Table Of ContentREGULARIZED REDUCED ORDER MODELS
FOR A STOCHASTIC BURGERS EQUATION
7 TRAIANILIESCU,HONGHULIU,ANDXUPINGXIE
1
0
2 ABSTRACT. In this paper, we study the numerical stability of reduced order
modelsforconvection-dominatedstochasticsystemsinarelativelysimpleset-
n
ting:astochasticBurgersequationwithlinearmultiplicativenoise. Ourprelim-
a
inaryresultssuggestthat, inaconvection-dominatedregime, standardreduced
J
order models yield inaccurate results in the form of spurious numerical oscil-
4
lations. Toalleviatetheseoscillations, weusetheLerayreducedordermodel,
whichincreasesthenumericalstabilityofthestandardmodelbysmoothing(reg-
]
n ularizing)theconvectivetermwithanexplicitspatialfilter. TheLerayreduced
y order model yields significantly better results than the standard reduced order
d modelandismorerobustwithrespecttochangesinthestrengthofthenoise.
-
u
l
f
.
s
c 1. INTRODUCTION
i
s Reducedordermodels(ROMs)arecommonlyusedinapplicationsthatrequire
y
h repeated numerical simulations of large, complex systems [33,53]. ROMs have
p beensuccessfulinthenumericalsimulationofvariousfluidflows[35,50]. Numer-
[
ical instability, usually in the form of unphysical numerical oscillations, is one of
1 themainchallengesforROMsoffluidflowsdescribedbytheNavier-Stokesequa-
v tions (NSE). There are several sources of numerical instability of ROMs for fluid
5
flows [15], such as (i) the convection-dominated (high Reynolds number) regime,
5
1 inwhichtheconvectionnonlineartermplaysacentralrole[3,35,50]; and(ii)the
1 inf-sup condition, which imposes a constraint on the ROM velocity and pressure
0
spaces [5,15]. To mitigate the spurious numerical oscillations created by these
.
1 sourcesofnumericalinstability,variousstabilizedROMshavebeenproposed(see,
0
e.g.,[2–6,8,16,25,31,38,52,54,60,62]forsuchexamples). Apromisingrecentde-
7
1 velopmentinthisclassofmethodsisregularizedROMs[57,61],whichuseexplicit
: spatialfilteringtoincreasethenumericalstabilityoftheROMapproximation.
v
i Recently,thedevelopmentofROMsforsystemsinvolvingrandomcomponents
X
has also received increased attention. For instance, ROMs for partial differen-
r
a tial equations (PDEs) subject to random inputs acting on the boundary as well as
PDEs with random coefficients have been considered in various contexts [12,13,
21–23,27,32,42,59]. However, ROMs for evolutionary PDEs driven by stochas-
tic processes such as Brownian motions seem to be much less investigated. To
our knowledge, only a few works are available [14]; see also [18], where a new
2010MathematicsSubjectClassification. 34F05,35R60,37L55,60H15.
Key words and phrases. Reduced-order modelling, Leray regularized model, stabilization
method,numericalinstability,stochasticBurgersequation,differentialfilter.
1
2 TRAIANILIESCU,HONGHULIU,ANDXUPINGXIE
stochastic parameterization framework is presented to address a related question
of parameterizing the unresolved high-frequency modes in terms of the resolved
low-frequencymodes.
Inthispaper,weconsiderROMswithinthecontextofnonlinearstochasticPDEs
(SPDEs) that are of relevance to fluid dynamics. The main purpose is to investi-
gatewithinasimplerelevantsetting—astochasticBurgersequation(SBE)driven
bylinearmultiplicativenoise—thestabilizationofthestandardGalerkinROM(G-
ROM)inaconvection-dominatedregime. Itisnumericallyillustratedthatspurious
oscillationsdevelopedinaG-ROMpersistasthenoiseisturnedon,andtheoscilla-
tionsworsenasthenoiseamplitudeincreases. ALerayregularizedROM(referred
to as L-ROM hereafter) is then tested. The L-ROM provides more accurate mod-
eling of the SBE dynamics by greatly reducing the artificial oscillations of the G-
ROM,especiallywhenthedimensionofthereducedmodelsarelow;cf.Figs.3–6.
