Reduced Basis Approximation and a Posteriori Error Estimation
for Affinely Parametrized Elliptic Coercive Partial Differential
Equations
Application to Transport and Continuum Mechanics
G. Rozza · D.B.P. Huynh · A.T. Patera
Abstract In this paper we consider (hierarchical, Lagrange)
reduced basis approximation and a posteriori error estima-
tion for linear functional outputs of affinely parametrized
elliptic coercive partial differential equations. The essen-
tial ingredients are (primal-dual) Galerkin projection onto
a low-dimensional space associated with a smooth “para-
metric manifold”—dimension reduction; efficient and ef-
fective greedy sampling methods for identification of opti-
mal and numerically stable approximations—rapid conver-
gence; a posteriori error estimation procedures—rigorous
and sharp bounds for the linear-functional outputs of in-
terest; and Offline-Online computational decomposition
strategies—minimum marginal cost for high performance
in the real-time/embedded (e.g., parameter-estimation, con-
This work was supported by DARPA/AFOSR Grants
FA9550-05-1-0114 and FA-9550-07-1-0425, the Singapore-MIT
Alliance, the Pappalardo MIT Mechanical Engineering Graduate
Monograph Fund, and the Progetto Roberto Rocca Politecnico di
Milano-MIT. We acknowledge many helpful discussions with
Professor Yvon Maday of University of Paris 6 and Luca Dedé
of MOX-Politecnico di Milano.
G. Rozza
Mechanical Engineering Department, Massachusetts Institute
of Technology, Room 3-264, 77
Mass Avenue, Cambridge,
MA 02142-4307, USA
e-mail: rozza@mit.edu
D.B.P. Huynh
Singapore-MIT Alliance, E4-04-10, National University
of Singapore, 4 Eng. Drive, Singapore, 117576, Singapore
e-mail: baophuong@nus.edu.sg
A.T. Patera
Massachusetts Institute of Technology, Room 3-266, 77 Mass
Avenue, Cambridge, MA 02142-4307, USA
e-mail: patera@mit.edu
trol) and many-query (e.g., design optimization, multi-
model/scale) contexts. We present illustrative results for heat
conduction and convection-diffusion, inviscid flow, and lin-
ear elasticity; outputs include transport rates, added mass,
and stress intensity factors.
1 Introduction and Motivation
In this work we describe reduced basis (RB) approximation
and a posteriori error estimation methods for rapid and re-
liable evaluation of input-output relationships in which the
output is expressed as a functional of a field variable that
is the solution of an input-parametrized partial differential
equation (PDE). In this particular paper we shall focus on
linear output functionals and affinely parametrized linear
elliptic coercive PDEs; however the methodology is much
more generally applicable, as we discuss in Sect. 2.
We emphasize applications in transport and mechanics:
unsteady and steady heat and mass transfer; acoustics; and
solid and fluid mechanics. (Of course we do not preclude
other domains of inquiry within engineering (e.g., electro-
magnetics) or even more broadly within the quantitative dis-
ciplines (e.g., finance).) The input-parameter vector typi-
cally characterizes the geometric configuration, the physical
properties, and the boundary conditions and sources. The
outputs of interest might be the maximum system tempera-
ture, an added mass coefficient, a crack stress intensity fac-
tor, an effective constitutive property, an acoustic waveguide
transmission loss, or a channel flowrate or pressure drop. Fi-
nally, the field variables that connect the input parameters
to the outputs can represent a distribution function, temper-
ature or concentration, displacement, pressure, or velocity.
The methodology we describe in this paper is motivated
by, optimized for, and applied within two particular con-
1
texts: the real-time context (e.g., parameter-estimation [54,
96, 154] or control [124]); and the many-query context (e.g.,
design optimization [107] or multi-model/scale simulation
[26, 49]). Both these contexts are crucial to computational
engineering and to more widespread adoption and applica-
tion of numerical methods for PDEs in engineering practice
and education.
