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 aONERA/DMPE,UniversitédeToulouse,31055Toulouse,FrancebISAE-SUPAERO,UniversitédeToulouse,31055Toulouse,France
a
b
s
t
r
a
c
t
Keywords: Fractionalderivative Fractionalcalculus Diffusiverepresentation Eigenvalueproblems Non-classicalmethod CompletelymonotonekernelThis 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:florian.monteghetti@onera.fr(F. Monteghetti),denis.matignon@isae.fr(D. Matignon),estelle.piot@onera.fr(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 isthenumberofquadra-turenodes.Section3presentsananalysisofthemethodinthecaseoffractionaloperators,whichhighlightsthedependency of
β
uponthe orderofthefractional operator. Numericalapplications andcomparisonsare gatheredinSection 4,where the Qβ,N methodiscomparedagainsttwoexisting methods,one optimization-basedandonequadrature-based.Section 5investigatestheuseofanonlinearleastsquaresminimizationtorefinethepolesandweightsgivenbythe Qβ,N method.
2. Definitionoftheproposedquadrature-baseddiscretizationmethod
The purposeof thissection isto introduce the proposed Qβ,N discretization method, where
β
is a scalar parameterto 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)
:=
∞
0e−ξtH
(
t)
μ(ξ )
dξ
(
t∈ R),
(1)where H istheHeavisideorunitstepfunction(H(t
)
=
1 fort>
0,nullelsewhere)andμ
∈
C((
0,∞))
isthediffusiveweight. Bydefinition,diffusivekernelsarelocallyintegrableon[
0,∞)
,i.e.h∈
L1loc
(
[
0,∞))
,sothatthediffusiveweightsatisfies∞
0μ(ξ )
1
+ ξ
dξ <
∞.
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) readsˆ
h(
s)
=
∞ 0μ(ξ )
s+ ξ
dξ.
Remark1.As definedherein,a diffusive kernelisa locallyintegrablecompletely monotone kernelon
(0,
∞)
.A diffusive kernelh isintegrableon(0,
∞)
ifandonlyif[19,Thm. 5.2.5]∞
0μ(ξ )
ξ
dξ <
∞,
whichisnotthecaseforthekernelsconsideredinthispaper,seethethreeexamplesbelow.
Remark2(Terminology).Inthispaper,weusethefollowingterminology:thediffusiverepresentation ofh istheidentity(1), whilethefunction
μ
iscalledthediffusiveweight.Thisslightlydiffersfrom[39] whereμ
iscalledthediffusive representa-tionofh.Thequantityμ
isalsoknownunderothernamessuchasspectralfunction [18] orrelaxationspectrum [29].Thecomputationalinterestofdiffusivekernelsisthat,formally,theconvolutionoperatoru
→
hu admitsthefollowing infinite-dimensionaltime-localrealization
⎧
⎪
⎪
⎨
⎪
⎪
⎩
∂
tϕ(
t, ξ )
= −ξ
ϕ(
t, ξ )
+
u(
t),
ϕ(
0, ξ )
=
0(ξ
∈ (
0,
∞)),
hu
(
t)
=
∞ 0ϕ(
t, ξ )μ(ξ )
dξ,
(2)whereu isacausalinput. Afunctionalframeworkforthisrealizationhasbeenproposed in[11,44]. Letusnowlistthree examplesofdiffusiveoperatorscoveredbythediscretizationmethodintroducedinSection2.2.
1. TheRiemann–Liouvillefractionalintegral,definedas[41,§ 2.3] [32]
Iαu
:=
Yαu
,
whereα
∈ (
0,
1)
andthefractionalkernelis
Yα
(
t)
:=
H
(
t)
(α)
t1−α, ˆ
Yα(
s)
=
1sα
.
(3)Theassociateddiffusiveweightis
μ
α(ξ )
:=
sin
(απ
)
πξ
α.
(4)2. Anotherdiffusivekernelisthezeroth-orderBesselfunctionofthefirstkind[31,§ 3.3]
J0
(
t)
H(
t)
= +
eit ∞ 0μ
1/2(ξ )
√
−ξ +
2ie −ξtdξ
+
e−it ∞ 0μ
1/2(ξ )
√
−ξ −
2ie −ξtdξ
(5)= +
2⎡
⎣
eit ∞ 0μ
1/2(ξ )
√
−ξ +
2ie −ξtdξ
⎤
⎦ ,
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))⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
∂
tϕ(
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 )
dξ,
(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=1μ
ne−ξntH(
t) (
t∈ R).