ItisfurtherillustratedthattheL-ROMismuchmorerobustthantheG-ROMwith
respect to the noise amplitude as revealed by the statistics of the corresponding
modelingerrors,whichhavesignificantlylowermeanandvariance;cf.Fig.7.
The rest of the paper is organized as follows. In Section 2, we outline the SBE
tobeusedinournumericalexplorationandderivethecorrespondingG-ROMand
the L-ROM based on the proper orthogonal decomposition. The performance of
the two ROMs is then tested and compared in Section 3 by placing the SBE in
a convection-dominated regime. Finally, some concluding remarks and potential
futureresearchdirectionsaregiveninSection4.
2. REDUCED ORDER MODELS FOR A STOCHASTIC BURGERS EQUATION
The viscous Burgers equation and its stochastic versions have been used previ-
ously to test new techniques in reduced order modeling and related contexts; see
amongmanyothers[17–19,40,41,49]. Inthispaper,wewillfocusonastochastic
Burgers equation (SBE) driven by linear multiplicative noise, which is presented
brieflyinSection2.1. Tofixideas,theROMsexploredinthispaperwillbederived
based on the proper orthogonal decomposition (POD). In Section 2.2, we outline
themainstepsinthederivationofthePODbasis. ThestandardGalerkinROMfor
theSBEisthenderivedinSection2.3. InSection2.4,wedeveloptheLerayROM,
which is a regularized ROM that aims at increasing the numerical stability of the
standardROMfortheSBE.
2.1. StochasticBurgersEquation(SBE). Inthispaper,wefocusonthefollow-
ingstochasticBurgersequation(SBE)drivenbylinearmultiplicativenoise:
(cid:0) (cid:1)
du= νu −uu dt+σu◦dW,
xx x t
(2.1) u(0,t)=u(1,t)=0, t ≥0,
u(x,0)=u (x), x∈(0,1),
0
where ν is a positive diffusion coefficient, W is a two-sided one-dimensional
t
Wiener process, σ is a positive constant which measures the “amplitude” of the
noise,andu issomeappropriateinitialdatumtobespecifiedbelow. Tofixideas,
0
REGULARIZEDREDUCEDORDERMODELSFORASTOCHASTICBURGERSEQUATION 3
the multiplicative noise term σu◦dW is understood in the sense of Stratonovich
t
[51].
SPDEs driven by linear multiplicative noise such as the SBE (2.1) arise in var-
ious contexts, including turbulence theory or non-equilibrium phase transitions
[9,26,48], the modeling of randomly fluctuating environment [7] in spatially-
extended harvesting models [20,34,46,47,55,56], or simply the modeling of pa-
rameterdisturbances[10].
2.2. Proper Orthogonal Decomposition (POD). We present in this section a
very brief account of the proper orthogonal decomposition (POD). The reader is
referred to, e.g., [35,50,58] for more details. The POD starts with the snapshots,
which, in this paper, are numerical approximations of the SBE (2.1) at different
time instances. The POD seeks a low-dimensional basis that approximates the
snapshots optimally with respect to a certain norm. In this paper, we employ the
commonlyusedL2-norm(see,e.g.,[41]foralternatives). Thesolutionofthemin-
imizationproblemisequivalenttothesolutionofaneigenvalueproblem[15]. The
PODsubspaceofagivendimensionrisspannedbythefirstrPODbasisfunctions,
which are the normalized functions {ϕ }r that correspond to the first r largest
j j=1
eigenvaluesoftheaforementionedeigenvalueproblem:
(2.2) Xr :=span{ϕ ,...,ϕ }.
