A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations

TL;DR: In this paper, a technique for the prediction of linear-functional outputs of elliptic partial differential equations with affine parameter dependence is presented, where the essential components are (i) rapidly convergent global reduced-basis approximations -Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest; and (iii) off-line/

Abstract: We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly convergent global reduced-basis approximations - (Galerkin) projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage - in which, given a new parameter value, we calculate the output of interest and associated error bound - depends only on N (typically very small) and the parametric complexity of the problem. In this paper we develop new a posteriori error estimation procedures for noncoercive linear, and certain nonlinear, problems that yield rigorous and sharp error statements for all N. We consider three particular examples: the Helmholtz (reduced-wave) equation; a cubically nonlinear Poisson equation; and Burgers equation - a model for incompressible Navier-Stokes. The Helmholtz (and Burgers) example introduce our new lower bound constructions for the requisite inf-sup (singular value) stability factor; the cubic nonlin-earity exercises symmetry factorization procedures necessary for treatment of high-order Galerkin summations in the (say) residual dual-norm calculation; and the Burgers equation illustrates our accommodation of potentially multiple solution branches in our a posteriori error statement. Numerical results are presented that demonstrate the rigor, sharpness, and efficiency of our proposed error bounds, and the application of these bounds to adaptive (optimal) approximation. © 2003 by the American Institute of Aeronautics and Astronautics, Inc.

Summary

1 Introduction

• The optimization, control, and characterization of an engineering component or system requires the prediction of certain “quantities of interest,” or performance metrics, which the authors shall denote outputs — for example deflections, heat transfer rates, or drags.
• The authors goal is the development of computational methods that permit rapid and reliable evaluation of this partial-differential-equation-induced input-output relationship in the limit of many queries — that is, in the design, optimization, control, and characterization contexts.
• The reduced-basis method recognizes that the field variable is not, in fact, some arbitrary member of the infinite-dimensional solution space associated with the partial differential equation; 1 of 18 American Institute of Aeronautics and Astronautics Paper 2003-3847 rather, it resides, or “evolves,” on a much lowerdimensional manifold induced by the parametric dependence.
• The work6,8, 9, 14,15,20 differs from these earlier efforts in several important ways: first, the authors develop global approximation spaces; second, they introduce rigorous a posteriori error estimators; and third, they exploit off-line/on-line computational decompositions (see2 for an earlier application of this strategy.).

2.1 Preliminaries

• (2) More general inner products and norms can (and should) be considered, as discussed in Section 2.4.2.
• The authors now introduce their parametrized bilinear form.

2.2.1 Weak Statement

• In the language of the introduction, s(µ) is their output, and u(µ) is their field variable.
• The authors assume that N is chosen sufficiently large that sN (µ) and uN (µ) may be effectively equated with s(µ) and u(µ), respectively.

2.2.2 Reduced-Basis Approximation

• The focus of the current paper is a posteriori error estimation.
• The authors shall thus take their reduced-basis approximation as given.
• The discrete inf-sup parameter associated with the latter may not be “good,” with corresponding detriment to the accuracy of uN (µ) and hence sN (µ).
• More sophisticated minimumresidual8,18 and in particular Petrov-Galerkin7,18 approaches restore stability, albeit at some additional complexity and cost.

2.3.1 Error Bound

• The authors note that their proof (or bound) does not exploit any special properties of e(µ) (or uN (µ)).
• It remains to develop their lower bound construction, β̂(µ), and to demonstrate that both β̂(µ) and ‖Yr( · ;µ)‖Y may be computed efficiently (that is, in complexity independent of N ).

2.3.2 Inf-Sup Lower Bound Construction

• Many of the most obvious eigenvalue approximation concepts are not relevant here, since the authors require a lower, not upper, bound.
• The authors can now state Proposition 2 The construction β̂(µ) of (42) satisfies the inequality (31).
• In essense, the equation for the error e(µ), (34), permits relaxations — and hence rigorous yet inexpensive bounds — that can not be directly applied to the original equation for u(µ), (24).

2.3.3 Offline/Online Computational Procedure

• The central computational aspect of their reduced-basis approach is an offline/online computational decomposition which separates the requisite calculations into two distinct stages.
• The authors develop here similar estimates for ∆N (µ) (and hence ∆sN (µ)).
• The authors also briefly address the associated offline complexity.
• (In many cases, domain decomposition may be exploited to further reduce the Q dependence — from 5 of 18 American Institute of Aeronautics and Astronautics Paper 2003-3847 quadratic to linear.