(8)IntheLaplacedomain,thisreads
ˆ
h(
s)
ˆ
hnum(
s)
=
N n=1μ
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, whosediffusiveweights
μ
aremonotoneon(0,
∞)
withasingularityatξ
=
0,whichleadstothefollowingassumption. Assumption3.Thediffusiveweightμ
∈
C((
0,∞))
hasapower-lawsingularityatξ
=
0,i.e.μ(ξ )
=
O
1ξ
α,
(9) withα
∈ (
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
functionusesthechangeofvariable
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),
thusyieldingξ
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−
v1 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.Usingβ,therepresentation(10) reads
h
(
t)
=
2β
1 −1 e−t 1+v 1−v 1 β(
1−
v)
−1−1β(
1+
v)
β1−1μ
1+
v 1−
v1 β dv
,
(13)whichleadstothedefinitionofthe Qβ,N discretizationmethodgivenbelow.
Definition4.The Qβ,N discretizationof(1) is(8) with
ξ
n:=
1+
vn 1−
vn1 β
,
μ
n:=
wn 2β
(
1+
vn)
1 β−1(
1−
v n)
−1− 1 βμ
(ξ
n) ,
(14)Intuitively,one mayexpectthebest valuefor
β
to bedependent onpropertiesofthediffusive weightμ
,suchasthe valueofα
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−1f−
qL∞([−1,1]).
Theconclusionfollowsfromapolynomialapproximationresult[13,Thm. I.VIII].
Giventheaboveresult,tofindtheoptimalvaluefor
β
inthe Qβ,N discretization,itissufficienttostudytheregularityoftheintegrandin(13).Letusnowfocusonthefractionalkernel(3),whichisthesimplestkernelthatsatisfies(9).Afirst convergenceresultissummarizedinthepropositionbelow.
2
Proposition6.Let
β >
0,N∈ N
∗,andYα,numbetheQβ,NdiscretizationofYα withα
∈ (
0,1).Ifβ
≤
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
Yα
(
t)
=
1 −1β
(
t,
v)
dv,
(17) withβ
(
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 aslongast
>
0,since x→
e−t 1+ 2x 1 β x1+ 1−βαisinfinitelydifferentiableat0+,withoutassumptionon
α
andβ
.Sincev→
e−t1+v 1−v
1 β
hasalimitasv
→ −
1+,theintegrandβ
(
t,
·)
iscontinuous ifandonlyif(15) holds.Furthermore,β
(
t,
·)
∈
C
∞(
[−
1,1])
FromProposition6,aconvergenceresulton
Yα− ˜
YαL1(,T)forany>
0 andT>
canbereadilydeduced,although
thisisnotsufficientfortime-domaincomputations.Indeed,foraninputu
∈
L2(0,
T)
∩
L∞(0,
T),
wehavethestraightforward estimate|
hu
(
T)
|
uL∞(0,T)≤
hL1(0,T),
(18)whichjustifiesaninterestinapproximatingtheL1 normofh.Thisrequiresanadditionalconstrainton
β
,seeProposition7.Proposition7.Let
β >
0,N∈ N
∗,andYα,numbetheQβ,NdiscretizationofYα withα
∈ (
0,1).Ifboth(15) andβ
≤
α
(19)hold,thenlimN→∞
Yα,numL1(0,T)=
YαL1(0,T)foranyT>
0.If,additionally,(16) holds,thenthisconvergenceisspectral. Proof. Let T>
0.Sinceα
∈ (
0,1),wehaveβ
∈
L1((0,
T)
× (−
1,1)).TheFubinitheoremyields YαL1(0,T)=
1 −1β
(
T,
v)
dv,
(20) whereβ
(
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,thenβ
(
T,
·)
∈
C
∞(
[−
1,1])
.2
Remark 8.Proposition 7 states convergence of the L1 norm, but not convergence in the L1 norm, i.e. lim
N→∞
Yα−
Yα,numL1(0,T)=
0 (which impliesconvergenceofthe L1 norm).This propositioncan thereforebe deemed insufficientinviewoftheestimate(18);however,itgivesasecondconstrainton
β,
namely(19),whichispracticallyuseful.Remark9(Frequencydomain).Asimilar studyin the frequencydomain reachesthe same conclusion.For anys
=
0 with(
s)
≥
0,wehaveˆ
Yα(
s)
=
2β
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 Yˆ
α(
s)
−
N n=1μ
n s+ ξ
n.
