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Reliable Real-Time Solution of Parametrized Partial
Dierential Equations: Reduced-Basis Output Bound
Methods
Christophe Prud’Homme, Dimitrios V. Rovas, Karen Veroy, Luc Machiels,
Yvon Maday, Anthony T. Patera, Gabriel Turinici
To cite this version:
Christophe Prud’Homme, Dimitrios V. Rovas, Karen Veroy, Luc Machiels, Yvon Maday, et al.. Reli-
able Real-Time Solution of Parametrized Partial Dierential Equations: Reduced-Basis Output Bound
Methods. Journal of Fluids Engineering, American Society of Mechanical Engineers, 2001, 124 (1),
pp.70-80. �10.1115/1.1448332�. �hal-00798326�
C. Prud’homme D. V. Rovas
K. Veroy
L. Machiels
Department of Mechanical
Engineering, Massachusetts
Institute of Technolo
gy,
Cambridge, MA 02139
Y. Maday
Laboratoire d’Analyse Numérique,
Université
Pierre et Marie Curie,
Boîte
courrier 187, 75252 Paris, Cedex 05, France
A. T. Patera
Department of Mechanical
Engineering, Massachusetts
Institute of Technology,
Cambridge, MA 02139
G. Turinici
ASCI-CNRS Orsay,
and INRA Rocquencourt M3N,
B.P. 105,
78153 LeChesnay Cedex France
Reliable Real-Time Solution of
Parametrized Partial Differential
Equations: Reduced-Basis Output
Bound Methods
We present a technique for the rapid and reliable prediction of linear-functional
outputs of elliptic (and parabolic) partial differential equations with affine
parameter dependence. The essential components are (i) (provably) rapidly
convergent global reduced-basis approximations—Galerkin projection onto a
space WN spanned by solutions of the gov-erning partial differential equation at
N selected points in parameter space; (ii) a poste-riori error estimation—
relaxations of the error-residual equation that provide inexpensive yet sharp and
rigorous 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; the method is thus ideally suited for the
repeated and rapid evaluations required in the context of parameter estimation,
design, optimization, and real-time control.
1 Introduction
The optimization, control, and characterization of an engineer-
ing component or system requires the prediction of certain ‘‘quan-
tities of interest,’’ or performance metrics, which we shall denote
outputs—for example deflections, maximum stresses, maximum
temperatures, heat transfer rates, flowrates, or lift and drags. These
outputs are typically expressed as functionals of field variables
associated with a parametrized partial differential equation which
describes the physical behavior of the component or system. The
parameters, which we shall denote inputs, serve to identify a par-
ticular ‘‘configuration’’of the component: these inputs may repre-
sent design or decision variables, such as geometry—for example,
in optimization studies; control variables, such as actuator
power—for example in real-time applications; or characterization
variables, such as physical properties—for example in inverse
problems. We thus arrive at an implicit input-output relationship,
evaluation of which demands solution of the underlying partial
differential equation.
Our 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 charac-
terization contexts. The ‘‘many queries’’ limit has certainly re-
ceived considerable attention: from ‘‘fast loads’’ or multiple right-
hand side notions 共e.g., Chan and Wan 关1兴, Farhat et al. 关2兴兲 to
matrix perturbation theories 共e.g., Akgun et al. 关3兴,Yip关4兴兲 to
continuation methods 共e.g., Allgower and Georg 关5兴, Rheinboldt
关6兴兲. Our particular approach is based on the reduced-basis
method, first introduced in the late 1970s for nonlinear structural
analysis 共Almroth et al. 关7兴, Noor and Peters 关8兴兲, and subse-
quently developed more broadly in the 1980s and 1990s 共Balmes
关9兴, Barrett and Reddien 关10兴, Fink and Rheinboldt 关11兴, Peterson
关12兴, Porsching 关13兴, Rheinboldt 关14兴兲. 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; rather, it resides, or ‘‘evolves,’’ on
a much lower-dimensional manifold induced by the parametric
dependence.
The reduced-basis approach as earlier articulated is local in
parameter space in both practice and theory. To wit, Lagrangian or
Taylor approximation spaces for the low-dimensional manifold
are typically defined relative to a particular parameter point; and
the associated a priori convergence theory relies on asymptotic
arguments in sufficiently small neighborhoods 共Fink and Rhein-
boldt 关11兴兲. As a result, the computational improvements—relative
to conventional 共say兲 finite element approximation—are often
quite modest 共Porsching 关13兴兲. Our work differs from these earlier
efforts in several important ways: first, we develop 共in some cases,
provably兲 global approximation spaces; second, we introduce rig-
orous a posteriori error estimators; and third, we exploit off-line/
on-line computational decompositions 共see Balmes 关9兴 for an ear-
lier application of this strategy within the reduced-basis context兲.
