1. L Machiels, Y Maday, IB Oliveira, AT Patera, and DV Rovas, Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, CR Acad Sci Paris Series I 331:153–158, 2000. 
doi:10.1016/S0764-4442(00)00270-6
This paper introduces the basic ingredients — global RB approximation spaces, a posteriori error estimators, and affine parameter dependence and associated offline-online computational procedures — in the context of a symmetric positive-definite (Laplacian) eigenproblem. Note that for the (nonlinear) eigenvalue example of this paper, the error bounds proposed are only asymptotic (as 
), and hence are not currently implemented in our software.
2.* C Prud'homme, DV Rovas, K Veroy, L Machiels, Y Maday, AT Patera, and G Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods, J Fluids Engineering, 124:70–80, 2002. doi:10.1115/1.1448332
This paper summarizes the basic ingredients — global RB approximation spaces, a posteriori error estimators, and affine parameter dependence and associated offline-online computational procedures — for the case of linear coercive elliptic PDEs (see also L Machiels, et.al., A blackbox reduced-basis output bound method for shape optimization, Proceeding of the 12th International Conference on Domain Decomposition Methods, DDM.org, 1999). Particular emphasis is on error estimation for both symmetric and non-symmetric problems and both compliant and non-compliant outputs; for non-symmetric or non-compliant problems, adjoint techniques are proposed. Examples include a thermal fin and an elastic truss system.
Note that our software is built on the rigorous "Method I'' error estimators; the "Method II'' error estimators are only asymptotic, and hence not currently implemented in our software.
3.* Y Maday, AT Patera, and G Turinici, Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations, CR Acad Sci Paris Series I 335:289–294, 2002.doi:10.1016/S1631-073X(02)02466-4
This paper develops the a priori theory for RB convergence in a particularly simple single-parameter context. The emphasis is two-fold: demonstration of uniform/global convergence of RB approximations — quantification of the "smooth, low-dimensional manifold'' argument; and motivation of an a priori parameter sample distribution — logarithmic — that performs quite well in practice.
4. K Veroy, C Prud'homme, DV Rovas, and AT Patera, A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, AIAA Paper 2003-3847, Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2003. full text available
This paper develops the first rigorous error bounds for noncoercive problems (the Helmholtz problem — see also Y Maday, et al., A Blackbox Reduced-Basis Output Bound Method for Noncoercive Linear Problems, in Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar Volume XIV (eds D Cioranescu and J-L Lions),
Elsevier Science B.V., pp. 533–569, 2002, and D Rovas, Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations, PhD Thesis, Massachusetts Institute of Technology, Cambridge, MA, 2002) and for nonlinear problems (a monotonic cubic nonlinearity); additionally, we discuss a first quasi-rigorous treatment for the Burgers equation. Errata: Note the Helmholtz deflation procedure described in this paper is not correct.
The proposals in this paper for the inf-sup lower bound (required for our error estimators for noncoercive equations), the treatment of greater-than-quadratic nonlinearities, and the error bounds for the Burgers equation, have all been subsequently improved: see Papers 12 and 18, Paper 13, and Papers 5 and 7 and 9, respectively.
This paper also introduces an adaptive sampling procedure which exploits our error bounds to construct very efficient RB samples 
and hence spaces 
— essentially a POD-like selection approach but in which only the retained snapshots (amongst a very large number of candidates) are actually computed on the FEM truth approximation space. This sampling approach is further articulated in Paper 7, Paper 8, and Paper 9.
5.* K Veroy, C Prud'homme, and AT Patera, Reduced-basis approximation of the viscous Burgers equation: Rigorous a posteriori error bounds, CR Acad Sci Paris Series I 337:619–624, 2003.doi:10.1016/j.crma.2003.09.023
This paper develops rigorous error bounds and associated offline-online computational procedures for the quadratically nonlinear Burgers equation. The essential new ingredient is a fully computational interpretation and associated Offline-Online development of the classical Brezzi-Rappaz-Raviart theory for analysis of variational approximations of nonlinear problems.