Forany
ω
m>
0,wereadilydeducethatiω
Yαˆ
(
iω
)
−
iω
nN=1μn
iω+ξn
L2(−ωm,ωm) hasthesameconvergenceproperties.Practicalchoiceof
β
Basedontheaboveresults,thefollowingrulescanbefollowedtochoose
β
inpractice. 1. Ifα
∈ (
0,1)∩ Q
suchthatα
=
n0 n1 withni∈ N
∗,thenβ
1:=
1 n1 (21) satisfiesthecondition (15),(16),and(19) sothatthe Qβ1,N methodyieldsaspectrallyconvergentapproximation.Thisvalueisalsosuitedfor
α
∈ (
0,1)∩ (R\Q)
withα
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,
namelyβ
2:=
min(α,
1−
α),
(22)whichyields atleasta convergent approximationfromPropositions6 and7.Section 3.2belowshowsthat
β
2 isthemostinterestingchoiceformoderatevaluesofN. 3.2. Numericalillustrations
Toinvestigatenumericallytheinfluenceof
β
ontheconvergenceofthe Qβ,N method,wedefine threeerrors.Thefirstoneisinthefrequencydomain
ε
∞,ωm:=
1−
ˆ
hnumˆ
h(
iω)
L∞(−ωm,ωm),
(23)with
ω
m>
0 agivenangularfrequency.Thesecondandthirdonesareinthetimedomain,namelyε
T:=
1−
hnum h(
T),
ε
1,T:=
hL1(0,T)−
hnumL1(0,T) hL1(0,T),
(24)withT
>
0.Fromnowon,wesetω
m=
T=
104,
(25)so that we considerbroadband approximationsof thekernel h.We first considerh
=
Yα, covered by theresults of Sec-tion3.1,withfourvaluesofα
andthenconcludewithh=
J0.Computationsaredonewithdoubleprecisionfloatingpoint.The case h
=
Yα withα
=
58
0.62
≥
12 is shown in Fig. 1. The choice
β
1=
18 achieves spectral convergence, withsaturation at double precision, asexpected from Section 3.1. The value
β
2=
1−
α
does not converge spectrally, butitprovidesabetterapproximationformoderatevaluesofN.Thevalue
β
3:=
max(α,
1−
α),
whichdoesnotsatisfy(15),istheleastinterestingoption.Thesensitivityoftheerrorsobtainedwith
β
= β
2 ishighlightedbythecurvescorrespondingto
β
=
0.99× β
2 andβ
=
1.01× β
2,whicharesignificantlyworseinthetimedomain.Theseerror plotshighlight that the time-domain norms do add information: here, the sensitivity to
β
2 cannot be seen inthefrequencydomainforinstance,whileitistheoppositeforothervaluesof
α
coveredbelow. TheupperrightplotofFig.1givesthemaximumpoleξ
max:=
maxn
ξ
n.
(26)Thisquantityisespeciallyimportantwhenusinganexplicitschemetoadvancetherealization(2) intime,sincethetime steptypicallyscalesas
O(ξ
max−1).
Theplotshowsthatξ
max=
O
(
N2
β
).
