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C. R. Acad. Sci. Paris, Ser. I 345 (2007) 473478

http://france.elsevier.com/direct/CRASS1/

Numerical Analysis

A successive constraint linear optimization method for lower boundsof parametric coercivity and infsup stability constants

D.B.P. Huynh a, G. Rozza b, S. Sen b, A.T. Patera b

a Singapore-MIT Alliance, National University of Singapore, Singapore 117576, Singaporeb Mechanical Engineering Department, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

Received 7 December 2006; accepted after revision 24 September 2007

Presented by Olivier Pironneau

Abstract

We present an approach to the construction of lower bounds for the coercivity and infsup stability constants required in a

posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. The method, based

on an OfflineOnline strategy relevant in the reduced basis many-query and real-time context, reduces the Online calculation

to a small Linear Program: the objective is a parametric expansion of the underlying Rayleigh quotient; the constraints reflect

stability information at optimally selected parameter points. Numerical results are presented for coercive elasticity and non-coercive

acoustics Helmholtz problems. To cite this article: D.B.P. Huynh et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

2007 Acadmie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Rsum

Une mthode doptimisation linaire de contraintes successives pour les bornes infrieures des constantes paramtriques

de coercivit et de stabilit infsup. Nous prsentons une mthode pour le calcul de bornes infrieures pour les constantes de

stabilit (de coercivit ou dinfsup) ncessaires pour les estimateurs derreur a posteriori, associes lapproximation par base

rduite dquations aux drives partielles ayant une dpendance affine en les paramtres. La mthodebase sur une strat-

gie hors-ligne/en-ligne intressante pour le calcul temps rel et les cas dvaluations multiplesrduit le calcul en-ligne un

problme doptimisation linaire peu coteux. La fonction objectif est un dveloppement paramtrique du quotient de Rayleigh.

Les contraintes traduisent la stabilit pour un ensemble optimal de paramtres. Nous prsentons des rsultats numriques pour

un problme dlasticit (coercif) ainsi que pour un problme dacoustique de type Helmholtz (non-coercif). Pour citer cet ar-

ticle : D.B.P. Huynh et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

2007 Acadmie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Version franaise abrge

On construit une borne infrieure de la constante de stabilit (de coercivit ou dinfsup (2)). La forme bilinaire

a, associe lquation aux drivees partielles pose sur un domaine , dpend de manire affine (voir (1)) dun

E-mail address: patera@mit.edu (A.T. Patera).

1631-073X/$ see front matter

2007 Acadmie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2007.09.019

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D.B.P. Huynh et al. / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 473478 475

where the q , 1 q Q, are continuous functions over D, and the aq , 1 q Q, are symmetric continuous bilinear

forms over Xe Xe. The assumption (1) is crucial in OfflineOnline strategies.Our problem statement is then: given D, find ue() Xe such that a(ue(), v;) = f (v), v Xe; then

evaluate the output se() = f (ue()). (In general, we can consider s() = (ue()) for bounded : Xe R; herewe restrict attention to the linear compliant case,

=f.)

We introduce a finite element truth approximation space of dimensionN, XN Xe, with inherited inner productand norm (w,v)XN (w,v)Xe and wXN wXe . We define

N() infwXN

supvXN

a(w,v;)wXNvXN

, N() infwXN

a(w,w;)w2

XN

, (2)

and N() supwXN supvXN[a(w,v;)/(wXNvXN)]. It immediately follows that e() N() andN() e(); we assume that N() > 0, D. Our truth FE approximation is then: given D, finduN() XN such that a(uN(), v;) = f (v), v XN; then evaluate the output sN() = f (uN()). We supposethat, to the desired accuracy, uN() is indistinguishable from ue().

Our interest is in reliable real-time and many-query response sN(): in the real-time context such as parame-ter estimation the premium is on reduced marginal cost; in the many-query context such as optimization the premium

is on reduced average cost. We consider reduced basis methods [7,1,9,6]: we develop Galerkin approximations to uN

and sN(), uN() WN XN and sN() = f (uN()), respectively, as well as (i) a rigorous and relatively sharpa posteriori error bound sN() such that |sN() sN()| sN(), D, and (ii) an OfflineOnline compu-tational strategy such that, in the Online stage, the marginal (and hence also asymptotic average) cost to compute

sN(), sN() depends only on N, the dimension ofWN, and Qand not on N.The error bound sN() takes the form [6] in the coercive case R( ;)2(XN) /LB() and in the non-coercive

case R( ;)2(XN) /LB(), where R(v;) f(v) a(uN(), v;), v XN, is the residual, (XN) denotes

the dual norm, and LB() and LB() are lower bounds for the coercivity and infsup constants. Our focus in this

paper is on a new lower bound (OfflineOnline) strategy for LB() and LB(). Our new approach is more efficient

and general than our earlier coercive [9,6] and non-coercive [6,10] proposals; also the new approach is much more

easily implemented.