We remark that, for general nonlinear problems, our error bounds can only be conditional; however, the necessary "proximity" criterion — related to the size of the residual (and hence 
) — can be rigorously verified online, and hence there is no loss in computational efficiency or rigor.
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6.* M Barrault, Y Maday, NC Nguyen, and AT Patera, An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations, CR Acad Sci Paris Series I 339:667–672, 2004.doi:10.1016/j.crma.2004.08.006
This paper relaxes the condition of affine parameter dependence: we propose a method for the treatment of linear elliptic coercive PDEs with general parametric dependence. The technique replaces nonaffine coefficient functions with an appropriately constructed collateral RB expansion and associated interpolaton (or "collocation") procedure that then permits an (effectively) affine offline-online computational decomposition. See Paper 13 for elaboration and extension.
The associated a posteriori error indicators are not upper bounds, and hence there is some loss in rigor relative to classical Galerkin treatment.
7.* K Veroy and AT Patera, Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: Rigorous reduced-basis a posteriori error bounds, Int. J. Numer. Meth. Fluids 47:773–788, 2005.doi:10.1002/fld.867
This paper extends our Brezzi-Rappaz-Raviart formulation for the Burgers equation of Paper 5 to the full incompressible Navier-Stokes equations (over divergence-free spaces); the paper also improves the requisite inf-sup lower bound construction. The method is applied to moderate-Grashof-number natural convection at zero Prandtl number in a cavity of aspect ratio four.
8.* MA Grepl and AT Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations, Mathematical Modelling and Numerical Analysis (M2AN), 39(1):157–181, 2005.doi:10.1051/m2an:2005006
This paper extends our methodology to time-dependent problems, in particular linear parabolic equations of the heat-equation type: we introduce RB approximations (primal and dual) for time-dependent outputs; rigorous a posteriori error bounds in energy and output norms; efficient offline-online computational decompositions; and adaptive greedy sampling procedures in [time, parameter] space.
See also Grepl thesis and DV Rovas, et al., Reduced Basis Output bounds methods for parabilic problems, IMA J Appl Math, 26(3):423-445, 2006 .
9.* NC Nguyen, K Veroy, and AT Patera, Certified real-time solution of parametrized partial differential equations, in Handbook of Materials Modeling, (S. Yip, editor), Springer (2005), pp. 1523–1558.full text available
This mini-review paper summarizes in pedagogical fashion our most recent approaches for non-coercive equations (Helmholtz) and quadratically nonlinear equations (the incompressible Navier-Stokes equations). Note that for the Navier-Stokes equations we now explicitly include the pressure through introduction of appropriate "Brezzi" supremizers in the velocity space, as discussed in more detail in Paper 14 (see also G. Rozza, Real-time reduced basis techniques in arterial bypass geometries, Proceedings of the Third M.I.T. Conference on Computational Fluid and Solid Mechanics, June 14-17, 2005, (K.J. Bathe, ed.) Elsevier, pp. 1284–1287).
Examples include a crack in a linear elastic medium subject to harmonic forcing; and Boussinesq natural convection at Pr = 0.7 (for which the velocity, pressure, and temperature are all now coupled) in a cavity.
This paper also reviews the adaptive sample construction procedure introduced in Paper 4.
10. MA Grepl, NC Nguyen, K Veroy, AT Patera, and GR Liu, Certified rapid solution of partial differential equations for real-time parameter estimation and optimization, in Real-time PDE-Constrained Optimization, (L. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes, and B. van Bloemen Waanders, eds.) SIAM Computational Science and Engineering Book Series, 2007, pp 197–215.full text available
This paper reviews our methodology for noncoercive (Helmholtz) elliptic equations, the Navier-Stokes equations, and parabolic equations, from the perspective of real-time and reliable computation.
