Rapid Reliable Solution
of the Parametrized Partial Differential Equations
of Continuum Mechanics and Transport

We include here selected recent publications of our group relevant to reduced basis approximation and a posteriori error estimation. The * to the right of the number indicates papers that develop techniques most closely aligned with the methodology implemented in the rbMIT © MIT software package and also Worked Problems.

Note this list has been recently pruned and re-ordered. We have archived the original list in the event that you might wish to refer to it.

60. M Yano and AT Patera, A space-time variational approach to hydrodynamic stability theory. Proceedings of the Royal Society A (accepted April 2013). full text (April 2013)
We present a hydrodynamic stability theory for incompressible viscous fluid flows based on a space-time variational formulation and associated generalized singular value decomposition of the (linearized) Navier-Stokes equations. We first introduce a linear framework applicable to a wide variety of stationary or time-dependent base flows: we consider arbitrary disturbances in both the initial condition and the dynamics measured in a "data" space-time norm; the theory provides a rigorous, sharp (realizable), and efficiently computed bound for the velocity perturbation measured in a "solution" space-time norm. We next present a generalization of the linear framework in which the disturbances and perturbation are now measured in respective selected space-time semi-norms; the semi-norm theory permits rigorous and sharp quantification of, for example, the growth of initial disturbances or functional outputs. We then develop a (Brezzi-Rappaz-Raviart) nonlinear theory which provides, for disturbances which satisfy a certain (rather stringent) amplitude condition, rigorous finite-amplitude bounds for the velocity and output perturbations. Finally, we demonstrate the application of our linear and nonlinear hydrodynamic stability theory to unsteady moderate Reynolds-number flow in an eddy-promoter channel.

59. S Vallaghé, The static condensation reduced basis element method for parabolic problems. Mathematical Models and Methods in Applied Sciences (submitted January 2013). full text available
We present a new approach for fast, flexible and reliable simulations of parameter-dependent parabolic problems with a component-based geometry. The static condensation reduced basis element (SCRBE) is a domain decomposition method with reduced basis (RB) approximation at the intradomain level to populate a Schur complement at the interdomain level. In the Offline stage, for a library of archetype subdomains we train RB spaces for a family of time-dependent problems; in the Online stage we assemble instantiated subdomains and solve the transient problem for the full system using static condensation. We provide system-level a posteriori error bounds and we present various numerical results to demonstrate the accuracy, flexibility and computational efficiency of our approach.

58. JL Eftang and AT Patera, Port Reduction in Component-Based Static Condensation for Parametrized Problems: Approximation and A Posteriori Error Estimation. International Journal for Numerical Methods in Engineering (submitted December 2012). full text available
We introduce a port (interface) approximation and a posteriori error bound framework for a general component-based static condensation method in the context of parameter-dependent elliptic partial differential equations. The key ingredients are (i) efficient empirical port approximation spaces — the dimensions of these spaces may be chosen small in order to reduce the computational cost associated with formation and solution of the static condensation system — and (ii) a computationally tractable a posteriori error bound realized through a non-conforming approximation to the true error and bound conditioners — the error in the system approximation or in an output quantity may be bounded with respect to the non-port-reduced full approximation.
Our approximation and a posteriori error bound framework is of particular computational relevance for the Static Condensation Reduced Basis Element (SCRBE) method for parameter-dependent partial differential equations. We provide several numerical examples with the SCRBE that serve to both illustrate our port approximation procedure and to demonstrate the efficacy of our port reduction error bounds.