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 expository review paper we focus on these new ingredi-
ents. In Section 2 we introduce an abstract problem formulation
and several illustrative instantiations. In Section 3 we describe, for
coercive symmetric problems and ‘‘compliant’’ outputs, the
reduced-basis approximation; and in Section 4 we present the as-
sociated a posteriori error estimation procedures. In Section 5 we
1
consider the extension of our approach to noncompliant outputs
and nonsymmetric operators; eigenvalue problems; and, more
briefly, noncoercive operators, parabolic equations, and nonaffine
problems. A description of the system architecture in which these
numerical objects reside may be found in Veroy et al. 关15兴.
2 Problem Statement
2.1 Abstract Formulation. We consider a suitably regular
domain ⍀傺R
d
, d⫽ 1, 2, or 3, and associated function space
X傺 H
1
(⍀), where H
1
(⍀)⫽
兵
v
苸 L
2
(⍀), ⵜ
v
苸 (L
2
(⍀))
d
其
, and
L
2
(⍀) is the space of square-X integrable functions over ⍀. The
inner product and norm associated with X are given by (•,•)
X
and
储
•
储
X
⫽(•,•)
X
1/2
, respectively. We also define a parameter set D
苸 R
P
, a particular point in which will be denoted
. Note that ⍀
does not depend on the parameter.
We then introduce a bilinear form a: X⫻ X⫻D→R, and linear
forms f: X→R,
l : X→R. We shall assume that a is continuous,
a(w,
v
;
)⭐
␥
(
)
储
w
储
X
储
v
储
X
⭐
␥
0
储
w
储
X
储
v
储
X
, ᭙
苸 D; further-
more, in Sections 3 and 4, we assume that a is coercive,
0⬍
␣
0
⭐
␣
共
兲
⫽ inf
w苸 X
a
共
w,w;
兲
储
w
储
X
2
, ᭙
苸 D, (1)
and symmetric, a(w,
v
;
)⫽ a(
v
,w;
); ᭙ w,
v
苸 X, ᭙
苸 D.We
also require that our linear forms f and
l be bounded; in Sections
3 and 4 we additionally assume a ‘‘compliant’’ output, f(
v
)
⫽
l (
v
), ᭙
v
苸 X.
We shall also make certain assumptions on the parametric de-
pendence of a, f, and
l . In particular, we shall suppose that, for
some finite 共preferably small兲 integer Q, a may be expressed as
a
共
w,
v
;
兲
⫽
兺
q⫽1
Q
q
共
兲
a
q
共
w,
v
兲
, ᭙ w,
v
苸 X,᭙
苸 D, (2)
for some
q
: D→R and a
q
: X⫻X→R, q⫽ 1,...,Q. This ‘‘sepa-
rability,’’ or ‘‘affine,’’ assumption on the parameter dependence is
crucial to computational efficiency; however, certain relaxations
are possible—see Section 5.3.3. For simplicity of exposition, we
assume that f and
l do not depend on
; in actual practice, affine
dependence is readily admitted.
Our abstract problem statement is then: for any
苸 D, find
s(
)苸 R given by
s
共
兲
⫽l
共
u
共
兲兲
, (3)
where u(
)苸 X is the solution of
a
共
u
共
兲
,
v
;
兲
⫽ f
共v
兲
, ᭙
v
苸 X. (4)
In the language of the Introduction, a is our partial differential
equation 共in weak form兲,
is our parameter, u(
) is our field
variable, and s(
) is our output. For simplicity of exposition, we
may on occasion suppress the explicit dependence on
.
2.2 Particular Instantiations. We indicate here a few in-
stantiations of the abstract formulation; these will serve to illus-
trate the methods 共for coercive, symmetric problems兲 of Sections
3 and 4.
2.2.1 A Thermal Fin. In this example, we consider the two-
and three-dimensional thermal fins shown in Fig. 1; these ex-
amples may be 共interactively兲 accessed on our web site.