2.4.2 Bound Conditioner

• This can be remedied by better choice of their bound conditioner.
• Χ1(µ) is high-wavenumber, and thus the authors may add a significant L2 contribution to their bound conditioner without adversely affecting the inf-sup parameter; this additional L2 term does, however, significantly improve their continuity constants — on which their lower bound construction is critically dependent, also known as The reason is simple.
• These arguments apply to Helmholtz problems generally; however, for larger ranges of frequency, the authors will need different bound conditioners for different subdomains of Dµ — if they wish to retain the Dµ-independence of J . 6 of 18 American Institute of Aeronautics and Astronautics Paper 2003-3847.

2.4.3 Deflation

• The second debilitating aspect of (55) is the − ln(εs) dependence.
• The latter should improve their bounds and effectivity; but, more importantly, it will remove the εs dependence from (55) — their regions will be generally larger, and will not shrink to zero as the authors approach resonances (or, at most, except very near resonances).
• It remains to address two issues concerning δDM (µ).

2.5.1 Model Helmholtz Problem: P = 2

• The authors assume that the boundary of the memrane is “pinned” except on the “stress-free” crack.
• The offline expense will be increased somewhat, not so much due to the λ2(µj), χ2(µj) (say for M = 1) — in particular, since J will now be much smaller — but rather due to the JQ2(M2 + MN)N operations required for the inner products associated with the deflation correction (60).
• Clearly, an elastic plate (and more realistic outputs) would be a much more relevant model; their methodology directly applies to this case as well.
• For a given Loc , the bilinear form is, apart from several scaling factors, identical to (53) of Section 2.4.1.
• In particular, unlike in Section 2.4, the authors can no longer characterize β(ω2, Loc) in terms of a few (more generally, denumerable) “resonance” eigenvalues — their lower bound constructions are now required.

2.5.2 The Inf-Sup Lower Bound

• First, the authors observe that the “correct” bound conditioner (I → II) considerably increases the size of the regions; furthermore, this effect will be even more dramatic for higher frequency ranges.
• Second, the authors observe that some deflation (II→ III) further improves the situation; and sufficient deflation (III → V) greatly improves the situation, in particular as they approach resonance.
• Note that although IV performs better than III, only with V do the authors have sufficient deflation in the sense that all dangerous modes are neutralized — it is clear from Figure 3 that three modes are “active” near the end of their segment (25, 0.4)(50, 0.6).
• In actual fact, β(µ) varies significantly only in the one direction perpendicular to the P − 1 dimensional “resonance” manifolds.

2.5.3 Error Bounds and Effectivity

• The authors consider here a point µTEST which lies within a region Rµj ,τ for all cases I, II, III, IV, and V.
• As expected from the arguments of Section 2.4.2, the bound conditioner has little effect on the effectivity.
• And, as expected from the arguments of Section 2.3.3, deflation has a modest 9 of 18 American Institute of Aeronautics and Astronautics Paper 2003-3847 (respectively, significant) positive effect on the error (respectively, effectivity).
• Second, round-off errors will become increasingly important, and ultimately dominant, in the very immediate vicinity of resonances; in particular, as the authors approach extremely close to a resonance, they may observe effectivities below unity.
• Exact orthogonalization recovers the theoretical result — ηN (µ) ≥ 1; in more realistic models, damping will provide the necessary “cut-off.”.

3.2.1 Weak Statement

• In the language of the introduction, s(µ) is their output, and u(µ) is their field variable.
• As for their Helmholtz problem, in actual practice the authors replace s(µ) and u(µ) with corresponding “truth” Galerkin approximations sN (µ) and uN (µ), respectively (see Section 2.2.1).

3.2.2 Reduced-Basis Approximation

• The focus of the current paper is a posteriori error estimation.
• The authors shall thus take their reduced-basis approximation as given.

3.3.2 Coercivity Lower Bound Construction

• The authors approach to the inf-sup lower bound, described in Section 2.3.2, can also be adapted to general coercive problems.
• For their purposes here, however, the authors consider a simple variant that exploits the monotonicity of α(µ).
• Further details on these and related bound conditioners for coercive problems may be found elsewhere.

3.3.3 Offline/Online Computational Procedure

• The authors nonlinear problem admits an offline/online decomposition quite similar to that for linear problems.
• The authors focus here will be on efficient (or as efficient as possible) treatment of these new terms.
• Obviously, näıve treatment of (96) directly yields N6 operations.
• In general, T(N,κ) ∼ N6/κ!: in relative terms, higher order (e.g., uκ) nonlinearities thus enjoy greater economies; however, in absolute terms, T(N,κ) will grow very rapidly with N for larger κ.