1 r
Note that the POD functions are orthogonal to each other with respect to the L2-
innerproduct(cid:104)·,·(cid:105)ontheunderlyingphasespace:
(2.3) (cid:104)ϕ ,ϕ (cid:105)=δ ,
i j ij
whereδ denotestheKronecker-delta. Notealsothatin(2.3)andtheremainderof
ij
the paper, the POD basis functions are considered as continuous functions on the
spatialdomain,sincetheyarelinearcombinationsoffiniteelementbasisfunctions.
2.3. GalerkinROM(G-ROM)forSBE. ThederivationofthePOD-basedGalerkin
ROM (G-ROM) follows the standard Galerkin approximation procedure with the
underlyingbasistakentobethePODbasis. Forthesakeofclarity,wesketchthis
derivation for the SBE (2.1) below. Given a positive integer r, the r-dimensional
PODGalerkinapproximationu oftheSBEsolutionutakesthefollowingform:
r
r
(2.4) u (x,t;ω):= ∑a (t;ω)ϕ (x),
r j j
j=1
wherethetime-varyingrandomcoefficients{a (t,ω)}r aredeterminedbysolv-
j j=1
ing:
(cid:10) (cid:11) (cid:10)(cid:0) (cid:1) (cid:11)
(2.5) du ,ϕ = ν(u ) −u (u ) ,ϕ dt+σ(cid:104)u ,ϕ (cid:105)◦dW, j=1,···,r.
r j r xx r r x j r j t
Theabovesystemcanberecastintothefollowingmoreexplicitformbyusingthe
expansionofu givenin(2.4)andtheorthogonalitypropertysatisfiedbythePOD
r
4 TRAIANILIESCU,HONGHULIU,ANDXUPINGXIE
basisfunctionsgivenin(2.3):
(2.6)
(cid:104) r r (cid:105)
(cid:10)(cid:0) (cid:11) (cid:10) (cid:11)
da = −ν ∑a (ϕ ) ,(ϕ ) + ∑ a a ϕ (ϕ ) ,ϕ dt+σa ◦dW,
j k k x j x k l k l x j j t
k=1 k,l=1
where j=1,···,r. This system of stochastic differential equations (SDEs) is the
r-dimensionalGalerkinROMfortheSBE(2.1).
2.4. Leray ROM (L-ROM) for SBE. To investigate fixes for G-ROM’s poten-
tial numerical instability in the convection-dominated regime of the SBE (2.1),
we draw inspiration from the deterministic case and consider regularized ROMs
(Reg-ROMs). TheseReg-ROMsbelongtothewideclassofstabilizedROMs(see,
e.g., [2–6,8,16,25,31,35,38,50,52,54,60,62] for such examples). What distin-
guishes the Reg-ROMs from the other stabilized ROMs is that they increase the
numerical stability of the model by using explicit spatial filtering, which is a rel-
atively new concept in the ROM field [57,60,61]. In this study, we will use the
simplest such Reg-ROM, the Leray ROM (L-ROM) [57,61], which is based on a
specificwayoffilteringtheconvectivetermintheSBE(2.1)asexplainedbelow.
TheLeraymodelwasfirstusedbyLeray[44]asatheoreticaltooltoprovelocal
existenceanduniquenessofweaksolutionsoftheNSE.TheLeraymodelhasbeen
used as a numerical tool in the simulation of convection-dominated deterministic
flowswithstandard(e.g.,finiteelement)numericalmethods[24,30,43]. Ithasalso
beenusedtoderiveReg-ROMsfordeterministicsystemsin[57,61].
TheextensionoftheL-ROMproposedin[57,61]tothestochasticproblem(2.1)
at hand is straightforward. There is only one crucial difference in its derivation
comparedtothederivationoftheG-ROMasoutlinedinSection2.3,whichconsists
ofreplacingthenonlineartermu (u ) in(2.5)byaregularizedtermu (u ) here.
r r x r r x
Thisregularized version, u , ofu isobtained basedon theusage ofthe following
r r
POD differential filter (DF)1 : Let δ be the radius of the DF. For a given u ∈Xr,
r
findu ∈Xr suchthat
r
(2.7) (cid:10)(cid:0)I−δ2∆(cid:1)u ,ϕ (cid:11)=(cid:104)u ,ϕ (cid:105), ∀j=1,...r.