1
The fins
consist of a vertical central ‘‘post’’ of conductivity k
˜
0
and four
horizontal ‘‘subfins’’ of conductivity k
˜
i
, i⫽ 1,...,4.Thefinscon-
duct heat from a prescribed uniform flux source q
˜
⬙
at the root
⌫
˜
root
through the post and large-surface-area subfins to the sur-
rounding flowing air; the latter is characterized by a sink tempera-
ture u
˜
0
and prescribed heat transfer coefficient h
˜
. The physical
model is simple conduction: the temperature field in the fin, u
˜
,
satisfies
兺
i⫽0
4
冕
⍀
˜
i
k
˜
i
ⵜ
˜
u
˜
•ⵜ
˜
v
˜
⫹
冕
⍀
˜
\
⌫
˜
root
h
˜
共
u
˜
⫺u
˜
0
兲
v
˜
⫽
冕
⌫
˜
root
q
˜
⬙
v
˜
,
᭙
v
˜
苸 X
˜
⬅H
1
共
⍀
˜
兲
, (5)
where ⍀
˜
i
is that part of the domain with conductivity k
˜
i
, and
⍀
˜
denotes the boundary of ⍀
˜
.
We now 共i兲 nondimensionalize the weak equations 共5兲, and 共ii兲
apply a continuous piecewise-affine transformation from ⍀
˜
to a
fixed 共
-independent兲 reference domain ⍀共Maday et al. 关16兴兲.
The abstract problem statement 共4兲 is then recovered for
⫽
兵
k
1
,k
2
,k
3
,k
4
,Bi,L,t
其
, D⫽
关
0.1,10.0
兴
4
⫻关0.01,1.0兴⫻关2.0,3.0兴
⫻关0.1,0.5兴, and P⫽7; here k
1
,...,k
4
are the thermal conductivi-
ties of the ‘‘subfins’’ 共see Fig. 1兲 relative to the thermal conduc-
tivity of the fin base; Bi is a nondimensional form of the heat
transfer coefficient; and, L, t are the length and thickness of each
of the ‘‘subfins’’ relative to the length of the fin root ⌫
˜
root
.Itis
readily verified that a is continuous, coercive, and symmetric; and
that the ‘‘affine’’ assumption 共2兲 obtains for Q⫽16 共two-
1
FIN2D: http://augustine.mit.edu/fin2d/fin2d.pdf and FIN3D: http://
augustine.mit.edu/fin3d
–
1/fin3d
–
1.pdf
Fig. 1 Two- and three-dimensional thermal fins
2
dimensional case兲 and Q⫽ 25 共three-dimensional case兲. Note that
the geometric variations are reflected, via the mapping, in the
q
(
).
For our output of interest, s(
), we consider the average tem-
perature of the root of the fin nondimensionalized relative to q
˜
⬙
,
k
˜
0
, and the length of the fin root. This output may be expressed as
s(
)⫽ l (u(
)), where l (
v
)⫽
兰
⌫
root
v
. It is readily shown that
this output functional is bounded and also ‘‘compliant’’:
l (
v
)
⫽ f (
v
), ᭙
v
苸 X.
2.2.2 A Truss Structure. We consider a prismatic microtruss
structure 共Evans et al. 关17兴, Wicks and Hutchinson 关18兴兲 shown in
Fig. 2; this example may be 共interactively兲 accessed on our web
site.
2
The truss consists of a frame 共upper and lower faces, in dark
gray兲 and a core 共trusses and middle sheet, in light gray兲. The
structure transmits a force per unit depth F
˜
uniformly distributed
over the tip of the middle sheet ⌫
˜
3
through the truss system to the
fixed left wall ⌫
˜
0
. The physical model is simple plane-strain 共two-
dimensional兲 linear elasticity: the displacement field u
i
, i⫽1,2,
satisfies
冕
⍀
˜
v
˜
i
x
˜
j
E
˜
ijkl
u
˜
k
x
˜
l
⫽⫺
冉
F
˜
t
˜
c
冊
冕
⌫
˜
3
v
˜
2
, ᭙
v
苸 X
˜
, (6)
where ⍀
˜
is the truss domain, E
˜
ijkl
is the elasticity tensor, and X
˜
refers to the set of functions in H
1
(⍀
˜
) which vanish on ⌫
˜
0
.We
assume summation over repeated indices.
