nonlinear least squaresminimization of the cost function (28) is not practical, dueto both its computational cost andsensitivity to the initial pole distribution (i.e. the initialdistributionof

(ξ

n

)

n).Thissectioninvestigatestheuseofanonlinearleastsquaresminimizationtorefinethepoles

andweights givenby the Qβ,N method(see Definition4),i.e.acombined useofanoptimizationandaquadraturerule.It

Fig. 17. Spectrumσ(Ah)for(32,33)obtainedbyminimizing(28) withthenonnegativityconstraintμ≥0.Upperboundchosenasξmax=104.Transport equationdiscretizedwith Np=80 nodes.Lowerboundξmin=10−10with( )N=400 (S(Ah)= −10−12);( )N=50 (S(Ah)= −10−12);( )N=20 (S(Ah)= −10−12).( )N=50 withξmin=10−15(S(Ah)= +5.4×10−16).

the numericalresultsshow that thecost function(37) isto bepreferredto build parsimoniousapproximationsasit can deliversubstantialimprovementswhenthenumberofquadraturenodesislow.

Section 5.1 defines the numerical methodology as well asthe covered cost functions, while Section 5.2 gathers the numericalresultsandconcludeswithpracticalguidelines.TheMATLABcodeisavailableonline.1

5.1. Numericalmethodologyandconsideredcostfunctions

The purpose of the numericalmethodology described below is to improvethe poles andweights given by the Qβ,N

methodbyminimizing agivencostfunction J ;fromnowon,thismethodologyisdenoted Q_β,OPT- J_N .Asimilarmethodology isusedin[26,§ 4.2] toimprovetheBirk–Songmethod(27).

Definition12.The Q_β,OPT- J_N discretizationof(1) is(8) wherethepolesandweightsarecomputedwiththefollowing three-stepmethod.

1. Choose N (numberof quadraturenodes),

β

(scalar parameterthat sets thechangeofvariable), and

(ξ,

μ

)

→

J

(ξ,

μ

)

(costfunction).

2. Computethepoles

(ξ

n

)

n andweights

(

μ

n

)

n usingtheQβ,N method(seeDefinition4).

3. Refinethecomputedpolesandweightsbyminimizing J underthelinearconstraints

0

≤ ξ

n

(

a)

ξ

n

≤ ξ

max (b)

μ

n

≥

0 (c)

(

n

∈ N),

(36)

startingwiththevaluesobtainedinstep2.

Theconstraints(36) are motivatedbythediscussionsoftheprevious sections:letussummarizetheirpurposes. Condi-tion(a)isrequiredforstability,asitpreventsanypoleofh

ˆ

num fromhavinganonnegativerealpart.Condition(b)ensures

that thepoles stay below theupper bound givenby the Qβ,N method,so that there isno time-step reduction withan

explicittime-integrationscheme.Condition(c)isoptionalbutcanbeenforcedwhenthediffusiveweight

ξ

→

μ

(ξ )

is non-negative asitenables toget an unpolluted andstablespectrum (see Section 4.5foran illustrationof theimpact ofthis constraint).

The useofa nonlinearleastsquaresoptimizationimpliesanadditionalfreedom inthedefinition ofthecostfunction, compared withthe linearleastsquaresconsidered inSection 4.2.The three studiedcost functionsareformulated in the frequencydomain.ThefirstoneisthatalreadyusedinSection4.2:

J

(ξ,

μ)

:=

K



k=1







N



n=1

μ

n i

ω

k

+ ξ

n

− ˆ

h

(

i

ω

k

)







2

,

(37)

withtheK angularfrequencieslogarithmicallyspacedin

[ξ

min

,

ξ

max

]

ω

k

= ξ

min



ξ

max

ξ

min

k−1 K−1

(

k

∈ J

1

,

K

K).

The second costfunction isformulated soasto cancelanintegrablesingularity of

ω

→ ˆ

h

(

i

ω

)

at

ω

=

0 (considerh

ˆ

= ˆ

Yα with

α

∈ (

0,1)):