www.elsevier.com/locate/cma

Comput. Methods Appl. Mech. Engrg. 196 (2007) 4184–4195

Proper orthogonal decomposition approach and error estimationof mixed finite element methods for the tropical Pacific Ocean

reduced gravity model

Zhendong Luo a, Jiang Zhu b, Ruiwen Wang b, I.M. Navon c,*

a School of Science, Beijing Jiaotong University, Beijing 100044, PR Chinab Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, PR China

c School of Computational Science and Department of Mathematics, Florida State University, Dirac Science Library Building,

#483, Tallahassee, FL 32306-4120, USA

Received 19 January 2007; received in revised form 9 March 2007; accepted 12 April 2007Available online 8 May 2007

Abstract

In this paper, the tropical Pacific Ocean reduced gravity model is studied using the proper orthogonal decomposition (POD) techniqueof mixed finite element (MFE) method and an error estimate of POD approximate solution based on MFE method is derived. POD is amodel reduction technique for the simulation of physical processes governed by partial differential equations, e.g., fluid flows or othercomplex flow phenomena. It is shown by numerical examples that the error between POD approximate solution and reference solution isconsistent with theoretical results, thus validating the feasibility and efficiency of POD method.� 2007 Elsevier B.V. All rights reserved.

Keywords: Proper orthogonal decomposition technique; Mixed finite element method; Error estimate; Numerical simulation

1. Introduction

The variability of fluid flow and fluid total layer thick-ness over tropical oceans is an important question in stud-ies of climate change and air–sea interaction. However, theaccurate assessment of fluid flow and fluid total layer thick-ness is greatly limited due to the lack of direct measure-ments and the insufficient knowledge of air–sea exchangeprocesses. The tropical Pacific Ocean reduced gravitymodel is a useful model to simulate fluid flow and fluidtotal layer thickness over tropical Pacific Ocean and ithas been extensively applied to studying the ocean dynam-ics in tropical regions (see Cane [1] and Seager et al. [2]).The model consists of two layers above the thermoclinewith the same constant density. The ocean below the ther-

0045-7825/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2007.04.003

* Corresponding author. Tel.: +1 850 644 6560; fax: +1 850 644 0098.E-mail address: navon@scs.fsu.edu (I.M. Navon).

mocline, with a higher density, is assumed to be sufficientlydeep so that its velocity vanishes (Fig. 1). The upper of thetwo active layers is a fixed-depth surface layer in which thethermodynamics are included. The surface layer communi-cates with the lower active layer through entrainment/detrainment at their interface and through frictional hori-zontal shearing. We assume that there is no density differ-ence across the base of the surface layer; that is, the surfacelayer is treated as part of the upper layer.

Following Seager et al. [2], the equations for the depth-averaged currents are

ouot � fv ¼ �g0 oh

ox þ sx

q0H þ Ar2u; ðx; y; tÞ 2 X� ð0; T 1Þ;ovot þ fu ¼ �g0 oh

oy þ sy

q0H þ Ar2v; ðx; y; tÞ 2 X� ð0; T 1Þ;

ohot þ H ou

ox þ ovoy

� �¼ 0; ðx; y; tÞ 2 X� ð0; T 1Þ;

8>>><>>>:

ð1:1Þ

with the boundary conditions

Fig. 1. Ocean depth.

Z. Luo et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4184–4195 4185

uðx; y; tÞjoX ¼ ubðx; y; tÞ; ðx; y; tÞ 2 oX� ð0; T 1Þ;vðx; y; tÞjoX ¼ vbðx; y; tÞ; ðx; y; tÞ 2 oX� ð0; T 1Þ;hðx; y; tÞjoX ¼ hbðx; y; tÞ; ðx; y; tÞ 2 oX� ð0; T 1Þ;

8><>: ð1:2Þ

and initial condition

uðx; y; 0Þ ¼ u0ðx; yÞ; vðx; y; 0Þ ¼ v0ðx; yÞ;hðx; y; 0Þ ¼ h0ðx; yÞ; ðx; yÞ 2 X; ð1:3Þ

where (u,v) is the horizontal velocity of the depth-averagedcurrents; h the total layer thickness; f the Coriolis force; H

the mean depth of the layer (constant); q0 the density ofwater; g 0 the reduced gravity; and A the horizontal eddyviscosity coefficient. The wind stress vector (sx,sy) is calcu-lated by the aerodynamic bulk formula

ðsx; syÞ ¼ qaCD

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU 2

wind þ V 2wind

qðU wind; V windÞ:

Here qa is the density of the air; CD the wind stress dragcoefficient; (Uwind,Vwind) the wind velocity vector; X de-notes the two dimensional rectangular domain, which ischosen from 30�S to 30�N in latitude and from 130�E to70�W in longitude; ub(x,y, t), vb(x,y, t), hb(x,y, t), u0(x,y),v0(x,y), and h0(x,y) are all given functions.

The seasonal net surface heat flux over tropical oceanshas been only simulated with Eqs. (1.1)–(1.3) augmentedby a thermodynamics equation as in Yu and O’Brien (see[3]). However, since the computational field over the trop-ical Pacific Ocean is very extensive, in order to obtain highresolution numerical solutions of fluid flow and fluid totallayer thickness over tropical oceans, gridding points mustbe taken with enough density, thus requiring a large num-ber of degree of freedom rendering computing very diffi-cult. Thus, an important problem is how to simplify thecomputational load and save time-consuming calculationsand resource demands in the actual computational processin a sense that guarantees a sufficiently accurate numerical

solution. Proper orthogonal decomposition (POD) is alsoknown as Karhunen–Loeve expansions in signal processingand pattern recognition (see [4]), principal componentanalysis in statistics (see [5]), and the method of empiricalorthogonal functions in geophysical fluid dynamics (see[6,7]) or meteorology (see [8]). The POD technique offersadequate approximation to represent fluid flow with areduced number of degrees of freedom, i.e., with lowerdimensional models (see [9]) in order to simplify the com-putation and save CPU and memory requirements. PODhas also found widespread applications in problems relatedto the approximation of large-scale models. Although thebasic properties of POD method are well established andstudies have been conducted to evaluate the suitability ofthis technique for various fluid flows (see [10–12]), its appli-cability and limitations for actual fluid flow and fluid totallayer thickness over the tropical Pacific Ocean are not welldocumented.