We now 共i兲 nondimensionalize the weak equations 共6兲, and 共ii兲
apply a continuous piecewise-affine transformation from ⍀
˜
to a
fixed 共
-independent兲 reference domain ⍀. The abstract problem
statement 共4兲 is then recovered for
⫽
兵
t
f
,t
t
,H,
其
, D⫽关0.08,1.0兴
⫻关0.2,2.0兴⫻关4.0,10.0兴⫻关30.0°,60.0°兴, and P⫽4. Here t
f
and t
t
are the thicknesses of the frame and trusses 共normalized relative to
t
˜
c
兲, respectively; H is the total height of the microtruss 共normal-
ized relative to t
˜
c
兲; and
is the angle between the trusses and the
faces. The Poisson’s ratio,
⫽0.3, and the frame and core Young’s
moduli, E
f
⫽75 GPa and E
c
⫽200 GPa, respectively, are held
fixed. It is readily verified that a is continuous, coercive, and
symmetric; and that the ‘‘affine’’ assumption 共2兲 obtains for Q
⫽44.
Our outputs of interest are 共i兲 the average downward deflection
共compliance兲 at the core tip, ⌫
3
, nondimensionalized by F
˜
/E
˜
f
;
and 共ii兲 the average normal stress across the critical 共yield兲 section
denoted ⌫
1
s
in Fig. 2. These compliance and noncompliance out-
puts can be expressed as s
1
(
)⫽ l
1
(u(
)) and s
2
(
)
⫽
l
2
(u(
)), respectively, where l
1
(
v
)⫽⫺
兰
⌫
3
v
2
, and
l
2
共v
兲
⫽
1
t
f
冕
⍀
s
i
x
j
E
ijkl
u
k
x
l
are bounded linear functionals; here
i
is any suitably smooth
function in H
1
(⍀
s
) such that
i
nˆ
i
⫽1on⌫
1
s
and
i
nˆ
i
⫽0on⌫
2
s
,
where nˆ is the unit normal. Note that s
1
(
) is a compliant output,
whereas s
2
(
) is ‘‘noncompliant.’’
3 Reduced-Basis Approach
We recall that in this section, as well as in Section 4, we assume
that a is continuous, coercive, symmetric, and affine in
—see
共2兲; and that
l (
v
)⫽ f (
v
), which we denote ‘‘compliance.’’
3.1 Reduced-Basis Approximation. We first introduce a
sample in parameter space, S
N
⫽
兵
1
,...,
N
其
, where
i
苸 D, i
⫽1,..., N; see Section 3.2.2 for a brief discussion of point dis-
tribution. We then define our Lagrangian 共Porsching 关13兴兲
reduced-basis approximation space as W
N
⫽span
兵
n
⬅u(
n
),n
⫽1,...,N
其
, where u(
n
)苸 X is the solution to 共4兲 for
⫽
n
.
In actual practice, u(
n
) is replaced by an appropriate finite ele-
ment approximation on a suitably fine truth mesh; we shall discuss
the associated computational implications in Section 3.3. Our
reduced-basis approximation is then: for any
苸 D, find s
N
(
)
⫽
l (u
N
(
)), where u
N
(
)苸 W
N
is the solution of
a
共
u
N
共
兲
,
v
;
兲
⫽l
共v
兲
, ᭙
v
苸 W
N
. (7)
Non-Galerkin projections are briefly described in Section 5.3.1.
3.2 A Priori Convergence Theory.
3.2.1 Optimality. We consider here the convergence rate of
u
N
(
)→u(
) and s
N
(
)→s(
)asN→⬁. To begin, it is stan-
dard to demonstrate optimality of u
N
(
) in the sense that
储
u
共
兲
⫺u
N
共
兲
储
X
⭐
冑
␥
共
兲
␣
共
兲
inf
w
N
苸 W
N
储
u
共
兲
⫺w
N
储
X
. (8)
共We note that, in the coercive case, stability of our 共‘‘conform-
ing’’兲 discrete approximation is not an issue; the noncoercive case
is decidedly more delicate 共see Section 5.3.1兲.兲 Furthermore, for
our compliance output,
s
共
兲
⫽s
N
共
兲
⫹l
共
u⫺ u
N
兲
⫽s
N
共
兲
⫹a
共
u,u⫺ u
N
;
兲
⫽s
N
共
兲
⫹a
共
u⫺ u
N
,u⫺ u
N
;
兲
(9)
from symmetry and Galerkin orthogonality. It follows that s(
)
⫺s
N
(
) converges as the square of the error in the best approxi-
mation and, from coercivity, that s
N
(
) is a lower bound for
s(
).
3.2.2 Best Approximation. It now remains to bound the de-
pendence of the error in the best approximation as a function of N.