There are many classical techniques for the estimation of smallest eigenvalues or smallest singular values. One

class of methods is based on Gershgorins theorem and variants [5]. A second class of methods is based on eigen-

function/eigenvalue (e.g., Rayleigh Ritz) approximation and subsequent residual evaluation [8,4]. Unfortunately, in

neither case can we obtain the rigor, sharpness, and online efficiency we require.

2. Successive Constraints Method (SCM)

We address the coercive case first. By way of preliminaries we define

Y

y = (y1, . . . , yQ) RQ

wy XNs.t. yq =

aq (wy , wy )

wy

2XN

, 1 q Q

. (3)

We further define the objective function J :D RQ R as J(; y) = Qq=1 q ()yq . We may then write ourcoercivity constant as N() = minyYJ(; y); we now consider relaxations.

We first define q infwXN[aq (w,w)/(w2XN)] and +q supwXN[aq (w,w)/(w2XN)], 1 q Q, andlet BQ

Qq=1[q , +q ] RQ. We also introduce the two parameter sets {1 D, . . . , J D} and CK

{1 D, . . . , K D}. For any finite-dimensional subset ofD, E (= or CK ), PM(; E) shall denote the M pointsclosest to in E (in the Euclidean norm); if card(E) M then we define PM(; E) = E, and ifM = 0 we definePM(; E) = .

For given CK , M N (stability constraints), and M+ N (positivity constraints), we define

YLB(;CK ) y BQ

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Qq=1

q ()yq N(), PM (;CK );Q

q=1 q ()yq 0, PM+ (; )

, (4)

and YUB(CK ) {y(k ), 1 k K} for y() argminyYJ(; y). We next defineLB(

;CK )

=min

yYLB(;CK )J(

;y) (5)

and UB(;CK ) = minyYUB(CK )J(; y). We then obtain

Proposition 1. For given CK (andM N, M+ N, ), LB(;CK ) N() UB(;CK ), D.

Proof. For the upper bound, clearly YUB Y; hence, UB(;CK ) N(). For the lower bound, Y YLB: forexample (we consider the stability constraints), for any y Y wy XN (see (3))

Qq=1

q ()yq =Q

q=1q ()

aq (wy , wy )

wy2XN inf

wXN

Qq=1

q ()aq (w,w)

w2XN

= (), D;

a similar argument applies to the positivity constraints. Hence N

() LB(;CK ). 2

We expect that ifCK is sufficiently large, (i) y() will be sufficiently close to a member ofYUB to provide a good

upper bound, and (ii) the stability and positivity constraints in YLB will sufficiently restrict y to provide a good lowerbound.

We note that (4), (5) is in fact a linear optimization problem or Linear Program (LP). Our LP (5) contains Qdesign variables (y = (y1, . . . , yQ)) and 2Q (y BQ) + M (stability) + M+ (positivity) inequality constraints. Theoperation count for the Online stage LB() is independent ofN.

But first we must determine CK and obtain the N(k ), 1 k K , by an Offline greedy algorithm. We first set

K = 1 and choose C1 = {1} arbitrarily; we also specify M , M+, , and a tolerance ]0, 1[. We then set K = 1and perform

While

max [(UB(;C

K ) LB(;C

K ))/UB(;C

K )] > :K+1 = argmax

[(UB(;CK ) LB(;CK ))/UB(;CK )]

CK+1 = CK K+1; K K + 1;end. Set Kmax = K.

The notable offline computations are (i) 2Q eigenproblems to form BQ, and Kmax eigenproblems to obtain

y(k), N(k), 1 k Kmax (for a judiciously chosen inner product, the latter can be efficiently calculated by

a Lanczos scheme [3]); (ii) O(NQKmax) operations to form YUB (we assume sparsity); and (iii) J Kmax LPs of size

O(Q + M + M+).We now consider the non-coercive case. We introduce operators Tq : XN XN as

(Tq w,v)XN

=aq (w,v),

v

XN, 1 q Q,

and Tw Qq=1 q ()Tq w. It is then readily demonstrated [10] thatN()

2 = infwXN

(Tw, Tw)XN/w2XN

,

which can be expanded as

N()

2 = infwXN

Q

q =1

Qq =q

(2 q q ) q()q

()

(Tq

w, Tq

w)XN/w2XN

;here q q is the Kronecker delta. We now identify

N()

2

N(),

(2 q q )q ()q (), 1 q q Q q (), 1 q Q Q(Q + 1)/2,

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D.B.P. Huynh et al. / C. R. Acad. Sci. Paris, Ser. I 345 (2007) 473478 477

and

(Tqw, Tq

v)XN, 1 q

q Q aq (w,v), 1 q Q,

and observe that

N()2 N() infwXN

Qq=1

q () aq

(w,w)w2XN

. (6)

We may thus directly apply our SCM procedure to (6), however our LP for (N())2 now has Q2/2 designvariables and (Q2 + M + M+) inequality constraints.