r j r j
Namely, the r-dimensional L-ROM approximation u of the SBE solution u
r
takesthefollowingform:
r
(2.8) u (x,t;ω):= ∑a (t;ω)ϕ (x),
r j j
j=1
wherethetime-varyingrandomcoefficients{a (t,ω)}r aredeterminedbysolv-
j j=1
ing:
(cid:10) (cid:11) (cid:10)(cid:0) (cid:1) (cid:11)
(2.9) du ,ϕ = ν(u ) −u (u ) ,ϕ dt+σ(cid:104)u ,ϕ (cid:105)◦dW, j=1,···,r.
r j r xx r r x j r j t
1Differentialfiltershavebeenusedinthesimulationofconvection-dominatedflowswithstandard
numericalmethods[28,29]. Inreducedordermodeling,theDFwasfirstusedin[57]andextended
in[61].
REGULARIZEDREDUCEDORDERMODELSFORASTOCHASTICBURGERSEQUATION 5
Since at each time instance t, the sought regularization ur(t,·;ω) lives in Xr, it
admitsthefollowingexpansion:
r
(2.10) u (t,x;ω)≡ ∑a (t;ω)ϕ (x),
r k k
k=1
wherea canbedeterminedbyusingtheexpansion(2.10)in(2.7),whichleadsto
j
r r r
(2.11) ∑a (t;ω)ϕ = ∑a (t;ω)ϕ = ∑a (t;ω)ϕ ,
k k k k k k
k=1 k=1 k=1
andthefilteredPODmodeϕ ,1≤k≤r,isdeterminedvia
k
(2.12) (cid:10)(cid:0)I−δ2∆(cid:1)ϕ ,ϕ (cid:11)=(cid:104)ϕ ,ϕ (cid:105), ∀j=1,...r.
k j k j
Consequently, in contrast to the G-ROM given in (2.6), the r-dimensional L-
ROMforSBE(2.1)isgivenby:
(2.13)
(cid:104) r r (cid:105)
(cid:10)(cid:0) (cid:11) (cid:10) (cid:11)
da = −ν ∑a (ϕ ) ,(ϕ ) + ∑ a a ϕ (ϕ ) ,ϕ dt+σa ◦dW,
j k k x j x k l k l x j j t
k=1 k,l=1
where j=1,···,r.
3. COMPUTATIONAL INVESTIGATION
In this section, we present a computational investigation on potential numeri-
cal instability of the standard G-ROM (2.6) for the SBE (2.1) and on a possible
alleviationofsuchinstabilityachievedbytheL-ROM(2.13).
Ithasbeenobservedinapreviousstudy[61]that,forthedeterministicBurgers
equation placed in a convection-dominated regime, the G-ROM yields excessive
spuriousoscillations,especiallywhenthedimensionoftheG-ROMislow. Similar
to [61], we set up the numerical experiments for the SBE (2.1) in a regime with a
small diffusion coefficient (ν =10−3) and a steep internal layer; see Section 3.1.
In Section 3.2, the emergence of such oscillations is confirmed in the current sto-
chastic setting as well. The improvement achieved by the L-ROM in the form of
significant reduction of the spurious oscillations is then presented in Section 3.3.
Finally, some preliminary statistical tests are presented in Section 3.4, which also
showstherobustnessoftheL-ROMwithrespecttothestrengthofthenoise.
3.1. Setup of the Numerical Experiments. In this section, we present a short
descriptionofthesetupofthenumericalexperiments.
NumericalDiscretizationoftheSBE.TheSBE(2.1)issolvedbyasemi-implicit
Eulerschemeasgivenin[18,Section6.1]. Forthereader’sconvenience,webriefly
describethenumericaldiscretizationbelow,andreferto[18,Section6.1]formore
details. Wealsoreferthereaderto[1,11,14,36,37,45]forothernumericalapprox-
imationschemesofnonlinearSPDEs.