57. M Yano, A Space-Time Petrov-Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations. SIAM Journal on Scientific Computing (submitted December 2012). full text available
We present a space-time certified reduced basis method for long-time integration of parametrized noncoercive parabolic equations with quadratic nonlinearity. We first consider a finite element discretization based on discontinuous Galerkin time integration and introduce associated Petrov-Galerkin space-time trial- and test-space norms which yield optimal and asymptotically mesh independent stability constants. We then employ an hp Petrov-Galerkin (or minimum residual) space-time reduced basis approximation. We provide the Brezzi-Rappaz-Raviart a posteriori error bounds which admit efficient offline-online computational procedures for the three key ingredients: the dual norm of the residual, an inf-sup lower bound, and the Sobolev embedding constant. The latter are based respectively on a more round-off resistant residual norm evaluation procedure, a variant of the successive constraint method, and a time-marching implementation of a fixed-point iteration of the embedding constant for the discontinuous Galerkin norm. Finally, we apply the method to a natural convection problem governed by the Boussinesq equations. The result indicates that the space-time formulation enables rapid and certified characterization of moderate-Grashof-number flows exhibiting steady periodic responses.

56. S Vallaghé and AT Patera, The Static Condensation Reduced Basis Element Method for a Mixed-Mean Conjugate Heat Exchanger Model. SIAM Journal on Scientific Computing (submitted August 2012). full text available
We propose a new approach for the simulation of conjugate heat exchangers. First, we introduce a dimensionality-reduced mathematical model for conjugate (fluid-solid) heat transfer: in the fluid channels, we consider a mixed-mean temperature defined on 1D filaments; in the solid we consider a detailed PDE conduction representation. We then propose a Petrov-Galerkin finite element (FE) numerical approximation which provides suitable stability and accuracy for our mathematical model. We next apply the static condensation reduced basis element (scRBE) method: a domain decomposition with parametric model order reduction at the intradomain level to populate a Schur complement at the interdomain level. We start by building a library of "components," each corresponding to a subdomain with a simple fluid channel geometry; for each component, we prepare Petrov-Galerkin reduced basis bubble approximations (and error bounds). We then form the linear system by static condensation, and solve for the temperature distribution in the whole domain. System-level error bounds are derived from matrix perturbation arguments; we also introduce a new output error bound which is sharper than the original scRBE estimator. We present numerical results for a 2D car radiator model which demonstrate the flexibility, accuracy and computational efficiency of our approach.
ERRATA In the equation after equation (5.8) the ${\bf u}^j_P$ should bear a $\tilde$.

55. DBP Huynh, DJ Knezevic, and AT Patera, A Static Condensation Reduced Basis Element Method: Complex Problems. Computer Methods in Applied Mechanics and Engineering (accepted February 2013). full text available
We extend the static condensation reduced basis element (scRBE) method to treat the class of parametrized complex Helmholtz partial differential equations. The main ingredients are (i) static condensation at the interdomain level, (ii) a conforming eigenfunction "port" representation at the interface level, (iii) the reduced basis (RB) approximation of nite element (FE) bubble functions at the intradomain level, and (iv) rigorous system-level error bounds which reflect RB perturbation of the FE Schur complement. We then incorporate these ingredients in an Offine-Online computational strategy to achieve rapid and accurate prediction of parametric systems formed as instantiations of interoperable parametrized archetype components from a Library. We introduce a number of extensions with respect to the original scRBE framework: first, primal-dual RB methods for general non-symmetric (complex) problems; second, stability constant procedures for weakly coercive problems (at both the interdomain level and intradomain level); third, treatment of non-port linear-functional outputs (as well as functions of outputs); fourth, consideration of particular components and outputs relevant to acoustic applications. We consider several numerical examples in acoustics (in particular focused on mufflers and horns) to demonstrate that the approach can handle models with many parameters and/or topology variations with particular reference to waveguide-like applications.

54. M Yano, AT Patera, and K Urban, A Space-Time Certified Reduced Basis Method for Burgers' equation. Mathematical Models and Methods in Applied Sciences (submitted July 2012). full text available
We present a space-time certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval (0, T] parametrized with respect to the Peclet number. We first introduce a Petrov-Galerkin space-time finite element discretization, which enjoys a favorable inf-sup constant that decreases slowly with Peclet number and final time T. We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi-Rappaz-Raviart a posteriori error bounds. We detail computational procedures that permit offine-online decomposition for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf-sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L²-error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T, unlike the exponentially growing estimate of the classical formulation for high Peclet number cases.