We first illustrate the real-time context: we can also char-
acterize this context as “deployed” or “in the field” or “em-
bedded.” A s an example of a real-time—and in fact, often
also many-query—application, we consider a crack in a crit-
ical structural component such as a composite-reinforced
concrete support (or an aircraft engine). We first pursue
Non-Destructive Evaluation (NDE) parameter estimation
procedures [17, 96, 154]—say by vibration or thermal tran-
sient analysis—to determine the location and configuration
of the delamination crack in the support. We then evaluate
stress intensity factors to determine the critical load for frac-
ture or the anticipated crack grow th due to fatigue. Finally
we modify “on site” the installation or subsequent mission
profile to prolong life. Safety and economics require rapid
and reliable response in the field.
We next illustrate the many-query context. As an ex-
ample of a general family of many-query applications, we
cite multiscale (temporal, spatial) or multiphysics models
in which behavior at a larger scale must “invoke” m any
spatial or temporal realizations of parametrized behavior
at a smaller scale. Particular cases (to which RB methods
have been applied) include stress intensity factor evaluation
[5, 62] within a crack fatigue growth model [63]; calculation
of spatially varying cell properties [26, 75] within homoge-
nization theory [25] predictions for macroscale composite
properties; assembly and interaction of many similar build-
ing blocks [79] in large (e.g., cardio-vascular [ 137]) biologi-
cal networks; or molecular dynamics computations based on
quantum-derived energies/forces [36]. In all these cases, the
number of input-output evaluations is often measured in the
tens of thousands.
Both the real-time and many-query contexts present a
significant and often unsurmountable challenge to “classi-
cal” numerical techniques such as the finite element (FE)
method. These contexts are often much better served by the
reduced basis approximations and associated a posteriori er-
ror estimation techniques described in this work. We note,
however, that the RB methods we describe do not replace,
but rather build upon and are measured (as regards accuracy)
relative to, a finite element model [22, 41, 126, 147, 158]:
the reduced basis approximates not the exact solution but
rather a “given” finite element discretization of (typically)
very large dimension N . In short, we pursue an algorithmic
collaboration rather than an algorithmic competition.
2 Historical Perspective and Background
The development of the reduced basis (RB) method can
perhaps be viewed as a response to the considerations and
imperatives described above. In particular, the parametric
real-time and many-query contexts represent not only com-
putational challenges, but also computational opportunities.
We identify two key opportunities that can be gainfully ex-
ploited:
Opportunity I. In the parametric setting, we restrict our at-
tention to a typically smooth and rather low-dimensional
parametrically induced manifold: the set of fields engen-
dered as the input varies over the parameter domain; in the
case of single parameter, the parametrically induced man-
ifold is a one-dimensional filament within the infinite di-
mensional space which characterizes general solutions to
the PDE. Clearly, generic approximation spaces are unnec-
essarily rich and hence unnecessarily expensive within the
parametric framework.
Opportunity II. In the real-time or many-query contexts, in
which the premium is on marginal cost (or equivalently as-
ymptotic average cost) per input-output evaluation, we can
accept greatly increased pre-processing or “Offline” cost—
not tolerable for a single or few evaluations—in exchange
for greatly decreased “Online” (or deployed) cost for each
new/additional input-output evaluation. Clearly, resource
allocation typical for “single-query” investigations will be
far from optimal for many-query and real-time exercises.
We shall review the development of RB methods in terms
of these two opportunities.
Opportunity I
Reduced Basis discretization is, in brief, (Galerkin) projec-
tion on an N -dimensional approximation space that focuses
(typically through Taylor expansions or Lagrange “snap-
shots”) on the parametrically induced manifold identified in
Opportunity I. Initial work grew out of two related streams
of inquiry: from the need for more effective, and perhaps
also more interactive, many-query design evaluation—[48]
considers linear structural examples; and from the need for
more efficient parameter continuation methods—[4, 98, 99,
101, 104, 105] consider nonlinear structural analysis prob-
lems. (Several modal analysis techniques from this era [92]
are also closely related to RB notions.)
The ideas present in these early somewhat domain-
specific contexts were soon extended to (i) general finite-
dimensional systems as well as certain classes of PDEs (and
ODEs) [19, 47, 76, 100, 106, 120, 131, 132], and (ii) a
variety of different reduced basis approximation spaces—
in particular Taylor and Lagrange [119] and more recently
Hermite [67] expansions. The next decade(s) saw further
2
expansion into different applications and classes of equa-
tions, such as fluid dynamics and the incompressible Navier-
Stokes equations [57, 66–69, 114].