3.4.1 Model Problems

• The authors model problem has already been specified in Sections 3.1 and 3.2.
• Note the nonlinearity will be most significant for µ1 and µ2 small.

3.4.2 Adaptive Reduced-Basis Approximation

• Given the higher powers of N that now appear in their complexity estimates, it is crucial (both as regards online and offline effort) to control N more tightly.
• The authors typically choose εpriord ε post d (µ) since their prior test sample is not exhaustive; and therefore, typically, Npost(µ) ≤ Nprior.
• The authors present in Table 3 the normalized error ‖e(µ∗)‖Y /‖u(µ∗)‖Y , as a function of N , for the (log) random and adaptive sampling processes (note that, in the results for the random sampling process, the sample SN is different for each N).
• The calculations were performed on a Pentium r©4 2.4GHz processor.
• Of course, in actual practice, the savings indicated in Table 3 can only be realized if their error estimators are true bounds (ηN (µ) ≥ 1), and good bounds (ηN (µ) ≈ 1).

4.2.1 Weak Statement

• For sufficiently large µ, (99), (110) — and the incompressible NavierStokes equations — have a unique solution; for smaller µ, the authors can encounter non-uniqueness — multiple solution branches may exist.
• As for their Helmholtz problem, in actual practice the authors replace s(µ) and u(µ) with corresponding “truth” Galerkin approximations sN (µ) and uN (µ), respectively (see Section 2.2.1).

4.2.2 Reduced-Basis Approximation

• Note uI(µn) refers to solutions of (99), (110), which are assumed to reside on a “first” branch; although the authors do not dwell here on possible bifurcation structure, other “parametric manifolds” (say, uII(µ)) may, in general, exist.
• The discrete inf-sup parameter associated with the latter may not be “good,” with corresponding detriment to the accuracy of uN (µ) and hence sN (µ).
• More sophisticated minimumresidual8,18 and in particular Petrov-Galerkin7,18 approaches restore stability, albeit at some additional complexity and cost.
• The authors comment that, for the case in which geometry is fixed and only viscosity varies, their reduced-basis approximation (and associated error estimation) procedure for the Burgers equation directly translates to the full incompressible Navier-Stokes equations — in particular, a divergence- (and hence pressure-) free formulation of the incompressible Navier-Stokes equations.

4.3.1 Preliminaries

• In what follows, the authors will explicitly highlight the N -dependence of β(µ), γ(µ), and σ(w;µ) only in those places where this dependence is either not obvious or potentially problematic.
• Note that T0 and the Tn are parameter-independent.

4.3.2 Error Bound

• In the quadratic case, the proof above is simpler and slightly sharper.
• In short, the error bound “sees” only the residual, which in turn “sees” only the branch-independent projection of uI(µ) (or uII(µ)), f(v).
• There is a dark side: one can not rigorously preclude the possibility that ‖uI(µ)−uN (µ)‖Y ≥ ΥN (µ).the authors.
• Clearly, in actual practice, the relative (and absolute) magnitude of ΥN (µ) will directly affect their comfort level in choosing (120).

4.3.2 Inf-Sup Lower Bound Construction

• Note if the authors include both (say, in the case of two branches) branches, uI(µn), uII(µn), in WN , then they will typically obtain good reduced-basis approximations to both branches — uIN (µ), uIIN (µ).
• The proof is almost identical to the proof of Proposition 2 for the Helmholtz inf-sup lower bound construction.

4.3.3 Offline/Online Computational Procedure

• All the elements of the offline/online procedure for the construction of Burgers a posteriori error bounds have already been introduced in the context of the Helmholtz and cubically nonlinear Poisson problems.
• Rather, the authors can make plausible continuity assumptions to construct these intervals, and then verify this condition, a posteriori , online.
• Second, the computationally most intensive online calculation (for large N) is precisely this ‖uN (µ)−uN (µj)‖L4(Ω) evaluation; however, by invoking the symmetry summation techniques developed in Section 3.3.3, the authors can reduce the relevant operation count to 124N 4 — typically not dominant for the small N realized by their adaptive sampling process.
• Third, for Burgers equation in R1, their reduced-basis approach is not competitive (even as regards marginal cost) with standard techniques, that is, direct computation of sN (µ).
• Their complexity estimates also apply to incompressible Navier-Stokes in R2,3, in which case the authors effect very considerable savings relative to finite element calculation of sN (µ).