53. DBP Huynh, A Static Condensation Reduced Basis Element Approximation: Application to three-dimensional acoustic muffler analysis. Proceedings of the International Conference on Advances in Computational Mechanics (accepted 2012). full text available
We apply the static condensation reduced basis element (scRBE) method to treat the class of parametrized complex Helmholtz partial differential equations. We construct a set of components of interoperable parametrized reference components in a Library to model a family of target models relevant to acoustic devices. The components in the Library are built upon the scRBE method by using RB formulation, and are compatible to each other by a set of common interfaces, or ports. We apply an Offline-Online computational strategy to achieve rapid and accurate prediction of any parametric systems formed from a set of components in a Library. We demonstrate that the approach can handle large scale models with many parameters and/or topology variations efficiency in several numerical examples. We show that significant computational savings can be obtained by the scRBE method.

52. JL Eftang and B Stamm, Parameter Multi-Domain "hp" Empirical Interpolation. International Journal for Numerical Methods in Engineering, 90(4):412–428, 2012. full text available | doi: 10.1002/nme.3327
In this paper, we introduce two parameter multi-domain "hp" techniques for the empirical interpolation method (EIM). In both approaches, we construct a partition of the original parameter domain into parameter subdomains: h-refinement. We apply the standard EIM independently within each subdomain to yield local (in parameter) approximation spaces: p-refinement. Further, for a particularly simple case we introduce a priori convergence theory for the partition procedure. We show through two numerical examples that our approaches provide significant reduction in the EIM approximation space dimension, and thus significantly reduce the computational cost associated with EIM approximations.

51. K Urban and AT Patera, An Improved Error Bound for Reduced Basis Approximation of Linear Parabolic Problems. Submitted to Mathematics of Computation, (accepted 2013). full text available
We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant β_δ, the inverse of which enters into error estimates: β_δ is unity for the heat equation; β_δ decreases only linearly in time for non-coercive (but asymptotically stable) convection operators. The latter in turn permits effective long-time a posteriori error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates. The paper contains a full analysis and various extensions for the formulation introduced briefly by K Urban and AT Patera [CR Acad Sci Paris Series I, 350(3-4):203–207, 2012] as well as numerical results for a model reaction-convection-diffusion equation.

50. AT Patera and EM Rønquist, Regression on Parametric Manifolds: Estimation of Spatial Fields, Functional Outputs, and Parameters from Noisy Data. CR Acad Sci Paris, Series I, 350(9-10):543-547, 2012. full text (revised May 2012) | doi:10.1016/j.crma.2012.05.002
In this Note we extend the Empirical Interpolation Method (EIM) to a regression context which accommodates noisy (experimental) data on an underlying parametric manifold. The EIM basis functions are computed Offline from the noise-free manifold; the EIM coefficients for any function on the manifold are computed Online from experimental observations through a least-squares formulation. Noise-induced errors in the EIM coefficients and in linear-functional outputs are assessed through standard confidence intervals and without knowledge of the parameter value or the noise level. We also propose an associated procedure for parameter estimation from noisy data.

49. JL Eftang, DBP Huynh, DJ Knezevic, EM Rønquist, and AT Patera, Adaptive Port Reduction in Static Condensation. Proceedings of 7th Vienna Conference on Mathematical Modelling — MATHMOD 2012 (submitted December 2011). full text available
We introduce a framework for adaptive reduction of the degrees of freedom associated with ports in static condensation (SC). We apply this framework to the SC reduced basis (RB) method and thus combine parametric model order reduction for the interior of a component with model order reduction on the ports in order to rapidly construct an approximate Schur complement linear system of reduced size. The port reduction framework invokes quasi-rigorous a posteriori error bounds for adaptation and allows a combination of empirical functions (snapshot-based) and eigenfunctions for the representation of the solution on the ports.