The POD method mainly provides a useful tool for effi-ciently approximating a large amount of data. The methodessentially provides an orthogonal basis for representingthe given data in a certain least squares optimal sense, thatis, it provides a way to find optimal lower dimensionalapproximations of the given data. In addition to beingoptimal in a least squares sense, POD has the property thatit uses a modal decomposition that is completely datadependent and does not assume any prior knowledge ofthe process used to generate the data. This property isadvantageous in situations where a priori knowledge ofthe underlying process is insufficient to warrant a certainchoice of basis. Combined with the Galerkin projectionprocedure, POD provides a powerful method for generat-ing lower dimensional models of dynamical systems thathave a very large or even infinite dimensional phase space.The fact that this method always searches for linear (oraffine) subspaces instead of curved submanifolds makes it

4186 Z. Luo et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4184–4195

computationally tractable. In many cases, the behavior of adynamic system is governed by characteristics or relatedstructures, even though the ensemble is formed by a largenumber of different instantaneous solutions. POD methodcan capture these temporal and spatial structures by apply-ing a statistical analysis to the ensemble of data.

In fluid dynamics, Lumley first employed the POD tech-nique to capture the large eddy coherent structures in a tur-bulent boundary layer (see [13]); this technique was furtherextended in [14], where a link between the turbulent struc-ture and dynamics of a chaotic system was investigated. InHolmes et al. [9], the overall properties of POD arereviewed and extended to widen the applicability of themethod. The method of snapshots was introduced by Siro-vich [15], and is widely used in applications to reduce theorder of POD eigenvalue problem. Examples of these areoptimal flow control problems [16–18] and turbulence[9,13,14,19,20].

In many applications, the POD method is used to gener-ate basis functions for a reduced order model (ROM),which can simplify and provide quicker assessment of themajor features of the fluid dynamics for the purpose of flowcontrol demonstrated by Kurdila et al. [18] or design opti-mization shown by Ly et al. [17]. This application is used ina variety of other physical applications, such as in [17],which demonstrates an effective use of POD for a chemicalvapor deposition (CVD) reactor.

In [21], while the tropical Pacific Ocean reduced gravitymodel is preliminarily dealt with POD method, an exacttheoretical analysis was not carried out, in particular anerror estimate of the POD approximate solution was notas yet derived. The objective of this paper is to investigatein depth to what extent can POD be successfully used toapproximate a mixed finite element (MFE) solution forthe tropical Pacific Ocean reduced gravity model. In partic-ular, we aim to provide an error estimate of the approxi-mate MFE solution, so that one could determine thenumber of required eigenmodes. Some numerical examplesare provided for validating the proposed theory.

2. Outline of proper orthogonal decomposition technique

The essential problem of POD is to identify the underly-ing, coherent structures of a collected ensemble of data.POD entails finding the optimal bases and constructing amodel of reduced dimension to approximate the originalensemble. Originally, POD was used as a data representa-tion technique. For model reduction of dynamical systems,POD may be used on data points derived from system tra-jectories obtained via experiments, numerical simulations,or analytical derivations.

2.1. Continuous case

Let Uið~xÞ ði ¼ 1; 2; . . . ; nÞ denote the set of n observa-tions (also called snapshots) of some physical process taken

at position~x ¼ ðx; yÞ. The average of the ensemble of snap-shots is given by

U ¼ hUi ¼ 1

n

Xn

i¼1

Uið~xÞ: ð2:1Þ

We form new ensemble by focusing on deviations from themean as follows:

V i ¼ U i � U : ð2:2Þ

We wish to find an optimal compressed description of thesequence of data (2.2). One description of the process is aseries expansion in terms of a set of basis functions. Intui-tively, the basis functions should in some sense be represen-tative of the members of the ensemble. Such a coordinatesystem, which possesses several optimality properties (tobe discussed in the sequel), is provided by the Karhunen–Loeve expansion (see [4]), where the basis functions areU, in fact, admixtures of the snapshots and are given by

U ¼Xn

i¼1

aiV ið~xÞ; ð2:3Þ

where the coefficients ai are to be determined such that Ugiven by (2.3) will most resemble the ensemble fV ið~xÞgn

i¼1.More specifically, POD seeks a function U such that

1

n

Xn

i¼1

jðV i;UÞj2; ð2:4Þ

subject to

ðU;UÞ ¼ kUk20 ¼ 1; ð2:5Þ

is minimized, where (Æ, Æ) and k Æk0 denote the usual L2-innerproduct and L2-norm, respectively (see Section 3.1).

It follows that (see, e.g. [22]) the basis functions are theeigenfunctions of the integral equationZ

XCð~x;~x0ÞUð~x0Þdx0 ¼ kUð~xÞ; ð2:6Þ

where the covariance kernel is given by

Cð~x;~x0Þ ¼ 1

n

Xn

i¼1

V ið~xÞV ið~x0Þ: ð2:7Þ

Substituting (2.3) into (2.6) yields the following eigenvalueproblem:

Xn

j¼1

Lijaj ¼ kai; i ¼ 1; 2; . . . ; n; ð2:8Þ

where Lij ¼ 1n ðV i; V jÞ, L = (Lij)n·n is a symmetric and non-

negative matrix. Thus we see that our problem amountsto solving for the eigenvectors of an n · n matrix where n

is the size of the ensemble of snapshots. Straightforwardcalculation (see also [22]) shows that the cost functional

1

n

Xn

i¼1

jðV i;UÞj2 ¼ ðkU;UÞ ¼ k

Z. Luo et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4184–4195 4187

is maximized when the coefficients ai (i = 1,2, . . . ,n) of (2.8)are the elements of the eigenvector corresponding to thelargest eigenvalue of L.