4.4 Numerical Results

• All results presented are for the adaptive sampling procedure.
• It is possible that deflation techniques — similiar to those introduced in the context of the Helmholtz problem in Section 2.4.3 — could considerably increase the effective inf-sup parameter, and hence considerably decrease J .
• The authors observe that the reduced-basis approximation converges very rapidly; that at least in this particular case, the “good” choice, (120), obtains — ‖e(µ)‖Y ≤ ∆N (µ), ∀ N ∈ N; that the effectivities are, as desired, quite close to unity; and that ΥN (µ) is (constant and) very large.
• (Recall that these results are for the adaptive sampling procedure; in the case of a random sample, condition (119) is not satisfied for all N .).

Figures

Table 1 The normalized error as a function of N for cases I, II, and V.

Table 2 The effectivity ηN (µ) as a function of N for cases I, II, and V.

Fig. 4 A (more or less) complete “paving” of Dµ for case V.

Fig. 3 Polygons Rµ j ,τ for cases I, II, III, and V.

Fig. 2 Contours of β(ω2, Loc) over Dµ for the (cracked membrane) model Helmholtz problem.

Table 4 Error bound and effectivity for the cubically nonlinear Poisson problem for an adaptive sample.

Table 3 Error bound for the cubically nonlinear Poisson problem for random and adaptive samples.

Table 5 Error, error bound, and effectivity for the cubically nonlinear Poisson problem for the N = 10 adaptive sample; µTEST,1 = (0.01; 0.1), µTEST,2 = (0.1; 1), and µTEST,3 = (1; 10).

Table 6 Error, error bounds, and effectivity as a function of N for the Burgers problem with µ = 1.0.

Table 7 Error, error bounds, and effectivity as a function of N for the Burgers problem with µ = .01.

Fig. 5 The inf-sup parameter β(µ), lower bound β̂(µ), and segments Rµ j ,τ=3/4 = [µj , µj+1] for the model Burgers problem.

Fig. 1 Membrane with crack of length Loc under harmonic excitation at frequency ω.

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A Posteriori Error Bounds for Reduced-Basis
Approximation of Parametrized Noncoercive and
Nonlinear Elliptic Partial Dierential Equations
Karen Veroy, Christophe Prud’Homme, Dimitrios V. Rovas, Anthony T.
Patera
To cite this version:
Karen Veroy, Christophe Prud’Homme, Dimitrios V. Rovas, Anthony T. Patera. A Posteriori Error
Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial
Dierential Equations. 16th AIAA Computational Fluid Dynamics Conference, 2003, Orlando, United
States. �hal-01219051�

A Posteriori Error Bounds fo r Reduced-Basis
Approximation of Parametr ized Noncoercive
and Nonlinear Elliptic Partial Differential
Equations
K. Veroy
∗
Massachusetts Institute of Technology, Cambridge, MA 02139
C. Prud’homme
∗
D.V. Rovas
†
University of Illinois – Urbana-Champaign, Urbana, IL 61801
and A.T. Patera
∗
We present a technique for the rapid and reliable prediction of linear–functional out-
puts of elliptic partial differential equations with affine parameter dependence. The
essential components are (i ) rapidly convergent global reduced–basis approximations —
(Galerkin) projection onto a space W
N
spanned by solutions of the governing partial dif-
ferential equation at N selected points in parameter space; (ii ) a posteriori error estimation
— relaxations of the error-residual equation that provide inexpensive yet sharp bounds
for the error in the outputs of interest; and (iii ) off-line/on-line computational procedures
— methods which decouple the generation and projection stages of the approximation
process. The operation count for the on–line stage — in which, given a new parameter
value, we calculate the output of interest and associated error bound — depends only on
N (typically very small) and the parametric complexity of t he problem.
In this paper we develop new a posteriori error estimation procedures for noncoercive
linear, and certain nonlinear, problems that yield rigorous and sharp error statements for
all N . We consider three particular examples: the Helmholtz (reduced-wave) equation; a
cubically nonlinear Poisson equation; and Burgers equation — a model for incompressible
Navier-Stokes. The Helmholtz (and Burgers) example introduce our new lower bound
constructions for the requisite inf-sup (singular value) stability factor; the cubic nonlin-
earity exercises symmetry factorization procedures necessary for treatment of high-order
Galerkin summations in the (say) r esidual dual-norm calculation; and the Burgers equa-
tion illustrates our accommodation of potentially multiple solution branches in our a
posteriori error statement. Numerical results are presented that demonstrate the rigor,
sharpness, and efficiency of our proposed error bounds, and the application of these
bounds to adaptive (optimal) approximation.
1 Introduction
The optimization, control, and characterization of
an engineering component or system requires the pre-
diction of certain “quantities of interest,” or per-
formance metrics, which we shall denote outputs —
for example deflections, heat transfer rates, or drags.
These outputs are typically expressed as functionals
of field variables associated with a parametrized par-
tial differential equation which describes the physical
behavior of the component or system. The parame-
∗
Department of Mechanical Engine ering , Room 3-264
†
Department of Mechanical and Industrial Engineering, MC
244
Copyright
c
 2003 by the American Institute of Aeronautics and
Astronautics, Inc. No copyright is asserted in the Uni ted States
under Title 17, U.S. Code. The U.S. Government has a royalty-
free license to exercise all rights under the copyright claimed herein
for Governmental Purposes. All other rights are reserved by the
copyright owner.
ters, which we shall denote inputs, serve to identify a
particular “configuration” of the component. We thus
arrive at an implicit input-output relationship, eval-
uation of which demands solution of the underlying
partial differential equation.
Our goal is the development of computational meth-
ods that permit rapid and reliable evaluation of this
partial-differential-equation-induced input-output re-
lationship in the limit of many queries — that is,
in the design, optimization, control, and character-
ization contexts. Our particular approach is based
on the reduced-basis method, first introduced in the
late 1970s for nonlinear structural analysis,
1, 11
and
subsequently developed more broadly in the 1980s
and 1990s.
2–4, 12, 13, 17
The reduced-basis method rec-
ognizes that the field variable is not, in fact, some
arbitrary member of the infinite-dimensional solution
space associated with the partial differential equation;
1 of 18
American Institute of Aeronautics and Astronautics Paper 2003-3847