The paper includes an example of real-time robust parameter estimation in the context of nondestructive evaluation of a crack in an elastic medium. We can rapidly deduce — from the response of the medium to tuned harmonic forcing — the "possibility" set of all crack lengths consistent with experimental measurements. No regularization or unverifiable prior assumptions are required: well-posedness is manifested in the shrinking of our "possibility" set as the experimental error tends to zero. The key computational enabler is the RB output prediction and associated a posteriori error estimator.
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11. IB Oliveira and AT Patera, Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations, Optimization and Engineering, 8(1):43-65, 2007. doi: 10.1007/s11081-007-9002-6
This paper considers the application of our RB approximations and associated error bounds to constrained optimization problems. Particular focus is placed on (i) efficient calculation of higher-order (gradient, Hessian) information, and (ii) rigorous incorporation of our error bounds — which ensures that our RB-generated optimizer is in fact feasible with respect to the truth finite element approximation. (For a single optimization offline effort will dominate — and our method is not efficient; however for many optimizations or for real-time applications, online effort will dominate — and our method becomes advantageous.)
The paper considers the example of optimization of a thermal fin with respect to heat transfer coefficient and geometry.
12. S Sen, K Veroy, DBP Huynh, S Deparis, NC Nguyen, and AT Patera, "Natural norm"a posteriori error estimators for reduced basis approximations. Journal of Computational Physics, 217 (1): 37–62, 2006.doi: j.jcp.2006.02.012
In this paper we propose a new "natural norm" formulation for our reduced basis error estimation framework that (a) improves our inf-sup lower bound construction (offline) and evaluation (online) -- a critical ingredient of our a posteriori error estimators, and (b) significantly sharpens our output error bounds, in particular (through deflation) for parameter
values corresponding to nearly singular solution behavior.
We apply the method to two illustrative problems: a coercive Laplacian heat conduction problem -- which becomes singular as the heat transfer coefficient tends to zero; and a noncoercive
Helmholtz acoustics problem -- which becomes singular as we approach resonance.
13. * MA Grepl, Y Maday, NC Nguyen, and AT Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Mathematical Modelling and Numerical Analysis (M2AN), 41(3):575–605, 2007.doi: 10.1051/m2an:2007031
This paper expands and elaborates upon Paper 6, and in particular extends our non-affine methodology to efficiently treat transcendentally nonlinear (in fact, semi-linear) elliptic and parabolic equations. RB methods based on standard Galerkin approaches are not efficient for greater-than-quadratic nonlinearities; the methods are particularly inefficient — the online complexity scales with the dimension of the finite element "truth" approximation space,
— for non-polynomial nonlinearities. Our "empirical interpolation" approach recovers -independence in the online stage for general nonlinearities.
See also Nguyen and Grepl theses. See also G Rozza, Reduced basis method for Stokes equations in domains with non-affine parametric dependence, Comp Vis Science, to appear 2007, Rozza thesis, and AE Løvgren,
Reduced basis modeling of hierarchical flow systems, PhD Thesis, Norwegian University of Science and Technology, Trondheim, Norway, 2005.
14. * G Rozza and K Veroy, On the stability of the reduced basis method for Stokes equations in parametrized domains. Computer Methods in Applied Mechanics and Engineering, 196(7): 1244-1260 2007.doi:10.1016/j.cma.2006.09.005
This paper focuses on proper treatment of the pressure in the incompressible Stokes system. It is very simple to construct parametric Stokes problems and simple RB approximations for which the discrete "Brezzi" inf-sup parameter is identically zero and the pressure error order unity. One possible solution is a divergence-free RB velocity space. This paper considers a second possible solution: the introduction of an enriched velocity space to ensure a good discrete Brezzi inf-sup parameter and associated good pressure approximation. Several options of varying degrees of rigor are proposed: in each case the velocity space is expanded to include "Brezzi" supremizers; the various options are distinguished by the sample selection and subsequent post-processing.