Giventhatthehigher
ξ
max,themorecostlythetimeintegration,onemayexpectthatahighervalueofξ
maxsystematicallyyields amoreaccurate discretization.However,thisneednot bethecase:althoughthisisindeedverifiedfor
β
1,β
2,andFig. 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
=
12. Here, the values
β
1,β
2, andβ
3 are identicalso that the Qβ,N discretizationenjoys spectral convergence,withdoubleprecisionon
ε
T reachedforaround 100variables.The sensitivitytoachangeinβ
aroundβ
2 canbeseeninbothfrequency-domainandtime-domainerrors:althoughthevaluesofξ
max remainclose,theapproximationsaresignificantlyworsefor0.99
× β
2 and1.01× β
2.Theconclusionsfor
α
=
27
0.28
≤
12,showninFig.3,areidenticalto
α
=
58.Theonlydifferenceisthatthesensitivity
to
β
2isonlyseeninthefrequency-domainnormε
∞,ωm.Fig.4showstheerrorsobtainedforα
=
√
2−1
√
2
0.29,avalueclose
to 27 butirrational. Themain differenceis theerrorobtainedfor
β
= β
1,whichis lessaccurate inthe frequencydomaincompared to
α
=
27.However,thehierarchybetweenthe Qβ,N methodsisidentical,andthe othererrorsare similar.The
choices
β
=
411 (justified byα
12
41)and
β
=
13 (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 deliverssatisfactoryconver-genceresultsforany
α
∈ (
0,1),rationalorirrational.Inaddition,thefactthatβ
2≥ β
1 impliesthattheQβ2,N methodyieldsa 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,wherenumericalappli-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)
withβ
=
14.However,theyproposetouseaGauss–JacobiquadratureruleinsteadofaGauss–Legendreone(therebyintroducing asingularityatv= −
1 intheintegrand),whichleadstothediscreterepresentation[4,Eq. (23)]ξ
n:=
1− ˜
vn 1+ ˜
vn4
,
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)withα
:=
1−
2α
.(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,whichfurthermorehaveawidevariationsincetheoretically
ξ
∈ (
0,∞)
.Toavoidthiscomputationaldifficultythemethodproceedsasfollows. 1. Thethreeinputparameters,namelyN∈ J
2,∞J
,ξ
min>
0,andξ
max> ξ
minarechosen.2. TheN poles
ξ
n arelogarithmicallyspacedin[ξ
min,
ξ
max]
:ξ
n= ξ
minξ
maxξ
minn−1 N−1
(
n∈ J
1,
NK).
3. Let A:=
(
iω
k+ ξ
n)
−1 k,n∈ C
K×N andb:=
hˆ
(
iω
k)
k∈ C
K,wheretheK angularfrequencies
ω
karealsologarithmically
spacedin
[ξ
min,
ξ
max]
.TheN weightsμ
n arecomputedwithalinearleastsquaresminimizationofJ
(μ)
:=
Cμ
−
d22=
K k=1 N n=1μ
n iω
k+ ξ
n− ˆ
h(
iω
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 weightsμ
nisenforcedthroughthedefinitionofC andd,whichseparatesrealandimaginaryparts.However,notethatthesignofeach
μ
n isunconstrained.Thistechnique isparticularlysuitedfortime-domainsimulations,where
ξ
max isnaturally known(frome.g. theminimumacceptabletimesteporthemaximumfrequencyofinterestinwavepropagationproblems);itcanalsohandlemorecomplex representations that involve additional poles. For a given N and
ξ
max, there is usually an optimal range for the lowerbound
ξ
min,whichgovernsthelong-timebehaviorofhnum,whichmustnotbechosen toosmall.Forthediffusive kernelsconsidered herein, a logarithmic spacing ofthe poles
ξ
n is satisfactory (alinear spacingyields poorerresults). In all theapplicationspresentedinthissection,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
hasbeenchosentobeβ
2 followingtheanalysisofSection3), 3fortheoptimizationmethod(namely N and theminimum and maximum poles
ξ
min andξ
max). In all the results presented below, the parametersξ
min andξ
max of theoptimizationmethodhavebeenempiricallychosentoyieldthebestresults. 4.3. Approximationerrors
InthespiritofSection3.2,herearegatheredcomparisonsoftheapproximationerrors.
ComparisonwiththeBirk–Songmethod. Acomparisonbetweenthe Qβ,N methodandtheBirk–Songmethod(27) isshown
inFig.6for
α
=
58 andFig.7for
α
=
1 2.Let usfirst consider the case
α
=
58.The behavior of
ε
∞,ωm highlights the accuracy of the Birk–Songmethod in theFig. 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 obtainedwithβ
= β
2.Allthemethodsarecloserfor
α
=
12,atleastinthetime domain,andthe method Qβ,N with
β
=
14 hasalmostidenticalconvergencepropertiestothatoftheBirk–Songmethod.
Asalreadymentionedwhencomparingthevarious Qβ,N methodsinSection3.2,thegraphsof
ε
∞,ωm,ε
T,andε
1,T alonearenotsufficienttocomparediscretizationmethods:onemusttakeintoaccountthevalueof
ξ
max,showninthetoprightplotofFigs.6and7.TheseplotsshowthatfortheBirk–Songmethodwehave
ξ
max=
O(
N 2β
)
withβ
=
14,i.e.