2.2. Discrete case

Alternatively, we also can consider the discrete Karh-unen–Loeve expansion to find an optimal representationof the ensemble of snapshots. In general, each sample ofsnapshots U ið~xÞ (defined on a set of m nodal points ~x)can be expressed as a m dimensional vector ~ui as follows:

~ui ¼ ðui1; ui2; . . . ; uimÞT; ð2:9Þ

where uij denotes the jth component of the vector~ui ði ¼ 1; 2; . . . ; nÞ. The mean of vector is given by

�uk ¼Xn

i¼1

uik; k ¼ 1; 2; . . . ;m: ð2:10Þ

We also can form a new ensemble by focusing on devia-tions from the mean value as follows:

vik ¼ uik � �uk; k ¼ 1; 2; . . . ;m: ð2:11Þ

Let the matrix A denote the new ensemble

A ¼

v11 v21 � � � vn1

v12 v22 � � � vn2

..

. ... ..

. ...

v1m v2m � � � vnm

0BBBB@

1CCCCA

m�n

;

where the discrete covariance matrix of the ensemble maybe written as

AAT~yk ¼ kk~yk; k ¼ 1; 2; . . . ;m; ~yk 2 Rm�m: ð2:12Þ

Thus, to compute the POD mode, one must solve a m · m

eigenvalue problem. For a discretization of an ocean prob-lem, the dimension often exceeds 104, so that a direct solu-tion of this eigenvalue problem is often not feasible. Wecan transform the eigenvalue problem into an n · n eigen-value problem (see [23]). In the method of snapshots, onethen solves the n · n eigenvalue problem

ATA~wk ¼ kk~wk; k ¼ 1; 2; . . . ; n; ~wk 2 Rn�n; ð2:13Þ

where the nonzero eigenvalues kk (1 6 k 6 n) are the samein (2.12). The eigenvectors may be chosen to be orthonor-mal, and the POD modes are given by ~/k ¼ A~wk=

ffiffiffiffiffikk

p. In

matrix form, with U ¼ ð~/1;~/2; . . . ;~/nÞ, and W ¼ ð~w1;~w2; . . . ; ~wnÞ, this becomes U = AW.

The eigenvalue problem (2.13) is more efficient than them · m eigenvalue problem (2.12) when the number of snap-shots n is much smaller than the number of states m.

3. POD technique and error estimate of MFE method for

tropical Pacific Ocean reduced gravity model

In this section, we apply the POD technique and MFEmethod to the upper tropical Pacific Ocean model

described in Section 1. This method provides a systematicway of creating a reduced basis space using the state ofthe system at different time instances. As in the generalreduced order basis methods, the states could come fromfull order numerical computations (also obtained from sys-tem trajectories obtained via experiments, or by analyticalderivations). Here, we apply the MFE methods to theupper tropical Pacific Ocean model for obtaining a fullorder numerical solution, then apply the POD techniqueto reconstruct the approximate solution and approximatethe solution of the reduced model. Finally, we comparethe error of the accurate solution with that of the approx-imate solution.

3.1. MFE method for the tropical Pacific Ocean reduced

gravity model

The Sobolev spaces along with their properties used inthis context are standard (cf. Ref. [24]). For example, forbounded domain X, we denote by Hm(X) (mP0) andL2(X) = H0(X) the usual Sobolev spaces equipped withthe semi-norm and the norm, respectively,

jvjm;X ¼Xjaj¼m

ZXjDavj2 dxdy

( )1=2

8v 2 H mðXÞ;

kvkm;X ¼Xm

i¼0

jvj2i;X

( )1=2

8v 2 H mðXÞ;

where a = (a1,a2), a1 and a2 are tow nonnegative integers,and jaj = a1 + a2. Especially, the subspace H 1

0ðXÞ ofH1(X) is denoted by

H 10ðXÞ ¼ fv 2 H 1ðXÞ; ujoX ¼ 0g:

Note that k Æk1 is equivalent to j Æ j1 in H 10ðXÞ. Let

L20ðXÞ ¼ fq 2 L2ðXÞ;

RX qdxdy ¼ 0g, which is the subspace

of L2(X). For the sake of convenience, we consider themixed variational formulation for (1.1) with the boundaryconditions

uðx; y; tÞjoX ¼ 0; vðx; y; tÞjoX ¼ 0; hðx; y; tÞjoX ¼ 0;

0 6 t 6 t1:

Therefore, the variational form for the tropical PacificOcean reduced gravity model can be written as:

Problem (I). Find ðu; v; hÞ 2 H10ðXÞ � H1

0ðXÞ � L20ðXÞ such

that

ðut;uÞ � f ðv;uÞ � g0ðh;uxÞ þAðru;ruÞ ¼ ðf1;uÞ8u 2 H 1

0ðXÞ;ðvt;wÞ þ f ðu;wÞ � g0ðh;wyÞ þAðrv;rwÞ ¼ ðf2;wÞ8w 2 H 1

0ðXÞ;ðht;qÞ þHðuxþ vy ;qÞ ¼ 0 8q 2 L2

0ðXÞ;uðx; y;0Þ ¼ u0ðx; yÞ; vðx; y;0Þ ¼ v0ðx; yÞ;hðx; y;0Þ¼ h0ðx; yÞ; ðx; yÞ 2 X;

8>>>>>>>>>>><>>>>>>>>>>>:

4188 Z. Luo et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4184–4195

where f1 = sx/(q0H) and f2 = sy/(q0H). Using the same asapproach in Ref. [25], we could check that Problem (I)has a unique solution.

In order to find the numerical solution for Problem (I),it is necessary to discretize Problem (I). We introduce afinite element approximation for the spatial variable andfinite difference scheme for the time derivative. Let I�h bea uniform regular triangulation of X (here X denotes thetwo dimensional rectangular domain, which is chosen from30�S to 30�N in latitude and from 130�E to 70�W in longi-tude in actual computation), i.e., for any K 2 I�h, put�hK = diam{K}, �h ¼ maxK2I�hf�hKg, then c�h 6 �hK 6 c1�h.Denote the time step increment by k = T1/N (T1 beingthe total time) and MFE approximation of (u,v,h) byðul

�h; vl�h; h

l�hÞ ¼ ðu�hðx; y; tlÞ; v�hðx; y; tlÞ; h�hðx; y; tlÞÞ, tl = lk (0 6

l 6 N). Define the finite element subspaces ofH 1

0ðXÞ and L20ðXÞ as follows, respectively,

X �h ¼ fu�h 2 H 10ðXÞ; u�hjK 2 P mðKÞ 8K 2 I�hg;