However, in these early methods, the approximation
spaces tended to be rather local and typically rather low-
dimensional in parameter (often a single parameter). In part,
this was due to the nature of the applications—parametric
continuation. But it was also due to the absence of a pos-
teriori error estimators and effective sampling procedures.
It is clear that in more global, higher-dimensional para-
meter domains the ad hoc reduced basis predictions “far”
from any sample points can not necessarily be trusted, and
hence a posteriori error estimators are crucial to reliabil-
ity (and ultimately, safe engineering interventions in par-
ticular in the real-time context). It is equally clear that in
more global, higher-dimensional parameter domains simple
tensor-product/factorial “designs” are not practicable, and
hence sophisticated sampling strategies are crucial to con-
vergence and computational efficiency.
1
Much current effort is thus devoted to development of
(i) a posteriori error estimation procedures and in partic-
ular rigorous error bounds for outputs of interest [121],
and (ii) effective sampling strategies in particular for higher
(than one) dimensional parameter domains [32, 33, 97, 136,
153]. The a posteriori error bounds are of course indispens-
able for rigorous certification of any particular reduced ba-
sis (Online) output prediction. However, the error estima-
tors can also play an important role in efficient and effective
(greedy) sampling procedures: the inexpensive error bounds
permit us first, to explore much larger subsets of the para-
meter domain in search of most representative or best “snap-
shots,” and second, to determine when we have just enough
basis functions. Just as in the finite element context [12],
the simultaneous emergence of error estimation and adap-
tive sampling/approximation capabilities is certainly not a
coincidence.
We note here that greedy sampling methods are similar in
objective to, but very different in approach from, more well-
known Proper Orthogonal Decomposition (POD) methods
[8, 24, 58, 73, 77, 93, 127–129, 144, 145, 157]. For reasons
that we shall explore, the former are applied in the (multi-
dimensional) parameter domain, while the latter are most
often applied in the (one-dimensional) temporal domain.
However, POD economizati on techniques can be, and have
successfully been, applied within the parametric RB con-
text [31, 40, 42, 49, 59, 83, 154]. (We shall conduct a brief
comparison of greedy and POD approaches—computational
cost and performance—in Sect. 8.1.4.)
1
Several early papers [102–104] did indeed discuss a posteriori error
estimation and even adaptive improvement/sampling of the RB space;
however, the approach could not be efficiently or rigorously applied
to partial differential equations due to the computational requirements,
the residual norms employed, and the absence of any stability consid-
erations.
Opportunity II
Early work on the reduced basis method certainly exploited
Opportunity II—but not fully. In particular, and perhaps at
least partially because of the difficult nonlinear nature of the
initial applications, early RB approaches did not fully de-
couple the underlying FE approximation—of very high di-
mension N —from the subsequent reduced basis projection
and evaluation—of very low dimension N . More precisely,
most often the Galerkin stiffness equations for the reduced
basis system were generated by direct appeal to the high-
dimensional FE representation: in nuts and bolts terms, pre-
and post-multiplication of the FE stiffness system by rectan-
gular basis matrices. As a result of this expensive projection
the computational savings provided by RB treatment (rela-
tive to classical FE evaluation) were typically rather modest
[98, 119, 120] despite the very small size of the ultimate
reduced basis stiffness system.
Much current work is thus devoted to full decoupling of
the FE and RB spaces through Offline-Online procedures:
the complexity of the Offline stage depends on N (the di-
mension of the FE space); the complexity of the Online
stage—in which we respond to a new value of the input
parameter—depends only on N (the dimension of the re-
duced basis space) and Q (which measures the parametric
complexity of the operator and data, as defined below). In
essence, in the Online stage we are guaranteed the accuracy
of a high-fidelity finite element model but at the very low
cost of a reduced-order model.