rather, it resides, or “evolves,” on a much lower-
dimensional manifold induced by the parametric de-
pendence.
The reduced-basis approach as earlier articulated is
local in parameter space in both practice and theory.
4
As a result, the computational improvements — rela-
tive to conventional (say) finite element approximation
— are often quite modest.
13
Our work
6, 8, 9, 14, 15, 20
differs from these earlier efforts in several important
ways: first, we develop global approximation spaces;
second, we introduce rigorous a posteriori error esti-
mators; and third, we exploit off-line/on-line compu-
tational decompositions (see
2
for an earlier application
of this strategy.) These three ingredients allow us —
for the restricted but important class of “parameter-
affine” problems — to reliably decouple the generation
and projection stages of reduced-basis approximation,
thereby effecting computational economies of several
orders of magnitude.
In this paper we develop new a posteriori error esti-
mation procedures for noncoercive linear, and certain
nonlinear, problems that — unlike our earlier “asymp-
totic” techniques
8, 15
— yield rigorous error statements
for all N. We consider three particular examples: the
Helmholtz (reduced-wave) equation (Section 2); a cu-
bically nonlinear Poisson equation (Section 3); and
Burgers equation (Section 4) — a model for incom-
pressible Navier-Stokes. The Helmholtz (and Burgers)
example introduce our new lower bound constructions
for the requisite inf-sup (singular value) stability fac-
tor; the cubic nonlinearity exercises symmetry factor-
ization procedures necessary for treatment of high-
order Galerkin summations in the (say) residual dual-
norm calculation; and the Burgers equation illustrates
our accommodation of potentially multiple solution
branches in our a posteriori error statement. Numer-
ical results are presented that demonstrate the rigor,
sharpness, and efficiency of our proposed error bounds,
and the application of these bounds to adaptive (opti-
mal) approximation.
2 Noncoercive Linear Problems:
Helmholtz Equation
2.1 Preliminaries
We consider a suitably regular domain Ω ⊂ R
d
,
1 ≤ d ≤ 3, with boundary ∂Ω. We then intro-
duce a Hilbert space Y with associated inner product,
( · , · )
Y
, and induced norm, k · k
Y
. We shall assume
that H
1
0
(Ω) ⊂ Y ⊂ H
1
(Ω), where H
1
(Ω) ≡ {v ∈
L
2
(Ω), ∇v ∈ (L
2
(Ω))
d
}, H
1
0
≡ {v ∈ H
1
(Ω)|v|
∂Ω
= 0},
and L
2
(Ω) is the space of square-integrable functions
over Ω. We shall further assume that
( ·, · )
Y
= ( ·, · )
H
1
(Ω)
,
k ·k
Y
= k ·k
H
1
(Ω)
,
(1)
where
(w, v)
H
1
(Ω)
≡
Z
Ω
∇w · ∇v + wv, ∀ w, v ∈ H
1
(Ω) ,
kvk
H
1
(Ω)
≡
Z
Ω
|∇v|
2
+ v
2
, ∀v ∈ H
1
(Ω) .
(2)
More general inner products and norms can (and
should) be considered, as discussed in Section 2.4.2.
We shall denote by Y
0
the dual space of Y . For a
g ∈ Y
0
, the dual norm is given by
kgk
Y
0
= sup
v ∈Y
g(v)
kvk
Y
. (3)
If we introduce the “representation” operator Y: Y
0
→
Y such that, for any g ∈ Y
0
,
(Yg, v)
Y
= g(v) , (4)
then
kgk
Y
0
= kYgk
Y
; (5)
this is simply a statement of the Riesz representation
theorem.
We now introduce our parametrized bilinear form.
We first define a parameter set D
µ
⊂ R
P
, a typical
point in which — our input P -tuple — shall be denoted
µ; we can then define, for any µ ∈ D
µ
, our bilinear
form a( · , · ; µ): Y × Y → R. We shall assume that
a satisfies a continuity and inf-sup condition for all
µ ∈ D, as we now state more precisely.
It shall prove convenient to state our hypotheses in
terms of a “supremizing” op e rator T
µ
: Y → Y . In
particular, for any given µ ∈ D
µ
, and any w ∈ Y ,
(T
µ
w, v)
Y
= a(w, v; µ), ∀ v ∈ Y ; (6)
it is readily shown that
T
µ
w = arg sup
v ∈Y
a(w, v; µ)
kvk
Y
. (7)
Furthermore, if we define the inf-sup (singular value)
and continuity c onstants as
β(µ) ≡ inf
w∈Y
sup
v ∈Y
a(w, v; µ)
kwk
Y
kvk
Y
(8)
and
γ(µ) ≡ sup
w∈Y
sup
v ∈Y
a(w, v; µ)
kwk
Y
kvk
Y
, (9)
then,
β(µ) = inf
w∈Y
σ(w; µ) , (10)
γ(µ) = sup
w∈Y
σ(w; µ) , (11)
where
σ(w; µ) ≡
kT
µ
wk
Y
kwk
Y
. (12)
2 of 18
American Institute of Aeronautics and Astronautics Paper 2003-3847

Our assumptions are then: for some positive constant
ε
s
, ε
s
≤ β(µ) ≤ γ(µ) < ∞, ∀ µ ∈ D
µ
.
We next define the bilinear form b(·, ·; µ): Y ×Y →
R as
b(w, v; µ) = (T
µ
w, T
µ
v)
Y
, ∀w, v ∈ Y . (13)
We then introduce the eigenproblem: Given µ ∈ D
µ
,
find χ
i
(µ) ∈ Y, λ
i
(µ) ∈ R, i = 1, . . . , ∞, such that
b(χ
i
(µ), v; µ) = λ
i
(µ)(χ
i
(µ), v)
Y
, ∀ v ∈ Y , (14)
kχ
i
(µ)k
Y
= 1 . (15)
We shall, for convenience, assume that the spectrum
is discrete (in actual practice we require only that the
first few modes belong to the discrete component). In
that case, we may assume that
b(χ
i
(µ), χ
j
(µ); µ) = λ
i
(µ)(χ
i
(µ), χ
j
(µ))
Y
= λ
i
(µ)δ
ij
,
(16)
where δ
ij
is the Kronecker-delta symbol; that 0 <
λ
1
(µ) ≤ λ
2
(µ) ≤ ···; and that Y = span {χ
i
(µ), i =
1, . . . , ∞}. Note that, from (10)-(14), β(µ) =
p
λ
1
(µ);
furthermore, γ(µ) is an upper bound for the spectrum.
We shall make the further assumption that a is
“affine in the parameter” in the sense that, for some
finite Q,
a(w, v; µ) =
Q
X
q=1
Θ
q
(µ) a
q
(w, v) , (17)
where Θ: D
µ
→ R
Q
are differentiable parameter-
dependent coefficient functions, and the a
q
: Y × Y →
R, 1 ≤ q ≤ Q, are parameter-independent bilinear
forms. We define, for future reference,
D
qp
= max
µ∈D
µ