ξ
max=
O(
N8),
whichimpliessignificantlylargervaluesthatthe Qβ2,N methodrecommendedfromtheanalysisofSection 3.2.Theimpact
oftheselargevaluesof
ξ
max isofconcernwhenusingexplicittime-marchingscheme,seeSection4.4.Comparisonwiththeoptimizationmethod. Theerrorsfor
α
=
12 and
α
=
58 areplottedinFigs.8and9,respectively.Giventhe
resultsofSection 3,onlythe Qβ,N methodwith
β
= β
2 isconsidered.Forthe(three-parameter)optimizationmethod,wechoose
ξ
max=
ω
m=
104andplottheerrorsforvariousvaluesofξ
min:theresultshowsthattheoptimalvalueofξ
mindoesstronglydependuponN,sothat
ξ
minisnotstraightforwardtochooseapriori.However,providedthatthevalueofξ
miniswell-chosen,theoptimizationmethodcanoutperformthe Qβ2,N methodonarangeofN,whichjustifiesitspopularityin
large-scaleapplicationswherethevalueofN iscritical.Notethat,bycontrastwithSection3.2,thecomparisonisrestricted to N
∈ J
1,50K
,sinceoutsideofthisinterval,themaximumpoleξ
max ofthe Qβ2,N methodissignificantlylargerthan104
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,thecaseξ
min=
10−14 isdifficulttoFig. 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 thatitisunsat-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 d12y(
t),
y(
0)
=
y0(
t>
0),
(30)where y is
˙
thestrongderivativeandd12 istheCaputoderivativedefinedin(6).Theexactsolutionof(30) canbeexpressedusingtheMittag-LefflerfunctionEα,β as[32,Ex. 1.6]
ye
(
t)
:=
y0λ
1− λ
2λ
1E1/2,1λ
2√
t− λ
2E1/2,1λ
1√
t,
(31)where
λ
1 andλ
2 aretherootsofs→
s2+
gs−
a.TheleftplotofFig.11showstheexactsolutionon[
0,tf]
withtf=
100,y0
=
1,andforboth g=
0 (i.e. standardODE) andg=
1 to highlighttheeffectofthefractional derivative.Toaccuratelyevaluate ye,werelyonthealgorithmproposedin[17].
Comparisonwiththeoptimizationmethod. Weseektocomputenumericalsolutionsof(30) witharelativeaccuracyof,say, 6%.WiththeQβ,N method,thesoleparameterofwhichis N,thisaccuracytargetisattainedforanyN
≥
6,sothatwesetN
=
6 fortheoptimizationmethodaswell. Thecorresponding numericalsolutions areshowninFig.11,whichalsoplots therelativeerrorforother valuesofξ
minandξ
max.Time-integrationisperformedusingafourth-ordereight-stageexplicitRunge–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) yieldsa stableresultonlyforN
≤
2.Forinstance,for N=
3,thestability timestepisfoundtobetmax
=
2.37×
10−3,whichisa significantreduction.Thiscanbe explainedby thelarge valuesof
ξ
max,alreadyhighlightedinSection4.3.Thistimestepreductioncouldbebalancedbyanaccuracyincrease.Toinvestigatethis,Fig.12plotsacomparisonbetweentheBirk–Song and Qβ2,N methodsatatimestep well-belowthe stabilitylimit, namely
t
=
10−3 for N
=
3.Onthisexample,the Qβ,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,andα
∈ (
0,1)[36,Thm. 7]. Tostudythestabilityof(32),werecastitintoanabstractCauchyproblemd X dt
(
t)
=
A
X(
t),
X(
0)
∈
H,
X:=
⎛
⎝
ψ
xϕ
⎞
⎠ ∈
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],whichleadstoA
X:=
⎛
⎜
⎝
Ax+
Bψ (
−
τ)
−
g I2$
∞ 0 [−ξ
ϕ(ξ )
+
x]μ
α(ξ )
dξ
dψ dθ−ξ
ϕ(ξ )
+
x⎞
⎟
⎠ ,
withstate-space H anddomain
D
(
A
)
givenbyH
:= C
2×
L2(
−
τ
,
0; C
2)
×
L2ξμα(ξ )(
0,
∞; C
2),
D
(
A
)
:=
&
(
x, ψ,
ϕ)
∈
Hψ
∈
H1(
−
τ
,
0; C
2)
−ξ
ϕ(ξ )
+
x∈
L2(1+ξ)μ α(ξ )(
0,
∞; C
2)
'
,
wheretheweightedL2spaces L2ξμα(ξ )andL 2 (1+ξ)μα(ξ )aredefinedas L2f
(
0,
∞) :=
⎧
⎨
⎩
ϕ
: (
0,
∞) → C
measurable ∞ 0|
ϕ(ξ )
|
2f(ξ )
dξ <
∞
⎫
⎬
⎭
,
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, thespectrumofA
consistsoftwodistinctparts:(a)isolatedeigenvalueswithfinitealgebraicmultiplicity;(b)anessential spectrumon(
−∞,
0)ifg=
0.ThisessentialspectrumimpliesthatS
(
A
)
:=
supλ∈σ(A)
(λ) =
0,
sothat(32) cannotbeexponentiallystable,butisindeedasymptoticallystable.