L�h ¼ fq�h 2 L20ðXÞ; q�hjK 2 P m�1ðKÞ 8K 2 I�hg;

(

where m P 1 is integer, Pm(K) polynomial subspace of de-grees 6m on K. Then, the fully discrete formulation forProblem (I) can be written as:

Problem (II). Find ðul�h; v

l�h; h

l�hÞ 2 X �h � X �h � L�h (l = 1,2,

. . .,N) such that

ðul�h;u�hÞ � kf ðvl

�h;u�hÞ � kg0ðhl�h;u�hxÞ þ kAðrul

�h;ru�hÞ

¼ kðf l1 ;u�hÞ þ ðul�1

�h ;u�hÞ 8u�h 2 X �h;

ðvl�h;w�hÞ þ kf ðul

�h;w�hÞ � kg0ðh�h;w�hyÞ þ kAðrvl�h;rw�hÞ

¼ kðf l2 ;w�hÞ þ ðvl�1

�h ;w�hÞ 8w�h 2 X �h;

ðhl�h; q�hÞ þ kHðu�hlx þ v�hy ; q�hÞ ¼ ðhl�1

�h ; q�hÞ 8q�h 2 L�h;

l ¼ 1; 2; . . . ;N ;

u0�h ¼ u0ðx; yÞ; v0

�h ¼ v0ðx; yÞ; h0�h ¼ h0ðx; yÞ;

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

where f l

1 ¼ f1ðtlÞ and f l2 ¼ f2ðtlÞ. Using the same as ap-

proach as in [26], we could check that Problem (II) has un-ique solution ðul

�h; vl�h; h

l�hÞ 2 X �h � X �h � L�h, and if solution

(u,v,h) 2 Hm+1(X) · Hm+1(X) · Hm(X) of Problem (I), thefollowing error estimates hold:

kuðx; y; tlÞ � ul�hk0 þ k1=2Pl

j¼1

krðuðx; y; tlÞ � ul�hÞk0

6 cð�hm þ kÞ;

kvðx; y; tlÞ � vl�hk0 þ k1=2Pl

j¼1

krðvðx; y; tlÞ � vl�hÞk0

6 cð�hm þ kÞ;

khðx; y; tlÞ � hl�hk0 6 cð�hm þ kÞ; l ¼ 1; 2; . . . ;N ;

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð3:1Þ

where c is a constant independent of �h and k, but depen-dent of (u,v,h).

3.2. POD technique for the tropical Pacific Ocean reduced

gravity model

In the construction described above in Section 2, thenumber n may be large, depending on the complexity ofthe dynamics represented in the ‘‘snapshots’’. In general,one should take n sufficiently large so that the snapshotsUi(x,y) contain all the salient features of the dynamicsbeing investigated. Thus, the POD basis functions Ui

(1 6 i 6 n), used with the original dynamics in a Galerkinprocedure, offer the possibility of achieving a high fidelitymodel albeit with perhaps a dimension n.

To apply the POD techniques to the upper tropical Paci-fic Ocean model in Section 1, we first solve Problem (II) atN time steps and obtain the solutions ðul

�h; vl�h; h

l�hÞ ðl ¼ 1; 2;

. . . ;NÞ of velocity field and upper layer thickness at anincrement of k = T1/N (for example, T1 = 1 year) day for(x,y) 2 X. And then we choose n (for example, n = 5, 20,or 30, in general, n� N) snapshots U iðx; yÞ ¼ ðui

�h; vi�h;

hi�hÞ ði ¼ 1; 2; . . . ; nÞ (which is useful and of interest for us)

from N group of solutions ðul�h; v

l�h; h

l�hÞð1 6 l 6 NÞ for Prob-

lem (II). These snapshots are discrete data over X. Using(2.10), (2.11), and (2.13) yields covariance matricesAT

u Au;ATv Av;A

Th Ah associated with ðui

�h; vi�h; h

i�hÞ ði ¼ 1; 2;

. . . ; nÞ. Since those matrices are all nonnegative Hermitianmatrices, they all have a complete set of orthogonal eigen-vectors with the corresponding eigenvalues arranged inascending order as ku

1 P ku2 P � � �P ku

n P 0, kv1 P kv

2

P � � �P kvn P 0, and kh

1 P kh2 P � � �P kh

n P 0, respec-tively. Then we construct POD basis elements Uu

i ðx; yÞ;Uv

i ðx; yÞ;Uhi ðx; yÞ such that

X PODu ¼ span Uu

1ðx; yÞ;Uu2ðx; yÞ; . . . ;Uu

nðx; yÞ� �

;

X PODv ¼ span Uv

1ðx; yÞ;Uv2ðx; yÞ; . . . ;Uv

nðx; yÞ� �

;

X PODh ¼ span Uh

1ðx; yÞ;Uh2ðx; yÞ; . . . ;Uh

nðx; yÞ� �

;

8>>><>>>:

ð3:2Þ

are defined as

Uuj ðx; yÞ ¼

Xn

i¼1

ajuiu

i�h; Uv

jðx; yÞ ¼Xn

i¼1

ajviv

i�h;

Uhi ðx; yÞ ¼

Xn

i¼1

ajhih

i�h; ð3:3Þ

where ajui; aj

vi; ajhi ð1 6 i 6 nÞ are the components of the

eigenvectors AuV ju=

ffiffiffiffiffiffikuj

p, AvV j

v=ffiffiffiffiffiffikvj

p, AhV j

h=ffiffiffiffiffiffikhj

pcorre-

sponding to the eigenvalues kuj ; k

vj ; k

hj , respectively.