In the context of affine parameter dependence, in which
the operator is expressible as the sum of Q products of
parameter-dependent functions and parameter-independent
operators, the Offline-Online idea is quite self-apparent
and indeed has been re-invented often [15, 66, 70, 114];
however, application of the concept to a posteriori error
estimation—note the Online complexity of both the output
and the output error bound calculation must be independent
of N —is more involved and more recent [64, 121, 122]. In
the case of n o n affine parameter dependence the develop-
ment of Offline-Online strategies is much less transparent,
and only in the last few years have effective procedures—in
effect, efficient methods for approximate reduction to affine
form—been established [18, 53, 138]. Clearly, Offline-
Online procedures are a crucial ingredient in the real-time
and many-query contexts.
We note that historically [47] and in this paper RB meth-
ods are built upon, and measured (as regards accuracy) rel-
ative to, underlying finite element discretizations (or related
spectral element approaches [79–82, 111]): the variational
framework provides a very convenient setting for approx-
imation and error estimation. However there are certainly
many good reasons to consider alternative settings: a sys-
tematic finite volume framework for RB approximation and
3
a posteriori error estimation is proposed and developed in
[60]. We do note that boundary and integral approximations
are less amenable to RB treatment or at least Offline-Online
decompositions, as the inverse operator will typically not be
affine in the parameter.
3 Current Status of Reduced Basis Methods
3.1 Affinely Parametrized Elliptic Coercive Problems
In this paper we shall consider the case of linear func-
tional outputs of affinely parametrized linear elliptic coer-
cive partial differential equations. This class of problems—
relatively simple, yet relevant to many important applica-
tions in transport (e.g., conduction and convection-diffusion)
and continuum mechanics (e.g., linear elasticity)—proves a
convenient expository vehicle for the methodology. We pro-
vide here a brief “table of contents” for the remainder of this
review.
In Sect. 4 (compliant problems) and at the conclusion of
the paper in Sect. 11 (non-compliant problems) we describe
the affine linear elliptic coercive setting; in Sect. 5 we con-
sider admissible classes of piecewise-affine geometry and
coefficient parametric variation; in Sect. 6 we introduce sev-
eral “working examples” which shall serve to illustrate the
formulation and methodology.
In Sect. 7.1 for compliant problems and subsequently
Sect. 11.2 for non-compliant problems we discuss (primal-
dual [117]) RB Galerkin projection [121] and optimality; in
Sect. 7.2 we describe (briefly) POD methods [8, 24, 58, 73]
and (more extensively) greedy sampling procedures [32, 33,
153] for optimal space identification; in Sect. 8.1 for one
parameter and Sect. 8.2 for many parameters we investigate
the critical role of parametric smoothness [47, 85] in con-
vergence theory and practice.
In Sect. 9 we present rigorous and relatively sharp a pos-
teriori output error bounds [3, 23, 108] for RB approxi-
mations [121, 142]; in Sect. 10 we develop the coercivity-
constant lower bounds [64] required by the a posteriori error
estimation procedures.
In Sect. 7.1 for the output prediction and Sect.
9.4 for the
output error bounds [84] we present the Offline-Online com-
putational strategies crucial to rapid real-time/many-query
(Online) response. In Sect. 8.2 we also provide a quantita-
tive comparison between RB (Offline and Online) and FE
computational performance.
Although this paper focuses on the affine linear elliptic
coercive case, the reduced basis approximation and a pos-
teriori error estimation methodology is much more general.
Furthermore, most of the basic concepts introduced in the
affine linear elliptic coercive case are equally crucial—with
suitable extension—to more general equations. In the next
section we briefly review the current landscape and provide
references for further inquiry.
3.2 Extensions and Generalizations
First, the reduced basis approach can also be readily ap-
plied to the more general case of affine linear elliptic non-
coercive problems.