∂Θ
q
∂µ
p
(µ)




, (18)
for 1 ≤ q ≤ Q, 1 ≤ p ≤ P . Furthermore, we as sume
that the a
q
are continuous in the sense that there exist
positive finite constants Γ
q
, 1 ≤ q ≤ Q, such that
|a
q
(w, v)| ≤ Γ
q
|w|
q
|v|
q
; (19)
here | ·|
q
: H
1
(Ω) → R are seminorms that satisfy
Q
X
q=1
|v|
2
q
!
1/2
≤ C
1/2
Y
kvk
Y
, ∀v ∈ Y , (20)
where C
Y
is a finite constant.
Finally, it directly follows from (6) and (17) that,
for any w ∈ Y , T
µ
w ∈ Y may be expressed as
T
µ
w =
Q
X
q=1
Θ
q
(µ) T
q
w , (21)
where, for any w ∈ Y , T
q
w, 1 ≤ q ≤ Q, is given by
(T
q
w, v)
Y
= a
q
(w, v), ∀ v ∈ Y . (22)
Note that the operators T
q
: Y → Y are independent
of the parameter µ.
2.2 Problem Formulation
2.2.1 Weak Statement
We introduce an output functional ` ∈ Y
0
and
“data” functional f ∈ Y
0
. Our weak statement of the
partial differential equation is then: Given µ ∈ D
µ
,
find
s = `(u(µ)) , (23)
where u(µ) ∈ Y satisfies
a(u(µ), v; µ) = f(v), ∀v ∈ Y . (24)
In the language of the introduction, s(µ) is our output,
and u(µ) is our field variable.
In actual practice, we shall replace (23)–(24) with a
truth approximation: Given µ ∈ D
µ
, find
s
N
(µ) = `(u
N
(µ)) ,
where u
N
(µ) ∈ Y
N
⊂ Y satisfies
a(u
N
(µ), v; µ) = f(v), ∀v ∈ Y
N
, (25)
and Y
N
is a finite element approximation subspace.
We assume that N is chosen sufficiently large that
s
N
(µ) and u
N
(µ) may be effectively equated with
s(µ) and u(µ), respectively. We shall thus distinguish
between Y
N
and Y only in our discussion of compu-
tational complexity. (Note that issues associated with
a possible continuous component to the spectrum of
(14) may be addressed by considering Y as the limit
of Y
N
, N → ∞.)
2.2.2 Reduced-Basis Approximation
The focus of the current paper is a posteriori error
estimation. We shall thus take our reduced-basis ap-
proximation as given. In particular, we assume that
we are provided with a reduced-basis approximation
to u(µ), u
N
(µ) ∈ W
N
, where
W
N
= span {ζ
n
≡ u(µ
n
), 1 ≤ n ≤ N} , (26)
S
N
= {µ
1
∈ D
µ
, . . . , µ
N
∈ D
µ
}, and u(µ
n
) satisfies
(24) (in practice, (25)) for µ = µ
n
. It follows that
u
N
(µ) may be expressed as
u
N
(µ) =
N
X
n=1
u
Nn
(µ) ζ
n
. (27)
The reduced-basis approximation to the output s(µ),
s
N
(µ), is given by s
N
(µ) = `(u
N
(µ)).
For the purposes of this paper, we shall consider only
standard Galerkin projections: a(u
N
(µ), v; µ) = f(v ),
∀ v ∈ W
N
. Howe ver, the discrete inf-sup param-
eter associated with the latter may not be “good,”
with corresponding detriment to the accuracy of u
N
(µ)
and hence s
N
(µ). More s ophisticated minimum-
residual
8, 18
and in particular Petrov-Galerkin
7, 18
ap-
proaches restore (guaranteed) stability, albeit at some
additional c omplexity and cost.
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American Institute of Aeronautics and Astronautics Paper 2003-3847