For three groups of basic functions Uui ðx; yÞ;Uv

i ðx; yÞ;Uh

i ðx; yÞ ði ¼ 1; 2; . . . ; nÞ, put

Z. Luo et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4184–4195 4189

us�h ¼ �uðx; yÞ þ

Pnj¼1

buj ðtsÞUu

j ðx; yÞ;

vs�h ¼ �vðx; yÞ þ

Pnj¼1

bvjðtsÞUv

jðx; yÞ;

hs�h ¼ �hðx; yÞ þ

Pnj¼1

bhj ðtsÞUh

j ðx; yÞ; s ¼ 1; 2; . . . ;

8>>>>>>><>>>>>>>:

ð3:4Þ

where buj ðtsÞ, bv

jðtsÞ, and bhj ðtsÞ (j = 1,2, . . .,n) are coeffi-

cients to determine; �uðx; yÞ, �vðx; yÞ, and �hðx; yÞ are the meanvalues of ðui

�h; vi�h; h

i�hÞ ði ¼ 1; 2; . . . ; nÞ, respectively. Note

that, if s = i(1 6 i 6 n), ðus�h; v

s�h; h

s�hÞ are the solutions for

Problem (II). Since the scales in model variables and arenot uniform, one may employ different modes to recon-struct the solutions. In order to reduce order for Problem(II), we apply the POD approximate solution

usM1¼ �uðx; yÞ þ

PM1

j¼1

buj ðtsÞUu

j ðx; yÞ;

vsM1¼ �vðx; yÞ þ

PM1

j¼1

bvjðtsÞUv

jðx; yÞ;

hsM2¼ �hðx; yÞ þ

PM2

j¼1

bhj ðtsÞUh

j ðx; yÞ; s ¼ 1; 2; . . . ;

8>>>>>>>><>>>>>>>>:

ð3:5Þ

where buj ðtsÞ and bv

jðtsÞ ðj ¼ 1; 2; . . . ;M1Þ, bhj ðtsÞ ðj ¼ 1; 2;

. . . ;M2Þ, �uðx; yÞ, �vðx; yÞ, and �hðx; yÞ are the same as Eq.(3.4). Substituting the solutions of Problem (II) with (3.4)and (3.5), respectively, we could obtain the following equa-tions, respectively.

Problem (III). Find ðbur ðtsÞ; bv

rðtsÞ; bhr ðtsÞÞ ðr ¼ 1; 2; . . . ; nÞ

such that

Fig. 2. Error of numerical solutions of different POD bases for 5, 20, and30 snapshots. (a) 5 snapshots; (b) 20 snapshots; and (c) 30 snapshots.

bur ðtsÞ� kf

Pnj¼1

bvjðtsÞðUv

j ;Uur Þ�kg0

Pnl¼1

bhl ðtsÞðUh

l ;UurxÞ

þkAPni¼1

bui ðtsÞðrUu

i ;rUur Þ¼ kðf s

1 ;Uur Þþbu

r ðts�1Þ;

bvrðtsÞþ kf

Pni¼1

bui ðtsÞðUu

i ;Uvr� kg0

Pnl¼1

bhl ðtsÞðUh

l ;UvryÞ

þkAPnj¼1

bvjðtsÞðrUv

j ;rUvrÞ¼ kðf s

2 ;UvrÞþbv

rðts�1Þ;

bhr ðtsÞþ kH

Pni¼1

bui ðtsÞðUu

ix;Uhr ÞþkH

Pnj¼1

bvjðtsÞðUu

jy ;Uhr Þ¼ bh

r ðts�1Þ;

r¼ 1;2; . . . ;n; s¼ 1;2; . . . ;

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:along with the initial condition

bur ð0Þ ¼ ðuðx; y; 0Þ � �uðx; yÞ;Uu

r ðx; yÞÞ;

bvrð0Þ ¼ ðvðx; y; 0Þ � �vðx; yÞ;Uv

rðx; yÞÞ;

bhr ð0Þ ¼ ðhðx; y; 0Þ � �hðx; yÞ;Uh

r ðx; yÞÞ; 1 6 r 6 n:

8>><>>:

ð3:6Þ

Problem (IV). Find ðbur ðtsÞ; bv

rðtsÞ; bh‘ðtsÞÞ (r = 1,2, . . . ,M1,

‘ = 1,2, . . . ,M2) such that

bur ðtsÞ� kf

PM1

j¼1

bvjðtsÞðUv

j ;Uur Þ� kg0

PM2

l¼1

bhl ðtsÞðUh

l ;UurxÞ

þkAPM1

i¼1

bui ðtsÞðrUu

i ;rUur Þ¼ kðf s

1 ;Uur Þþbu

r ðts�1Þ;

r¼ 1;2; . . . ;M1; s¼ 1;2; . . . ;

bvrðtsÞþ kf

PM1

i¼1

bui ðtsÞðUu

i ;Uvr�kg0

PM2

l¼1

bhl ðtsÞðUh

l ;UvryÞ

þkAPM1

j¼1

bvjðtsÞðrUv

j ;rUvrÞ¼ kðf s

2 ;UvrÞþbv

rðts�1Þ;

r¼ 1;2; . . . ;M1; s¼ 1;2; . . . ;

bh‘ðtsÞþ kH

PM1

i¼1

bui ðtsÞðUu

ix;Uh‘Þþ kH

PM1

j¼1

bvjðtsÞðUu

jy ;Uh‘Þ¼ bh

‘ðts�1Þ;

‘¼ 1;2; . . . ;M2; s¼ 1;2; . . . ;

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:along with the initial condition

bur ð0Þ ¼ ðuðx; y; 0Þ � �uðx; yÞ;Uu

r ðx; yÞÞ; r ¼ 1; 2; . . . ;M1;

bvrð0Þ ¼ ðvðx; y; 0Þ � �vðx; yÞ;Uv

rðx; yÞÞ; r ¼ 1; 2; . . . ;M1;

bh‘ð0Þ ¼ ðhðx; y; 0Þ � �hðx; yÞ;Uh

‘ðx; yÞÞ; ‘ ¼ 1; 2; . . . ;M2:

8><>:

ð3:7Þ

4190 Z. Luo et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4184–4195

There is the following error estimate between the solu-tions for Problem (III) and the solutions for Problem(IV), whose proof is provided in Appendix A.

Theorem 1. If maxfðPn

i¼M1þ1kui Þ

1=2; ðPn

i¼M1þ1kvi Þ

1=2g 6 �h=cand k is sufficiently small, then the error estimate between the

solutions for full basic Problem (II) and the solutions for thereduced order basic Problem (IV) is

kððus�h � us

M1Þ; ðvs

�h � vsM1Þ; ðhs

�h � hsM1ÞÞk0

6 C5

Xn

i¼M1þ1

ðkui þ kv

i Þ þXn

i¼M2þ1

khi

!;

where C5 ¼ 3C4 ðC4 see Appendix A) and s = 1,2, . . ..