2
The canonical example is the ubiqui-
tous Helmholtz (reduced-wave equation) relevant to time-
harmonic acoustics [141], elasticity [95], and electromag-
netics. (We also note that a special formulation for quadratic
output functionals [62, 141]—important in such applica-
tions as acoustics (transmission loss outputs) and linear elas-
tic fracture theory (stress intensity factor outputs)—is per-
force non-coercive.) With respect to the elements we con-
sider in the current paper on coercive problems, the key
new methodological challenges for non-coercive problems
are the development of (i) discretely stable primal-dual RB
approximations [86, 134], and (ii) efficient Offline-Online
computational procedures [64, 142] for the construction of
lower bounds for the (no longer coercivity, but rather) inf-
sup constant [ 9] required by the a posteriori error estima-
tors. The possibility of resonances and near-resonances can
adversely affect the efficiency of both the RB approximation
and the RB error bounds, often limiting the dimensionality
or extent of the parameter domain.
RB-like “snapshot” ideas are also found in some of the
many Reduced Order Model (ROM) approaches in the tem-
poral domain [14, 38, 39, 89, 116, 130, 145, 155, 156]: POD
sampling procedures are often invoked, and more recently
greedy sampling approaches have also been considered [21].
Thus, combination of “parameter + time” approaches—
essentially the marriage of ROM in time with RB in parame-
ter, sometimes referred to as PROM (Parametric ROM)—is
quite natural [31, 40, 42, 46, 59, 83, 143]. The exploration
of the “parameter + time” framework in the important con-
text of affine linear (stable) parabolic PDEs—such as the
heat equation and the passive scalar convection-diffusion
equation (also the Black-Scholes equation of derivative the-
ory [118])—is carried out in [52, 55, 60, 133]; many of the
primal-dual approximations, greedy (or, better yet, greedy +
POD) sampling strategies, a posteriori error estimation con-
cepts, and Offline-Online computational strategies described
here for elliptic PDEs admit ready extension to the parabolic
case.
The reduced basis methodology, in both the elliptic and
parabolic cases, can also be extended to problems with non-
affine parametric variation. The strategy is ostensibly sim-
ple: reduce the nonaffine operator and data to approximate
affine form, and then apply the methods developed for affine
operators described in this paper. However, this reduction
2
The special issues associated with saddle problems [28, 29], in par-
ticular the Stokes equations of incompressible flow, are addressed for
divergence-free spaces in [57, 67, 114] and non-divergence-free spaces
in [135, 139].
4
must be done efficiently in order to avoid a proliferation
of parametric functions and a corresponding degradation of
Online response time. This extension is based on the Empir-
ical Interpolation Method (EIM) [18]: a collateral RB space
for the offending nonaffine coefficient functions; an interpo-
lation system that avoids costly (N -dependent) projections;
and several (from less rigorous/simple to completely rig-
orous/very cumbersome) a posteriori error estimators. The
EIM within the context of RB treatment of elliptic and par-
abolic PDEs with nonaffine coefficient functions is consid-
ered in [52, 53, 95, 138, 148]; the resulting approximations
preserve the usual Offline-Online efficiency—the complex-
ity of the Online stage is independent of N .
The reduced basis approach and associated Offline-
Online procedures can be applied without serious compu-
tational difficulties to quadratic (and arguably cubic [34,
153]) nonlinearities. Much work focuses on the station-
ary incompressible (quadratically nonlinear) Navier-Stokes
equations [29, 50, 57] of incompressible fluid flow: suitable
stable approximations are considered in [57, 67, 114, 123,
137, 139]; rigorous a posteriori error estimation—within the
general Brezzi-Rappaz-Raviart (“BRR”) a posteriori frame-
work [30, 34]—is considered in [45, 97, 151, 152]. The lat-
ter is admittedly quite complicated, and presently limited
to very few parameters—a Reynolds number and perhaps a
Prandtl number or aspect ratio.
Symmetric eigenproblems associated with (say) the
Laplacian [10] or linear elasticity operator are another im-
portant example of quadratic nonlinearities. Reduced basis
formulations for one or two lowest eigenvalues (as relevant
in structural mechanics) and for the first “many” eigenvalues
(as relevant in quantum chemistry) are developed in [84] and
[35, 36, 113], respectively. Here, implicitly, the interpreta-
tion of the BRR theory is unfortunately less compelling due
to the (guaranteed) multiplicity of often nearby solutions;
hence the a posteriori error estimators for eigenvalue prob-
lems [84, 113] are currently less than satisfactory.