48. JL Eftang, MA Grepl, AT Patera, and EM Rønquist, Approximation of Parametric Derivatives by the Empirical Interpolation Method. Foundations of Computational Mathematics, published online June 2012. full text available | doi: 10.1007/s10208-012-9125-9
We introduce a general a priori convergence result for the approximation of parametric derivatives of parametrized functions. We consider the best approximations to parametric derivatives in a sequence of approximation spaces generated by a general approximation scheme, and we show that these approximations are convergent provided that the best approximation to the function itself is convergent. We also provide estimates for the convergence rates. We present numerical results with spaces generated by a particular approximation scheme — the Empirical Interpolation Method — to confirm the validity of the general theory.

47. N Jung, AT Patera, B Haasdonk, and B Lohmann, Model Order Reduction and Error Estimation with an Application to the Parameter-Dependent Eddy Current Equation. Mathematical and Computer Modelling of Dynamical Systems 17(6):561–582, 2011. doi: 10.1080/13873954.2011.582120
In product development, engineers simulate the underlying partial differential equation many times with commercial tools for different geometries. Since the available computation time is limited, we look for reduced models with an error estimator that guarantees the accuracy of the reduced model. Using commercial tools the theoretical methods proposed by G Rozza, DBP Huynh and AT Patera [Arch Comput Methods Eng 15:229–275] lead to technical difficulties. We present how to overcome these challenges and validate the error estimator by applying it to a simple model of a solenoid actuator that is a part of a valve.

46. K Urban and AT Patera, A New Error Bound for Reduced Basis Approximation of Parabolic Partial Differential Equations. CR Acad Sci Paris Series I, 350(3-4):203–207, 2012. full text available | doi:10.1016/j.crma.2012.01.026
We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant \beta_\delta: \beta_\delta is bounded from below by unity for the heat equation; \beta_\delta grows only linearly in time for non-coercive (asymptotically stable) convection operators. The latter in turn permits eective long-time a posteriori error bounds for reduced basis approximations, in sharp contrast to classical exponentially growing energy estimates.

45. DBP Huynh, DJ Knezevic, and AT Patera, A Static Condensation Reduced Basis Element Method: Approximation and A Posteriori Error Estimation. Mathematical Modelling and Numerical Analysis, 47(1): 213–251, 2013. full text available (15 May 2012) | doi:10.1051/m2an/2012022
We propose a new reduced basis element-cum-component mode synthesis approach for parametrized elliptic coercive partial differential equations. In the Offline stage we construct a Library of interoperable parametrized reference components relevant to some family of problems; in the Online stage we instantiate and connect reference components (at ports) to rapidly form and query parametric systems. The method is based on static condensation at the interdomain level, a conforming eigenfunction "port" representation at the interface level, and finally reduced basis (RB) approximation of finite element (FE) bubble functions at the intradomain level. We show under suitable hypotheses that the RB Schur complement is close to the FE Schur complement: we can thus demonstrate the stability of the discrete equations; furthermore, we can develop inexpensive and rigorous (system-level) a posteriori error bounds. We present numerical results for model many-parameter heat transfer and elasticity problems with particular emphasis on the Online stage; we discuss flexibility, accuracy, computational performance, and also the effectivity of the a posteriori error bounds.

44. A Quarteroni, G Rozza, and A Manzoni, Certified Reduced Basis Approximation for Parametrized Partial Differential Equations and Applications. Journal of Mathematics in Industry 2011, 1:3 (3 June 2011). http://www.mathematicsinindustry.com/content/1/1/3
Reduction strategies, such as model order reduction (MOR) or reduced basis (RB) methods, in scientific computing may become crucial in applications of increasing complexity. In this paper we review the reduced basis method (built upon a high-fidelity "truth" finite element approximation) for a rapid and reliable approximation of parametrized partial differential equations, and comment on their potential impact on applications of industrial interest. The essential ingredients of RB methodology are: a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform a competitive Offline-Online splitting in the computational procedure, and a rigorous a posteriori error estimation used for both the basis selection and the certification of the solution. The combination of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). After a brief excursus on the methodology, we focus on linear elliptic and parabolic problems, discussing some extensions to more general classes of problems and several perspectives of the ongoing research. We present some results from applications dealing with heat and mass transfer, conduction-convection phenomena, and thermal treatments.