15. É Cancès, C Le Bris, Y Maday, NC Nguyen, AT Patera, and GSH Pau, Feasibility and competitiveness of a reduced basis approach for rapid electronic structure calculations in quantum chemistry. Centre de Recherches Mathématiques: CRM Proceedings and Lecture Notes, 41: 15–57, 2007.
In this paper we consider the application of the RB approach to several model problems in quantum mechanics: a Hartree Fock approximation of diatomic hydrogen; and a simple Density-Functional Theory Kohn-Sham approximation to an idealized one-dimensional crystalline solid. The paper focuses on several of the new difficulties associated with RB treatment of electronic structure (eigenproblem-like) calculations: efficient evaluation of higher-order and non-polynomial nonlinearities (see Paper 13); economical representation of fields defined over high-dimensional product spaces; and (related) efficient incorporation of mutual orthogonality constraints.
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16. * DBP Huynh and AT Patera, Reduced basis approximation and a posteriori error estimation for stress intensity factors. International Journal for Numerical Methods in Engineering, 72(10):1219–1259, 2007. doi: 10.1002/nme.2090
This paper introduces a new formulation for quadratic outputs: we first transform a (coercive or non-coercive) elliptic equation with quadratic outputs to a non-coercive elliptic equation (of twice the size) with linear compliant outputs; we then consider RB approximation and a posteriori error estimation for this expanded system. The output error bounds for the expanded linear system are both simpler and considerably sharper than the output bounds for the original quadratic representation.
We derive and apply this new "expanded" approach to the calculation of the Energy Release Rate and hence Stress Intensity Factor (SIF) associated with Mode-I linear elastic cracks (ultimately relevant to fracture). Many-query and real-time reliable prediction of the SIF can be crucial: for fatigue-induced evolution of a nascent crack; for "in-the-field" assessment of (brittle) failure. We present RB results for three examples: a heterogeneous two-layer notch; a center-crack test specimen; and a hole-crack configuration. (The truth FE model is based on singularity-enriched spaces.)
17. AT Patera and E. Rønquist, Reduced Basis Approximations and a posteriori error estimation for a Boltzmann model. Computer Methods in Applied Mechanics and Engineering, 196(29-30): 2925–2942, 2007.doi:10.1016/j.cma.2007.02.008
In this paper we consider RB approximation and a posteriori error estimation for a simple one-dimensional (in space) linearized BGK Boltzmann model for flow in a channel at transitional Knudsen numbers: the inputs are the Knudsen number and accommodation coefficient; the output is the channel flowrate. This first-order hyperbolic ("non-local") problem is first consistently reformulated through a "Strange Upwind Petrov Galerkin" procedure as a second-order non-symmetric coercive elliptic problem (in one space dimension, no variational crimes are committed): this stable representation -- with suitable mappings to treat the infinite molecular-velocity domain -- then serves to define both the (spectral element) truth approximation and the subsequent RB treatment. The RB output approximation converges rapidly; and the RB a posteriori error estimators (based on an ad hoc coercivity lower bound) is respectable -- though for smaller RB errors the effectivities can occasionally be large.
18. * DBP Huynh, G Rozza, S Sen, and AT Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. CR Acad Sci Paris Series I 345:473–478, 2007. doi: 10.1016/j.crma.2007.09.019
This paper presents an approach to the construction of lower bounds for the coercivity and inf-sup stability constants required in a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. The method, based on an Offline-Online 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.
The method is simple and general to implement, the Offline stage is based on standard eigenproblems that can be efficiently treated by the Lanczos method, and the Online Linear Program is typically of modest size.
Numerical results are presented for an (coercive) elasticity problem and an (non-coercive) acoustics Helmholtz problem.