At present, the theory is restricted to the case in which P⫽ 1, D
⫽
关
0,
max
兴
, and
a
共
w,
v
;
兲
⫽a
0
共
w,
v
兲
⫹
a
1
共
w,
v
兲
, (10)
where a
0
is continuous, coercive, and symmetric, and a
1
is con-
tinuous, positive semi-definite (a
1
(w,w)⭓0, ᭙w苸X), and sym-
metric. This model problem 共10兲 is rather broadly relevant, for
2
TRUSS: http://augustine.mit.edu/simple
–
truss/simple
–
truss.pdf
Fig. 2 A truss structure
3
example to variable orthotropic conductivity, variable rectilinear
geometry, variable piecewise-constant conductivity, and variable
Robin boundary conditions.
We now suppose that the
n
, n⫽ 1,...,N, are logarithmically
distributed in the sense that
ln
共
¯
n
⫹1
兲
⫽
n⫺ 1
N⫺ 1
ln
共
¯
max
⫹1
兲
, n⫽ 1,...,N, (11)
where
¯
is an upper bound for the maximum eigenvalue of a
1
relative to a
0
. 共Note
¯
is perforce bounded thanks to our assump-
tion of continuity and coercivity; the possibility of a continuous
spectrum does not, in practice, pose any problems.兲 We can then
prove 共Maday et al. 关19兴兲 that, for N⬎N
crit
⬅e ln(
¯
max
⫹1),
inf
w
N
苸 W
N
储
u
共
兲
⫺w
N
储
X
⭐
共
1⫹
max
¯
兲
储
u
共
0
兲
储
X
⫻exp
再
⫺
共
N⫺ 1
兲
共
N
crit
⫺1
兲
冎
, ᭙
苸 D. (12)
We observe exponential convergence, uniformly 共globally兲 for all
in D, with only very weak 共logarithmic兲 dependence on the
range of the parameter (
max
). 共Note the constants in 共12兲 are for
the particular case in which (•,•)
X
⫽a
0
(•,•).兲
The proof exploits a parameter-space 共nonpolynomial兲 interpo-
lant as a surrogate for the Galerkin approximation. As a result, the
bound is not always ‘‘sharp:’’ we observe many cases in which the
Galerkin projection is considerably better than the associated in-
terpolant; optimality 共8兲 chooses to ‘‘illuminate’’ only certain
points
n
, automatically selecting a best ‘‘subapproximation’’
among all 共combinatorially many兲 possibilities. We thus see why
reduced-basis state-space approximation of s(
) via u(
) is pre-
ferred to simple parameter-space interpolation of s(
) 共‘‘con-
necting the dots’’兲 via (
n
,s(
n
)) pairs. We note, however, that
the logarithmic point distribution 共11兲 implicated by our
interpolant-based arguments is not simply an artifact of the proof:
in numerous numerical tests, the logarithmic distribution performs
considerably 共and in many cases, provably兲 better than other more
obvious candidates, in particular for large ranges of the parameter.
Fortunately, the convergence rate is not too sensitive to point se-
lection: the theory only requires a log ‘‘on the average’’ distribu-
tion 共Maday et al. 关19兴兲; and, in practice,
¯
need not be a sharp
upper bound.
The result 共12兲 is certainly tied to the particular form 共10兲 and
associated regularity of u(
). However, we do observe similar
exponential behavior for more general operators; and, most im-
portantly, the exponential convergence rate degrades only very
slowly with increasing parameter dimension, P. We present in
Table 1 the error
兩
s(
)⫺ s
N
(
)
兩
/s(
) as a function of N,ata
particular representative point
in D, for the two-dimensional
thermal fin problem of Section 2.2.1; we present similar data in
Table 2 for the truss problem of Section 2.2.2. In both cases, since
tensor-product grids are prohibitively profligate as P increases, the
n
are chosen ‘‘log-randomly’’ over D: we sample from a multi-
variate uniform probability density on log(
). We observe that, for
both the thermal fin (P⫽7) and truss (P⫽ 4) problems, the error
is remarkably small even for very small N; and that, in both cases,
very rapid convergence obtains as N→⬁. We do not yet have any
theory for P⬎ 1. But certainly the Galerkin optimality plays a
central role, automatically selecting ‘‘appropriate’’ scattered-data
subsets of S
N
and associated ‘‘good’’ weights so as to mitigate the
curse of dimensionality as P increases; and the log-random point
distribution is also important—for example, for the truss problem
of Table 2, a non-logarithmic uniform random point distribution
for S
N
yields errors which are larger by factors of 20 and 10 for
N⫽ 30 and 80, respectively.