Let uschose g
=
2 and try to recover this stability result numerically, by computing the spectrum ofA
h, afinite-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,thespec-trumissatisfactoryintheregionofinterest,sothatanywitnessedspectralpollutionstemsfromtheapproximationofthe fractionalderivative.
Thefractional derivativeisapproximatedwithN variables
ϕ
n,sothatthematrixA
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,wehavethemostimportantpropertythatS
(
A
h)
:=
maxλh∈σp(Ah)
(λ
h)
isnegativeforallvaluesofN.Hence,thestabilityresultisverifiednumerically.
LetusnowturntotheoptimizationmethoddescribedinSection4.2.Letusset
ξ
max=
104 so thatthetworemainingfreeparametersare N andthelowerbound
ξ
min.Fig.14plotsthespectraobtainedusingtwovaluesforN andξ
min,namelysmallones(N
=
20 andξ
min=
10−15)andlargeones(N=
400 andξ
min=
10−10).Theleftplotshowsthatthestructureofthespectrumisapparentlyidenticaltothatobtainedwiththe Qβ,N method,withareasonablyconvergedpointspectrum.
However, thezoomgivenintherightplotshowsthattheessentialspectrumispolluted.Significantlyforastabilitystudy, thespectrumcanbecomeslightlyunstable,seethepositivevalueofS
(
A
h)
forξ
min=
10−15:although S(
A
h)
remainsclosetozero,itssigndependsonthechoiceof
ξ
minandN. Thisimpliesthattheoptimizationmethodisnot suitedtocomputethespectrumof
A
inthisexample.TheBirk–Songmethod(27) alsoenjoysspectralaccuracy,seeFig.15.AtN
=
11,thespectrumiswell-convergedandthe essential spectrum isboth non-polluted andstable. Thissuggeststhe conjecturethat spectral correctness isexhibitedbyFig. 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 I2J0x
(
t−
τ
0),
(35)where
τ
0≥
0 and A, B,g,andτ
aregivenby(33).Similarlyto(32),thediffusiverepresentationof J0givenby(5) enablestoformulatean 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 valuesofτ
0,namelyτ
0=
0and
τ
0=
2τ
.Thespectrumexhibitstwostraightlinesthatstartfrom−
i and+
i,whicharethecutschosentoextendtheLaplacetransform
ˆ
J0tothelefthalf-plane.Thisdiscretizationenablesustoconcludethatthecaseτ
0=
0 isasymptoticallystablewhilethecase
τ
0=
2τ
isunstable.Constrainedoptimization. The pollution ofthe essentialspectrum visibleinthe rightplotofFig. 14can be linked tothe unconstrained natureofthelinearleastsquaresoptimizationdescribedinSection4.2.Morespecifically,itiscausedbythe negativityofsomeweights
μ
n (whichisnotnecessarilyapracticalconcernfortime-domaincomputations).Analternativeisthereforetouse,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 beobtainedforsomepoorlychosenparameterssuchasξ
min=
10−15,ξ
max=
104,andN=
50.Thecostofthisoptimizationalgorithm,aswell asits difficultytoconvergeforsometriplets
(ξ
min,
ξ
max,
N),
isapracticalchallenge tothecomputationofstabilityregions.
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).Thissectioninvestigatestheuseofanonlinearleastsquaresminimizationtorefinethepolesandweights 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 Qβ,OPT- JN .Asimilarmethodology isusedin[26,§ 4.2] toimprovetheBirk–Songmethod(27).
Definition12.The Qβ,OPT- JN 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)ensuresthat 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=1μ
n iω
k+ ξ
n− ˆ
h(
iω
k)
2,
(37)withtheK angularfrequencieslogarithmicallyspacedin
[ξ
min,
ξ
max]
ω
k= ξ
minξ
maxξ
mink−1 K−1
(
k∈ J
1,
KK).
The second costfunction isformulated soasto cancelanintegrablesingularity of