Ateachtimestepthenonlinearityuu =(u2) /2andthenoisetermσu◦dW are
x x t
treatedexplicitly,andtheothertermsaretreatedimplicitly. TheLaplacianoperator
6 TRAIANILIESCU,HONGHULIU,ANDXUPINGXIE
is discretized using the standard second-order central difference approximation.
Theresultingsemi-implicitschemereadsasfollows:
(3.1) un+1−un=(cid:16)ν∆ un+1+σ2un−1∇ (cid:0)(un)2(cid:1)(cid:17)∆t+σζ un√∆t,
j j d j 2 j 2 d j n j
where un is the discrete approximation of u(j∆x,n∆t), ∆x the mesh size of the
j
spatial discretization, and ∆t the time step. The discretized Laplacian ∆ and the
d
discretizedspatialderivative∇ in(3.1)aregivenby
d
un −2un+un (un )2−(un)2
∆ un= j−1 j j+1; ∇ (cid:0)(un)2(cid:1)= j+1 j , j∈{1,···,N −2}.
d j (∆x)2 d j ∆x x
Theboundaryconditionsin(3.1)areun=un =0,whereN isthetotalnumber
0 Nx−1 x
ofgridpointsusedforthediscretizationofthespatialdomain[0,1]. Theζ in(3.1)
n
are random variables drawn independently from a normal distribution N (0,1).
Note that the additional drift term σ2un/2 in the RHS of (3.1) is due to the con-
j
version of the Stratonovich noise term σu◦dW into its Itoˆ form. Throughout the
t
paper,thesimulationsoftheSBE(2.1)areperformedfor∆t=10−4andN =1025
x
sothat∆x≈9.8×10−4. Thediffusioncoefficientν issettobe0.001. Thevalues
oftheparameterσ willbespecifiedbelow.
Choice of the Initial Data. The initial condition is chosen to be a mollified and
slightlyshiftedversionofthestepfunctionusedin[40],whichisgivenby
(cid:90) ∞
(3.2) u (x)= ξ(y)φ (x−y)dy, x∈[0,1].
0 ε
−∞
Here, ξ is the step function defined by ξ(x)=1 if x∈(0.05,0.55) and ξ(x)=0
otherwise. Themollifierφ isgivenbyφ (x)= 1φ(x)with
ε ε ε ε
(cid:40)Cexp(cid:0)− 1 (cid:1) if|x|<1,
φ(x)= (1−x2)
0 otherwise,
andthenormalizationconstantC ischosensuchthat(cid:82)1 φ(x)dx=0. Throughout
−1
ournumericalexperiments,theparameterε inthemollifierφ issettobeε =0.01.
ε
The modification adopted here is mainly intended to enforce the compatibility
oftheinitialandboundaryconditionattheleftboundarypoint(x=0)andtoavoid
anypotentialregularityissuesthatmayariseinournumericaldiscretizationofthe
SBEin(3.1)duetothediscontinuityinthestepfunction.
As will be seen below, by choosing such a step-function like initial profile and
by setting the diffusion constant ν sufficiently small, the SBE exhibits interesting
transientdynamicsthatwillturnouttobeagoodlaboratorytostudythepotential
instabilityoftheG-ROM;cf.Fig.1.
NumericalIntegrationoftheROMs. ThediscretizationoftheG-ROM(2.6)and
the L-ROM (2.13) are carried out by using a standard Euler-Maruyama scheme
(see, e.g., [39, p. 305]). For instance, the corresponding G-ROM discretization is
REGULARIZEDREDUCEDORDERMODELSFORASTOCHASTICBURGERSEQUATION 7
givenby:
an+1−an=(cid:104)−ν ∑r an(cid:10)(cid:0)(ϕ ) ,(ϕ ) (cid:11)+σ2an
j j k k x j x 2 j
k=1
(3.3)
r (cid:105) √
+ ∑ anan(cid:10)ϕ (ϕ ) ,ϕ (cid:11) ∆t+σζ an ∆t, j=1,···,r,
k l k l x j n j
k,l=1
where, as in (3.1), ζ are random variables drawn independently from a normal
n
distribution N (0,1), and n = 1,···,N, with N being the total number of time
steps.