Combining (3.1) with Theorem 1 could yield in the fol-lowing result.

Theorem 2. If maxfðPn

i¼M1þ1kui Þ

1=2; ðPn

i¼M1þ1kvi Þ

1=2g 6 �h=cand k is sufficiently small, then the error estimate between the

solutions for Problem (I) and the solutions for the reduced

order basic Problem (IV) is

ðuðtlÞ � ulM1Þ; ðvðtlÞ � vl

M1Þ; ðhðtsÞ � hl

M1Þ

� ���� ���0

6 cð�hm þ kÞ þ C5

Xn

i¼M1þ1

ðkui þ kv

i Þ þXn

i¼M2þ1

khi

!;

l ¼ 1; 2; . . . ; n:

Fig. 3. Fluid total layer thickness for Problem (II) (black curve), 4 POD bases f7 POD bases for 30 snapshots for month of June (b) and December (d). (a) C

Remark 1. When one computes actual problems, one mayobtain the ensemble of snapshots from physical system tra-jectories by drawing samples from experiments and inter-polation (or data assimilation). For example, for weatherforecast, one can use previous weather prediction resultsto construct the ensemble of snapshots, then restructurethe POD basis for the ensemble of snapshots by Section2, and finally combine it with a Galerkin projection toderive a reduced order dynamical system, i.e., one needsonly to solve Problem (IV) which has only few degrees offreedom, but it is unnecessary to solve Problem (II). Thus,the forecast of future weather change can be quickly simu-lated, which is of major importance for actual real-lifeapplications.

Remark 2. In general, c0 is a very small value so thatexp(kc0n) approaches 1 in Theorem 1, and taking m = 1or 2 is sufficient in actual numerical simulations. Sinceour methods employ some MFE solutions ðui

�h; vi�h; h

i�hÞ

ði ¼ 1; 2; . . . ; nÞ for Problem (II) as assistant analysis, theerror estimates in Theorem 2 are correlated to the griddingscale �h and time step size k. However, using same argumentas in Remark 1, the assistant ðui

�h; vi�h; h

i�hÞ ði ¼ 1; 2; . . . ; nÞ

could be substituted with the interpolation functions ofexperimental and previous results, it is unnecessary to solveProblem (II), it is only necessary to directly solve Problem(IV) such that Theorem 1 is satisfied. Since Problem (IV) is

or 5 snapshots (green curve), 7 POD bases for 20 snapshots (red curve), andase on June and (b) case on December.

Fig. 4. Profiles of currents for Problem (II) (blue vector) for month of June (a) and December (c) and profiles of currents for Problem (IV) using 7 PODbases for 20 snapshots (red curve) for month of June (b) and December (d). (a) Currents for Problem (II) on June (b) currents for Problem (IV) on June (c)currents for Problem (II) on December (d) currents for Problem (IV) on December. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

Z. Luo et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4184–4195 4191

4192 Z. Luo et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4184–4195

only dependent on M1 and M2, and is independent of thegridding scale �h and time step size k, and, in general, M1

and M2� n, it is only necessary to solve Problem (IV) withvery few freedom degrees.

4. Some numerical examples

In this section, we present numerical computations relatedto the approaches presented in the previous paragraphs. Inthis study, we applied the model to the tropic Pacific Oceandomain X (which is a domain 30�S–30�N, 130�E–70�W),using parameters values of f = 2(7.29E�5)sin(x,y), H =150 m, q0 = 1.2 kg m�3, qa = 1025 kg m�3, g 0 = 3.7 · 10�2,A = 750 m2 s�1, and CD = 1.5 · 10�3 for Problem (I). Thischosen model domain allows for all possible equatoriallytrapped waves to be excited by the applied wind forcing(see [3]). The no-normal flow and no-slip conditions areapplied at these solid boundaries. We choose the uniform reg-ular triangulation with �h = 0.5� and the time mesh size to bek = 100s. The model is driven by the Florida State University(FSU) climatology monthly mean winds (cf. [27]), and thedata are projected onto each time step by a linear interpola-tion and onto each grid point by a bilinear interpolation.

We choose 5, 20, and 30 group of solutions (i.e., snap-shots) solving Problem (II) for 1 year. It is shown by com-puting that eigenvalues ku4, kv4, and kh4 are all less than10�3 if the number of snapshots is 5, while eigenvaluesku8, kv8, and kh8 are all less than 10�3 if the number of snap-shots is 20 and 30, which are consistent with the errorsbetween numerical solutions obtained with a different num-ber of POD optimal bases for Problem (IV) and solutionsobtained with Problem (II), where the red curve representsthe error of fluid total layer thickness, the blue curve repre-sents the error of fluid velocity in the longitude direction,and the green curve represents the error of fluid velocityin the latitude direction in Fig. 2.1 It is shown by compar-ing results for Problem (II) and POD reduced model thatthe computational load for velocity vector and fluid totallayer thickness with Problem (IV) is sizably reduced, andthe error between them does not exceed 10�3. And theresults of the error for the actual example are consistentwith the theoretical results obtained by computing withTheorem 2. This also shows that finding the approximatesolutions for the tropical Pacific Ocean reduced gravitymodel with Problem (IV) is computationally very effective.

We obtain the solution for fluid total layer thickness h

depicted graphically in Fig. 3a and b, respectively, wherethe green curves represent numerical solution for 5 snap-shots used 4 POD bases to solve reduced Problem (IV),the red curves represent numerical solution for 20 snap-shots used 7 POD bases to solve reduced Problem (IV),the purple curves represent numerical solution for 20 snap-shots used 7 POD bases to solve reduced Problem (IV), and

1 For interpretation of the references to colour in Fig. 2, the reader isreferred to the web version of this article.

the black curves represent the solutions solving Problem(II) on June and on December.

We also obtain the profiles of currents for fluid velocity(u,v) depicted graphically in Fig. 4b and d, respectively, for20 snapshots using 7 POD bases to solve Problem (IV), andFig. 4a and c are, respectively, the solutions solving Prob-lem (II) for June and for December.