3. Numerical results

As a coercive example, we consider a linear elastic orthotropic material [2] in two dimensions (plane stress):

d = dv = 2. The original domain is a rectangle ]0, L[ ]0, 1[: we map this domain to a fixed reference domain ]0, Lref = 1[ ]0, 1[; subsequently L shall appear only in the parameter-dependent coefficient functions. Ourfunction space X

e

is given by {v H1

()|v|D = 0}2

, where

D

is the left boundary of ; XN

Xe

is a linearfinite element space of dimension N= 1296.We consider P = 4 parameters: 1 Ex2 /Ex1 , where Ex2 and Ex1 are the orthotropic Youngs moduli; 2 ,

the Poisson ratio; 3 G/Ex1 , where G is the tangential modulus; and 4 L. The parameter domain is given byD [0.05, 1.0] [0.1, 0.3225] [0.02, 0.50] [0.1, 15].

Our affine development (1) then applies for Q = 6 and

1() = 14(1 22)

, a1(v,w) =

v1

x1

w1

x1,

2()

=2

1 22

, a2(v,w)

= v2

x2

w1

x1 +v1

x1

w2

x2 ,3() = 34, a3(v,w) =

v1

x2

w1

x2

4() = 3, a4(v,w) =

v1

x2

w2

x1+ v2

x1

w1

x2

,

5() =3

4, a5(v,w) =

v2

x1

w2

x1,

6()

=14

1 22, a6(v,w)

= v2

x2

w2

x2.

We choose for our inner product (w,v)XN a(w,v;ref = {1.0, 0.1, 0.5, 1.0}) + min(w,v)L2 , where min is theminimum of the Rayleigh quotient a(w,w;ref)/(w,w)L2 ; this choice of inner product ensures a good spectrum forthe Lanczos algorithm [3].

We apply our greedy algorithm for = .75 and a random sample of size J = 1500. For M = (effectivelyM+ = Kmax in the Online stage) and M+ = 0 we obtain Kmax = 30. However, we can significantly reduce M (andOnline effort) with only a slight increase in Kmax (and Offline effort): for M = 12, M+ = 0, we obtain Kmax = 50.(Note that since M+ = 0, our lower bound is a constructive proof that a is coercive over .) For this optimal set ofparameters, we observe that our Online lower bound LB() is 100 times faster than direct computation of N().

As a non-coercive example, we consider a simple acoustics (microphone probe) Helmholtz problem [10] in two

dimensions: d = 2, dv = 1. The original domain is a channel probe followed by a microphone plenum of height

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1/4 + L: we map this domain to a fixed reference domain (Lref= 1) = [1/2, 1] [0, 1/4] [0, 1] [1/4, 5/4];subsequently L shall appear only in the parameter-dependent coefficient functions. Our function space Xe is given by

{v H1()|v|D = 0} where D is the left boundary of ; XN Xe is then a quadratic finite element space ofdimensionN= 4841 over a triangulation of .

We consider P

=2 parameters: 1

L, and 2

=k2 (the reduced frequency squared); the parameter domain

D [0.3, 0.6] [3.0, 6.0] lies between the first and second resonances. Our affine development (1) then applies [10]for Q = 5 and hence Q = Q(Q + 1)/2 = 15. For our inner product we choose (w,v)XN a0(w,v) + min(w,v)L2 ,where a0(w,v) a(w,v; (0.45, 0)) and min is the minimum of the Rayleigh quotient a0(w,w)/(w,w)L2 .

We apply our greedy algorithm for = 0.75 and a random sample of size J = 3671. For the case M = (effectively, M = Kmax in the Online stage) and M+ = 4 we obtain Kmax = 22: this choice minimizes the Offlineeffort. If we wish to minimize the Online effort we can choose M = 4 (and M+ = 4)note the Online effort isindependent of Kmaxwhich then yields Kmax = 98: in this case, our Online lower bound LB() is 160times faster than direct computation of N(). For a better OfflineOnline balance we can consider M = 16,M+ = 4 which yields Kmax = 25: in this case, our Online lower bound LB() is 137 times faster than directcomputation of N(). We have also applied the SCM approach to the (in fact, coercive) parameter domainconsidered in [10]; the SCM performs better than the natural norm approach of [10], and is also much simpler to

implement.

Acknowledgements

We thank Professor Claude Le Bris of ENPC for helpful comments. This work was supported by DARPA/AFOSR

Grant FA9550-05-1-0114 and the Singapore-MIT Alliance.

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