2.2.3 Error Estimation: Objective
We now wish to develop a posteriori error bounds
∆
N
(µ) and ∆
s
N
(µ) such that
ke(µ)k
H
1
(Ω)
≤ ∆
N
(µ) , (28)
and
|s(µ) −s
N
(µ)| = |`(e(µ))| ≤ ∆
s
N
(µ) , (29)
where e(µ) ≡ u(µ) − u
N
(µ). For the purposes
of this paper, we shall focus on the H
1
(Ω) bound,
∆
N
(µ), in terms of which ∆
s
N
(µ) can be expressed as
kY`k
Y
∆
N
(µ); the latter may be significantly improved
by the introduction of adjoint techniques.
5, 15
It shall prove convenient to introduce the notion of
effectivity, defined (here) as
η
N
(µ) ≡
∆
N
(µ)
ke(µ)k
H
1
(Ω)
. (30)
Our certainty requirement (28) may be stated as
η
N
(µ) ≥ 1, ∀ µ ∈ D
µ
. However, for efficiency, we
must also require η
N
(µ) ≤ C
η
, where C
η
≥ 1 is a con-
stant independent of N and µ; preferably, C
η
is close to
unity, thus ensuring that we cho os e the smallest N —
and hence most economical — reduced-basis approxi-
mation consistent with the specified error tolerance.
2.3 A Posteriori Error Estimation
2.3.1 Error Bound
We assume that we are given a
ˆ
β(µ) such that, for
the given inner product ( ·, ·)
Y
≡ (·, ·)
H
1
(Ω)
(which in
our previous papers
14, 2 0
would be denoted a “bound
conditioner”),
β(µ) ≥
ˆ
β(µ) ≥ (1 − τ) ε
s
, ∀µ ∈ D
µ
, (31)
where τ ∈ ]0, 1[ . We then define our error bound as
∆
N
(µ) ≡
kYr( · ; µ)k
Y
ˆ
β(µ)
, (32)
where
r(v; µ) = f(v) − a(u
N
(µ), v; µ), ∀ v ∈ Y , (33)
is the residual associated with u
N
(µ). Note it follows
from (24) that (33) may be restated as
a(e(µ), v; µ) = r(v; µ), ∀ v ∈ Y , (34)
where we recall that e(µ) ≡ u(µ) −u
N
(µ).
We can then state
Proposition 1 For the error bound ∆
N
(µ) of (32),
the effectivity satisfies
1 ≤ η
N
(µ) ≤
γ(µ)
(1 −τ ) ε
s
, ∀ µ ∈ D , (35)
for all N ∈ N.
Proof It follows from (4), (6), and (34) that
kYr( · ; µ)k
Y
= kT
µ
e(µ)k
Y
. (36)
Furthermore, from (12) we know that
ke(µ)k
Y
=
kT
µ
e(µ)k
Y
σ(e(µ); µ)
, (37)
and hence from (1), (30), (32), (36), and (37)
η
N
(µ) =
σ(e(µ); µ)
ˆ
β(µ)
. (38)
The result then directly follows from (10), (11), (31),
and (38). 
We note that our proof (or bound) does not exploit
any special properties of e(µ) (or u
N
(µ)).
It remains to develop our lower bound construc-
tion,
ˆ
β(µ), and to demonstrate that both
ˆ
β(µ) and
kYr( · ; µ)k
Y
may be computed efficiently (that is, in
complexity independent of N).
2.3.2 Inf-Sup Lower Bound Construction
Many of the most obvious eigenvalue approximation
concepts are not relevant here, since we require a lower,
not upper, bound. We thus develop a construction
particularly s uited to our context.
We assume that we are given a set of J parameter
points, L
J
≡ {µ
1
∈ D
µ
, . . . , µ
J
∈ D
µ
}, and associated
set of polygonal regions R
µ
j
,τ
, 1 ≤ j ≤ J, where
R
µ,τ
≡ {µ ∈ D
µ
|B
µ
q
(µ) ≤
τ
C
Y
β(µ), 1 ≤ q ≤ Q} ,
(39)
and
B
µ
q
(µ) = Γ
q
P
X
p=1
D
qp
|µ
p
− µ
p
| ; (40)
we further ass ume that
J
[
j=1
R
µ
j
,τ
= D
µ
. (41)
We then define J : D
µ
→ {1, . . . , J} such that, for a
given µ, R
µ
J (µ)
,τ
is that region (or a selected region)
which contains µ.
Our lower bound is then: Given µ ∈ D
µ
,
ˆ
β(µ) = β(µ
J (µ)
) −C
Y
B
µ
J (µ)
max
(µ) , (42)
where
B
µ
max
(µ) = max
q∈{1 ,. ..,Q}
B
µ
q
(µ) (43)
for B
µ
q
(µ) defined in (40).
We can now state
Proposition 2 The construction
ˆ
β(µ) of (42) satis-
fies the inequality (31).
4 of 18
American Institute of Aeronautics and Astronautics Paper 2003-3847