3.2. G-ROMResults: SpuriousOscillations. Inthissection, weassesstheper-
formanceoftheG-ROMinitsabilitytoreproducetheSBE’sspatio-temporalfield
forafixednoiseamplitudeσ =0.3andanarbitrarilyfixedrealizationofthenoise.
ThestatisticalrelevanceoftheresultspresentedinthissectionisconfirmedinSec-
tion3.4.
For this purpose, we first simulate the SBE (2.1) over the time interval [0,1]
followingthenumericalsetuppresentedinSection3.1andconstructthePODbasis
functions used in the derivation of the G-ROM (2.6). In Fig. 1, the numerically
simulated spatio-temporal field of the SBE (2.1) as well as the initial profile and
thefinaltimesolutionprofileareplotted.
FIGURE 1. The numerically simulated spatio-temporal field of
the SBE (2.1) with σ =0.3 forced by an arbitrary realization of
the noise (left panel), and the initial profile given by (3.2) with
ε =0.01 (right panel, solid line) as well as the solution profile at
timet =1(rightpanel,dashedline).
ToconstructthePODbasisfunctionsusedinthederivationoftheG-ROM(2.6),
we collected 101 equally spaced snapshots (without subtracting the centering tra-
jectory) from the simulated SBE spatio-temporal field, and we used the method
of snapshots [58]. For illustration purposes, we plot four POD basis functions in
Fig.2.
8 TRAIANILIESCU,HONGHULIU,ANDXUPINGXIE
' ' ' '
1 3 5 7
2 2
0
1
1 1
-0.5
0
0 0
-1 -1 -1 -1
-1.5 -2 -2 -2
0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1
x x x x
FIGURE 2. A few POD basis functions constructed based on the
spatio-temporalfieldplottedinFig.1.
ThetestsfortheG-ROMarecarriedoutwithdimensionr=6aswellasr=10;
the results are plotted in Fig. 3. In both cases, the percentage of the total kinetic
energy contained in the first r modes is already high: 98.5% for r=6 and 99.3%
for r=10. Despite such a high percentage of energy captured by the first r POD
modes, the corresponding G-ROM exhibits very strong spurious oscillations, as
can be observed from both the reconstructed spatio-temporal fields and the final-
timesolutionprofilesinFig.3. Ontheotherhand,aninspectionontheevolutionof
theprojecteddynamicsontoeachPODmoderevealsthattheG-ROMisperform-
ing actually quite well in modeling the dynamics of the first two modes, while its
performancedeterioratesforhigherfrequencymodes;seeFig.4forthecaser=6.
For the SBE problem studied here, as the dimension of the G-ROM increases,
theoverallaccuracyalsoimproves,ascanalreadybeseeninFig.3. Notealsothat
theG-ROMperformanceimprovesasthediffusioncoefficientν increases(results
not shown). This behavior is expected since increasing ν increases the diffusion
effects,which,inturn,reducesthesteepnessofthelocalizedinternallayer. These
numerical results suggest the convection-dominated regime to be a primary cause
of the G-ROM’s numerical instability observed here, just as in the deterministic
case[61].
3.3. L-ROM Results: Alleviation of G-ROM’s Spurious Oscillations. In this
section, we illustrate that the G-ROM’s spurious oscillations such as those illus-
tratedintheprevioussectioncanbealleviatedbyusingtheL-ROM(2.13)derived
in Section 2.4 when the spatial filtering parameter δ is appropriately calibrated;
cf.(2.12).