3.3 Computational Procedure. The theoretical and empiri-
cal results of Sections 3.1 and 3.2 suggest that N may, indeed, be
chosen very small. We now develop off-line/on-line computa-
tional procedures that exploit this dimension reduction.
We first express u
N
(
)as
u
N
共
兲
⫽
兺
j⫽ 1
N
u
Nj
共
兲
j
⫽
共
uគ
N
共
兲兲
T
គ
, (13)
where uគ
N
(
)苸 R
N
; we then choose for test functions
v
⫽
i
, i
⫽1,...,N. Inserting these representations into 共7兲 yields the de-
sired algebraic equations for uគ
N
(
)苸 R
N
,
Aគ
N
共
兲
uគ
N
共
兲
⫽Fគ
N
, (14)
in terms of which the output can then be evaluated as s
N
(
)
⫽Fគ
N
T
uគ
N
(
). Here Aគ
N
(
)苸 R
N⫻N
is the SPD matrix with entries
A
Ni, j
(
)⬅a(
j
,
i
;
), 1⭐i,j⭐N, and Fគ
N
苸 R
N
is the ‘‘load’’
共and ‘‘output’’兲 vector with entries F
Ni
⬅ f (
i
), i⫽1,..., N.
We now invoke 共2兲 to write
A
Ni, j
共
兲
⫽a
共
j
,
i
;
兲
⫽
兺
q⫽1
Q
q
共
兲
a
q
共
j
,
i
兲
, (15)
or
Aគ
N
共
兲
⫽
兺
q⫽1
Q
q
共
兲
Aគ
N
q
,
where the Aគ
N
q
苸 R
N⫻N
are given by A
Ni, j
q
⫽a
q
(
j
,
i
), i⭐i,j
⭐N,1⭐q⭐Q. The off-line/on-line decomposition is now clear.
In the off-line stage, we compute the u(
n
) and form the Aគ
N
q
and
Fគ
N
: this requires N 共expensive兲 ‘‘a’’ finite element solutions and
O(QN
2
) finite-element-vector inner products. In the on-line stage,
for any given new
, we first form Aគ
N
from 共15兲, then solve 共14兲
for uគ
N
(
), and finally evaluate s
N
(
)⫽ Fគ
N
T
uគ
N
(
): this requires
O(QN
2
)⫹ O(2/3 N
3
) operations and O(QN
2
) storage.
Thus, as required, the incremental, or marginal, cost to evaluate
s
N
(
) for any given new
—as proposed in a design, optimiza-
tion, or inverse-problem context—is very small: first, because N is
very small, typically O(10)—thanks to the good convergence
properties of W
N
; and second, because 共14兲 can be very rapidly
Table 1 Error, error bound „Method I…, and effectivity as a
function of
N
, at a particular representative point
«D, for the
two-dimensional thermal fin problem „compliant output…
N
兩
s(
)⫺ s
N
(
)
兩
/s(
) ⌬
N
(
)/s(
)
N
(
)
10
1.29⫻10
⫺2
8.60⫻10
⫺2
2.85
20
1.29⫻10
⫺3
9.36⫻10
⫺3
2.76
30
5.37⫻10
⫺4
4.25⫻10
⫺3
2.68
40
8.00⫻10
⫺5
5.30⫻10
⫺4
2.86
50
3.97⫻10
⫺5
2.97⫻10
⫺4
2.72
60
1.34⫻10
⫺5
1.27⫻10
⫺4
2.54
70
8.10⫻10
⫺6
7.72⫻10
⫺5
2.53
80
2.56⫻10
⫺6
2.24⫻10
⫺5
2.59
Table 2 Error, error bound „Method II…, and effectivity as a
function of
N
, at a particular representative point
«D, for the
truss problem „compliant output…
N
兩
s(
)⫺ s
N
(
)
兩
/s(
) ⌬
N
(
)/s(
)
N
(
)
10
3.26⫻10
⫺2
6.47⫻10
⫺2
1.98
20
2.56⫻10
⫺4
4.74⫻10
⫺4
1.85
30
7.31⫻10
⫺5
1.38⫻10
⫺4
1.89
40
1.91⫻10
⫺5
3.59⫻10
⫺5
1.88
50
1.09⫻10
⫺5
2.08⫻10
⫺5
1.90
60
4.10⫻10
⫺6
8.19⫻10
⫺6
2.00
70
2.61⫻10
⫺6
5.22⫻10
⫺6
2.00
80
1.19⫻10
⫺6
2.39⫻10
⫺6
2.00
4