19. * NC Nguyen, MA Grepl, AT Patera, and GR Liu, An "Uncertainty Region" Reduced Basis Approach to Parameter Estimation for Linear Parabolic Partial Differential Equations. Inverse Problems 2007 (submitted). full text available
We present an inverse problem procedure for the efficient calculation of approximations to the parametric "uncertainty region." The uncertainty region
is the compact set of all parameter values consistent with interval (experimental)
output measurements; we provide a series of approximate uncertainty regions from
less rigorous and very fast -- for initial design and real-time assessment -- to fully rigorous and less fast -- for
final confirmation. The dependence of the uncertainty region on the magnitude of the
experimental error distinguishes identifiable and non-identifiable inverse problems; for
the former, our approach can further provide quantitative and sharp prediction of the
unknown parameter value, the associated (experimentally and numerically induced)
uncertainty, and the sensitivity to measurement protocol.
The parameter estimation method depends
critically on the Reduced Basis (RB) output bound evaluator for the forward problem:
the RB output prediction provides the very fast response needed (to effect the many
queries required) to construct the approximate uncertainty region; the RB output
bound ensures that we control (optimize) and rigorously incorporate the numerical
error in our inverse analysis. We develop the technique for (parametrically coercive)
linear parabolic partial differential equations such as the transient heat conduction
equation; as an example, we consider detection and characterization of de-lamination
cracks in Fiber-Reinforced Polymer surface treatment of concrete structures.
20. * S Boyaval, Reduced-Basis Approach for Homogenization Beyond the Periodic Setting.
SIAM Multiscale Modeling and Simulation 7(1):466–494, 2008.
doi: 10.1137/070688791
This paper considers the development of the reduced basis method for the computation of averaged coefficients for the homogenization of elliptic partial differential equations. In the (non-periodic) homogenization context, as in many multiscale frameworks, we must effect a large number of similar calculations on a microscale cell for property and geometric parameter values induced by macroscale "distribution" functions. This multi-query computational task can be very efficiently effected by the offline-online reduced basis approach; the reduced basis method further provides rigorous a posteriori error bounds for the effective properties — which can in turn be integrated into error estimators for the ultimate macroscale predictions.
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21. * G Rozza, DBP Huynh, and AT Patera, Reduced Basis Approximation and A Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations — Application to Transport and Continuum Mechanics.
Arch Comput Methods Eng (to appear September 2008).
Available online at doi: 10.1007/s11831-008-9019-9
This paper provides a comprehensive review of (Lagrange) reduced basis approximation and a posteriori error estimation for linear coercive elliptic PDEs. We also provide a summary of the "state of the art" of RB methods for PDEs more generally as well as an extensive bibliography.
We discuss the formulation of geometric and coefficient parametric variations consistent with affine parameter dependence; we describe primal-dual RB Galerkin approximation and associated energy and output optimality results; we compare several sampling strategies — in particular, POD and greedy methods — for generation of effective RB spaces; we provide theoretical and computational evidence of RB convergence in the single and many parameter cases; we summarize our RB a posteriori (output) error bounds and associated theoretical effectivity results; we present the Successive Constraint Method for construction of the coercivity constant lower bounds required by our a posteriori error estimators; we describe the Offline-Online computational procedures and associated operation counts for RB output approximation and a posteriori error estimation; we provide computational results — convergence, effectivity, and performance — for several problems in transport (conduction, advection-diffusion), inviscid flow (added mass), and elasticity (Stress Intensity Factors, effective properties).
22. * CN Nguyen, G Rozza, & AT Patera, Reduced Basis Approximation and A Posteriori Error Estimation for the Time-Dependent Viscous Burgers Equation. Calcolo, 2008 (submitted). full text available
In this paper we present rigorous a posteriori L2 error bounds for reduced basis approximations of the unsteady viscous Burgers equation in one space dimension. The key new ingredient is accurate solution-dependent (Online) calculation of the exponential-in-time stability factor by the Successive Constraint Method. Numerical results indicate that the a posteriori error bounds are practicable for reasonably large times — many convective scales — and reasonably large Reynolds numbers — O(100) or larger. Work in progress (in collaboration with Dr Paul Fischer) considers the full unsteady incompressible Navier-Stokes equation.
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