Wechoosetheoptimalvalueofthisfreeparameterδ tobethevaluethatmini-
mizestheL2-errorofthecorrespondingL-ROMinreconstructingtheSBE’sspatio-
temporalfield. Inournumericalexperiments,wefindtheoptimalvaluebytrialand
error. Toreducethenumericalefforts,especiallyinviewofthestatisticaltestgiven
in the next section, all the numerical results related to the L-ROM (2.13) are ob-
tained for δ =0.12, which is a nearly optimal δ value for the r =10 and σ =0
REGULARIZEDREDUCEDORDERMODELSFORASTOCHASTICBURGERSEQUATION 9
FIGURE3. Spatio-temporalfield,uG:=∑rj=1ajϕj,reconstructed
from the numerical simulation of the G-ROM (2.6) with dimen-
sionr=6(leftpanel)andr=10(middlepanel),respectively. The
noise path is the same as that used to generate the SBE’s spatio-
temporalfieldplottedinFig.1;σ =0.3. Thecorrespondingsolu-
tionprofilesattimet =1areshownintherightpanel.
a a a
1 2 3
0.4 0.2
SBE
-0.4
G-ROM 0.2
-0.5
0 0
-0.6
-0.7 -0.2
-0.2
0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1
t t t
a a a
4 5 6
0.2
0.2 0.2
0 0 0
-0.2
-0.2 -0.2
0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1
t t t
FIGURE 4. The time series aj, 1≤ j≤r, as modeled by the G-
ROM (2.6) with dimension r=6 (blue curves). Also plotted are
the SBE solution projections onto the first r POD modes (black
curves).
case.2 The L-ROM results corresponding to those plotted in Figs. 3 and 4 for the
G-ROM are shown in Figs. 5 and 6, respectively. As can be observed from these
results, the spurious oscillations are indeed significantly reduced in the L-ROM
2Wehavecheckedthat,undertheparametersettingusedtogenerateFigs.5and6,theδ value
wechose(δ =0.12)isclosetotheoptimalδ valuesforbothther=6andr=10cases.
10 TRAIANILIESCU,HONGHULIU,ANDXUPINGXIE
dynamics,andanimprovementinthemodelingoftheSBE’sspatio-temporalfield
isalsoachieved.
It is also interesting to note that although the regularization used in the L-
ROMsuccessfullyreducesthespuriousoscillationobservedintheG-ROM’shigh-
frequencymodes,itleadstoaslightdeteriorationonthemodelingoftheprojected
dynamics onto the first POD mode as can be seen by comparing the upper left
panelsofFig.6andFig.4. Thisdeteriorationisalsoobservedeveniftheoptimal
δ value is used. Of course, the deterioration is reduced when the dimension of
theL-ROMisincreased. Weintendtofurtherinvestigatethisissue(togetherwith
potentialL-ROMimprovements)inaseparatecommunication.
FIGURE 5. Results corresponding to Fig. 3 for the L-
ROM (2.13), where the spatio-temporal field uL := ∑rj=1ajϕj is
reconstructedfromthenumericalsimulationof(2.13)withdimen-
sionr=6(leftpanel)andr=10(middlepanel).
3.4. RobustnessoftheL-ROMresults. Inthissection,wepresentsomefurther
numericalresultsregardingthestatisticalrelevanceoftheresultsgiveninSections
3.2 and 3.3. We also investigate the effect of the magnitude of the noise on the
results.
Forthis purpose, the performancesofthe G-ROMand L-ROMareassessed by
usingtherelativeL2-errorscomputedasfollows:
(cid:113)
(cid:82)1(cid:82)1|u(·,·;ω)−u (·,·;ω)|2dxdt
0 0 r
(3.4) E(ω)= ×100%,
(cid:113)
(cid:82)1(cid:82)1|u(·,·;ω)|2dxdt
0 0
whereforeachsamplepathω,u(·,·;ω)denotesthesolutiontotheSBE(2.1),and
u (·,·;ω)denotesthesolutiontoeithertheG-ROM(2.6)ortheL-ROM(2.13)with
r
dimensionr.
We consider 13 noise magnitude σ values equally spaced between 0 and 0.6.
For each of these σ values, we perform 3000 numerical simulations of the fine
resolutiondiscretizationoftheSBE(toobtainu)andthetwoROMs(toobtainu ).
r
ThedimensionoftheROMsischosentober=10,andtheparameterδ usedinthe