Nonpolynomial nonlinearities (in the operator and also
output functional) for both elliptic and parabolic PDEs may
be considered. The Empirical Interpolation Method can be
extended to address this important class of problems [36,
53, 113]: the nonlinearity is treated in a collateral reduced
basis expansion, the coefficients of which are then obtained
by interpolation relative to the reduced basis approximation
of the field variable; the usual Offline-Online efficiency can
be maintained—Online evaluation of the output is indepen-
dent of N . (For alternative approaches to nonlinearities in
the ROM context, see [16, 39, 115].) Unfortunately, for this
difficult class of problems we can not yet cite either rigorous
a posteriori error estimators or particularly efficient sam-
pling procedures. (It perhaps not surprising that initial work
in RB methods [98, 101], which focused on highly nonlinear
problems, attempted neither complete Offline-Online decou-
pling nor rigorous error estimation.)
Finally, we mention two other topics of current research
interest. First, the “reduced basis element method” [79–82]
is a marriage of reduced basis and domain decomposition
concepts that permits much greater geometric complexity
and also provides a framework for the integration of mul-
tiple models. Second, (at least linear) hyperbolic problems
are also ripe for further development: although there are
many issues related to smoothness and stability, there are
also proofs-of-concept in both the first order [60, 111] and
second order [ 74] contexts which demonstrate that RB ap-
proximation and a posteriori error estimation can be gain-
fully applied to hyperbolic equations.
4 Elliptic Coercive Parametric PDEs
We consider the following problem: G iven µ ∈ D ⊂ R
P
,
evaluate
s
e
(µ) = !(u
e
(µ)),
where u
e
(µ) ∈ X
e
(Ω) satisfies
a(u
e
(µ), v;µ) = f (v), ∀v ∈X
e
. (1)
The superscript
e
refers to “exact.” Here µ is the input
parameter—a P -tuple; D is the parameter domain—a sub-
set of R
P
; s
e
is the scalar output; ! is the linear output
functional; u
e
is the field variable; Ω is a suitably regular
bounded spatial domain in R
d
(for d = 2 or 3) with bound-
ary ∂Ω; X
e
is a Hilbert space; and a and f are the bilinear
and linear forms, respectively, associated with our PDE.
We shall exclusively consider second-order partial differ-
ential equations, and hence (H
1
0
(Ω))
ν
⊂ X
e
⊂ (H
1
(Ω))
ν
,
where ν = 1 (respectively, ν = d) for a scalar (respec-
tively, vector) field. Here H
1
(Ω) ={v ∈ L
2
(Ω) | ∇v ∈
(L
2
(Ω))
d
},H
1
0
(Ω) ={v ∈ H
1
(Ω)|v|
∂Ω
= 0}, and L
2
(Ω)
={v measurable |
!
Ω
v
2
finite}. We associate to X
e
an inner
product and induced norm (equivalent to the H
1
(Ω) norm),
the choice of which shall be described below.
We shall assume that the bilinear form a(·, ·;µ): X
e
×
X
e
→ R is continuous and coercive over X
e
for all µ in D.
(We provide precise definitions of our continuity and coer-
civity constants and conditions below.) We further assume
that f is a bounded linear functional over X
e
. Under these
standard hypotheses on a and f ,(1) admits a unique solu-
tion.
We shall further presume for most of this paper that we
are “in compliance” [112]. In particular, we assume that
(i) a is symmetric—a(w, v;µ) = a(v,w;µ), ∀w, v ∈ X
e
,
∀µ ∈ D, and furthermore (ii) ! = f . This assumption will
greatly simplify the presentation while still exercising most
of the important RB concepts; furthermore, many important
engineering problems are in fact “compliant” (see Sect. 6).
5
![Fig. 28 Elastic Crack problem: ĜNN (µ) and the error bar interval [ĜNN (µ)−∆ĜN (µ), ĜNN (µ)+∆ĜN (µ)] for µ1 ∈ [0.1,0.45], µ2 = 0.25, and N = 30](/figures/fig-28-elastic-crack-problem-gnn-u-and-the-error-bar-2g112kh9.webp)