These profiles demonstrate that the results of the numer-ical simulations coincide with both the theory and theactual cases. Especially, the POD reduced model Problem(IV) is a reduced method to apply the existing datum tosimulating future phenomena, which has far fewer(2M1 + M2� n) degrees of freedom of Problem (II).Therefore, the POD reduced method is very suitable fordealing with large-scale science engineering computations,and could simplify computing and reduces both CPU andmemory requirements in the actual computational processin a sense that guarantees a sufficiently accurate numericalsolution.

5. Conclusions

In this paper, we have employed the POD and Galerkintechniques to study the reduced format for the tropicalPacific Ocean reduced gravity model and to reconstructPOD optimal orthogonal bases of ensembles of data whichare compiled from transient solutions derived by using theMFE equation system. We have also combined the PODbases with a Galerkin projection procedure, thus yieldinga new reduced model of lower dimensional order and ofhigh accuracy for the tropical Pacific Ocean reduced grav-ity model. We have then proceeded to derive error esti-mates between our reduced format approximate solutionsand the usual full order MFE numerical solutions, andhave shown using numerical examples that the errorbetween the POD approximate solution of reduced formatand the full MFE solution is consistent with the theoreticalerror results obtained, thus validating both feasibility andefficiency of our reduced format. Future work in this areawill aim to extend the reduced format, implementing it toa realistic sea forecast system and to more complicatedPDEs, for instance, the nonlinear shallow water equationsystem consisting of water dynamics equations, silt trans-port equation and the equation of bottom topographychange. We have shown both by theoretical analysis as wellas by numerical examples that the reduced format pre-sented herein has extensive perspective applications.

Acknowledgments

Z. Luo and J. Zhu acknowledge the support of the Na-tional Science Foundation of China Nos. 10471100 and40437017, and Beijing Jiaotong University Science Tech-nology Foundation. I.M. Navon acknowledges supportof NSF Grant No. ATM-0201808.

Z. Luo et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4184–4195 4193

Appendix A

The proof of Theorem 1 is as follows.Subtracting Problem (IV) and (3.7) from Problem (III)

and (3.6) yields the following error equations:

bur ðtsÞ � kf

Xn

j¼M1þ1

bvjðtsÞðUv

j ;Uur Þ � kg0

Xn

l¼M2þ1

bhl ðtsÞðUh

l ;UurxÞ

þ kAXn

i¼M1þ1

bui ðtsÞðrUu

i ;rUur Þ ¼ kðf s

1 ;Uur Þ þ bu

r ðts�1Þ;

r ¼ M1 þ 1;M1 þ 2; . . . ; n; s ¼ 1; 2; . . . ; ðA:1Þ

bvrðtsÞ þ kf

Xn

i¼M1þ1

bui ðtsÞðUu

i ;UvrÞ � kg0

Xn

l¼M2þ1

bhl ðtsÞðUh

l ;UvryÞ

þ kAXn

j¼M1þ1

bvjðtsÞðrUv

j ;rUvrÞ ¼ kðf s

2 ;UvrÞ þ bv

rðts�1Þ;

r ¼ M1 þ 1;M1 þ 2; . . . ; n; s ¼ 1; 2; . . . ; ðA:2Þ

bh‘ðtsÞ þ kH

Xn

i¼M1þ1

bui ðtsÞðUu

ix;Uhr Þ

þ kHXn

j¼M1þ1

bvjðtsÞðUu

jy ;Uhr Þ ¼ bh

r ðts�1Þ;

‘ ¼ M2 þ 1;M2 þ 2; . . . ; n; s ¼ 1; 2; . . . ; ðA:3Þ

along with the initial condition

bur ð0Þ ¼ ðuðx; y; 0Þ � �uðx; yÞ;Uu

r ðx; yÞÞ;r ¼ M1 þ 1;M1 þ 2; . . . ; n;

bvrð0Þ ¼ ðvðx; y; 0Þ � �vðx; yÞ;Uv

rðx; yÞÞ;r ¼ M1 þ 1;M1 þ 2; . . . ; n;

bh‘ð0Þ ¼ ðhðx; y; 0Þ � �hðx; yÞ;Uh

‘ðx; yÞÞ;‘ ¼ M2 þ 1;M2 þ 2; . . . ; n:

Eqs. (A.1)–(A.3) can be written as in the following vec-tor format:

bsu

bsv

bsh

0B@

1CA ¼ k

�AD1 fB g0C1

�fBT �AD2 g0C2

�HCT1 �HCT

2 O

0B@

1CA

bsu

bsv

bsh

0B@

1CA

þ k

F s1

F s2

0

0B@

1CAþ bs�1

u

bs�1v

bs�1h

0B@

1CA;

ðA:4Þ

where O is a (n �M2) · (n �M2) zero matrix, 0 is a(n �M2)-dimensional zero vector, and

D1 ¼Z

XðrUu

M1þ1; . . . ;rUunÞ

TðrUuM1þ1; . . . ;rUu

nÞdxdy;

D2 ¼Z

XðrUv

M1þ1; . . . ;rUvnÞ

TðrUvM1þ1; . . . ;rUv

nÞdxdy;

C1 ¼Z

XðUh

M2þ1; . . . ;UhnÞ

T ðUuM1þ1Þx; . . . ; ðUu

nÞx� �

dxdy;

C2 ¼Z

XðUh

M2þ1; . . . ;UhnÞ

T ðUvM1þ1Þy ; . . . ; ðUv

nÞy� �

dxdy;

B ¼Z

XðUv

M1þ1; . . . ;UvnÞ

TðUuM1þ1; . . . ;Uu

nÞdxdy;

F s1 ¼ ððf s

1 ;UuM1þ1Þ; . . . ; ðf s

1 ;UunÞÞ

T;

F s2 ¼ ððf s

2 ;UvM1þ1Þ; . . . ; ðf s

2 ;UvnÞÞ

T;

bsu ¼ ðb

uM1þ1ðtsÞ; . . . ; bu

nðtsÞÞT;bs

v ¼ ðbvM1þ1ðtsÞ; . . . ; bv

nðtsÞÞT;bs

h ¼ ðbhM2þ1ðtsÞ; . . . ; bh

nðtsÞÞT:

Since it is well known (see [9]) that

Uw1 ; . . . ;U

wMj;Uw

Mjþ1; . . . ;Uwn

� � 0; . . . ;0;Uw

Mjþ1; . . . ;Uwn

� �� ��2

0

¼Xn

i¼Mjþ1

kwi ; w¼ u; v;h; j¼ 1;2;

ðA:5Þ

we obtain

ðus�h � us

M1Þ; ðvs

�h � vsM1Þ; ðhs

�h � hsM2Þ

� ���� ���0

6 bsu; b

sv; b

sh

� T��� ��� Xn

i¼M1þ1

ðkui þ kv

i Þ þXn

i¼M2þ1

khi

!1=2

;

s ¼ 1; 2; . . . ; ðA:6Þ

where k Æk is the norm of matrices or vector. From the in-verse inequality (see [28] or [29]) and matrix normal prop-erty, we obtain

kC1k 6 ðUhM2þ1; . . . ;Uh

nÞT

��� ���0ðUu

M1þ1; . . . ;UunÞx

��� ���0

6 c�h�1 ðUhM2þ1; . . . ;Uh

nÞT

��� ���0ðUu

M1þ1; . . . ;UunÞ

��� ���0

6 c�h�1Xn

i¼M1þ1

kui

Xn

i¼M2þ1

khi

!1=2

; ðA:7Þ

kC2k 6 c�h�1Xn

i¼M1þ1

kvi

Xn

i¼M2þ1

khi

!1=2

; ðA:8Þ

kBk6 ðUvM1þ1; . . . ;U

vnÞ

T��� ���

0ðUu

M1þ1; . . . ;UunÞ

��� ���0

Xn

i¼M1þ1

kui

Xn

i¼M1þ1

kvi

!1=2

;

ðA:9Þ

kF s1k 6 kf s

1k0 ðUuM1þ1; . . . ;Uu

n��� ���

0

¼ kf s1k0

Xn

i¼M1þ1

kui

!1=2

; ðA:10Þ

4194 Z. Luo et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4184–4195

kF s2k 6 kf s

2k0 ðUvM1þ1; . . . ;Uv

n��� ���

0

¼ kf s2k0

Xn

i¼M1þ1

kvi

!1=2

: ðA:11Þ

Then, multiplying (A.4) by ððbsuÞ

T; ðbs

vÞT; ðbs

hÞTÞ, one could

get

bsu

bsv

bsh

0B@

1CA

T bsu

bsv

bsh

0B@

1CA ¼ k

bsu

bsv

bsh

0B@

1CA

T �AD1 fB g0C1

�fBT �AD2 g0C2

�HCT1 �HCT

2 O

0B@

1CA

bsu

bsv

bsh

0B@

1CA

þ k

bsu

bsv

bsh

0B@

1CA

T F s1

F s2

0

0B@

1CAþ

bsu

bsv

bsh

0B@

1CA

Tbs�1

u

bs�1v

bs�1h

0B@

1CA:ðA:12Þ

Noting that AðbsuÞ

TD1bsu > 0 and Aðbs

vÞTD2b

sv > 0, if

maxXn

i¼M1þ1

kui

!1=2

;Xn

i¼M1þ1

kvi

!1=28<:

9=; 6 �h=c;

which is reasonable, by using matrix normal property, and(A.4) and (A.7)–(A.12), one can obtain

ðbsu; b

sv; b

shÞ

T�� �� 6 c0k ðbs

u; bsv; b

shÞ

T�� ��þ Cs

0k

þ ðbs�1u ; bs�1

v ; bs�1h Þ

T�� ��; ðA:13Þ

where

c0 ¼ 2fXn

i¼M1þ1

kui

Xn

i¼M1þ1

kvi

!1=2

þ 2g0Xn

i¼M2þ1

khi

!1=2

þ 2HXn

i¼M2þ1

khi

!1=2

;

and

Cs0 ¼ kf s

1kXn

i¼M1þ1

kui

!1=2

þ kf s2k

Xn

i¼M1þ1

kvi

!1=2

:

Summing (A.13) from 1 to s, if k is sufficiently small, suchthat 1 � c0k 6 1/2, yields

ðbsu; b

sv; b

shÞ

T�� �� 6 C2

Xn

i¼M1þ1

kvi

!1=2

þ C1

Xn

i¼M1þ1

kui

!1=2

þ c0kXn

i¼0

ðbiu; b

iv; b

ihÞ

T�� ��þ ðb0

u; b0v ; b

0hÞ

T�� ��;

ðA:14Þ

where ðb0u;b

0v ;b

0hÞ ¼ ðb

uM1þ1ð0Þ; . . . ;bu

nð0Þ;bvM1þ1ð0Þ; . . . ;bv

nð0Þ;bh

M2þ1ð0Þ; . . . ; bhnð0ÞÞ, C1 ¼ 2k

Pni¼0kf i

1k, and C2 ¼2kPn

i¼0kf i2k. Noting that

ðb0u; b

0v ; b

0hÞ

T�� �� 6 ðkuðx; y; 0Þk0 þ k�uk0Þ ðUu

M1þ1; . . . ;UunÞ

��� ���0

þ ðkvðx; y; 0Þk0 þ k�vk0Þ ðUvM1þ1; . . . ;Uv

n��� ���

0

þ ðkhðx; y; 0Þk0 þ k�hk0Þ ðUuM2þ1; . . . ;Uh

n��� ���

0

6 C3

Xn

i¼M1þ1

kui

!1=2

þXn

i¼M1þ1

kvi

!1=224

þXn

i¼M2þ1

khi

!1=235; ðA:15Þ

where C3 ¼ maxfkuðx; y; 0Þk0 þ k�uk0; kvðx; y; 0Þk0 þ k�vk0;khðx; y; 0Þk0 þ k�hk0g. Using discrete Gronwall inequalityfor (A.14), one could obtain

ðbsu;b

sv;b

shÞ

T�� ��6 C4

Xn

i¼M1þ1

kui

!1=2

þXn

i¼M1þ1

kvi

!1=2

þXn

i¼M2þ1

khi

!1=224

35;

ðA:16Þ

where C4 ¼ expðnkc0ÞmaxfC1 þ C3;C2 þ C3;C3g. Com-bining (A.6) with (A.16) and using Cauchy inequality yieldsTheorem 1.

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