43. KC Hoang, BC Khoo, GR Liu, NC Nguyen, and AT Patera, Rapid identification of material properties of the interface tissue in dental implant systems using reduced basis method. Journal of Inverse Problems in Science and Engineering (submitted March 2011). full text available
This paper proposes a rapid inverse analysis approach based on the reduced basis method and the Levenberg-Marquardt-Fletcher algorithm to identify the "unknown" material properties: Young's modulus and stiffness-proportional Rayleigh damping coefficient of the interfacial tissue between a dental implant and the surrounding bones. In the forward problem, a finite element approximation for a three-dimensional dental implant-bone model is first built. A reduced basis approximation is then established by using a Proper Orthogonal Decomposition (POD)-Greedy algorithm and the Galerkin projection to enable extremely fast and reliable computation of displacement responses for a range of material properties. In the inverse analysis, the reduced basis approximation for the dental implant-bone model are incorporated in the Levenberg-Marquardt-Fletcher algorithm to enable rapid identification of the unknown material properties. Numerical results are presented to demonstrate the efficiency and robustness of the proposed method.

42. DBP Huynh, DJ Knezevic, and AT Patera, Certified Reduced Basis Model Characterization: a Frequentistic Uncertainty Framework. Computer Methods in Applied Mechanics and Engineering. Computer Methods in Applied Mechanics and Engineering 201–204:13–24, 2012. full text available |doi:10.1016/j.cma.2011.09.011
We introduce a frequentistic validation framework for assessment — acceptance or rejection — of the consistency of a proposed parametrized partial differential equation model with respect to (noisy) experimental data from a physical system. Our method builds upon the Hotelling T ² statistical hypothesis test for bias first introduced by Balci & Sargent [Am J Math Manage Sci 4(3-4):375–406, 1984] and subsequently extended by McFarland & Mahadevan [CMAME 197(29–32):2467–2479, 2008] . Our approach introduces two new elements: a spectral representation of the misfit which reduces the dimensionality and variance of the underlying multivariate Gaussian but without introduction of the usual regression assumptions; a certified (verified) reduced basis approximation — reduced order model — which greatly accelerates computational performance but without any loss of rigor relative to the full (finite element) discretization. We illustrate our approach with examples from heat transfer and acoustics, both based on synthetic data. We demonstrate that we can efficiently identify possibility regions that characterize parameter uncertainty; furthermore, in the case that the possibility region is empty, we can deduce the presence of "unmodeled physics" such as cracks or heterogeneities.
ERRATA Corollary 3.1 is stated in a misleading way. We should state instead in the following fashion:
If $s^{PS} = s_{h}$ then ${\cal H}$ of (30) is true; Proposition 2.1 then applies. If ${\cal H}$ of (30) is false, then (31) (of Corollary 3.1) demonstrates that $s^{PS} \neq s_{h}$.

41. JL Eftang, DBP Huynh, DJ Knezevic, and AT Patera, A Two-Step Certified Reduced Basis Method. Journal of Scientific Computing 51(1):28–58, 2012. full text available | doi: 10.1007/s10915-011-9494-2
In this paper we introduce a two-step Certified Reduced Basis (RB) method. In the first step we construct from an expensive finite element “truth” discretization of dimension \calN an intermediate RB model of dimension $N \ll {\cal N}$. In the second step we construct from this intermediate RB model a derived RB (DRB) model of dimension M ≤ N. The construction of the DRB model is effected at cost ${\cal O} (N)$ and in particular at cost independent of ${\cal N}$; subsequent evaluation of the DRB model may then be effected at cost ${\cal O} (M)$. The DRB model comprises both the DRB output and a rigorous a posteriori error bound for the error in the DRB output with respect to the truth discretization.
The new approach is of particular interest in two contexts: focus calculations and hp-RB approximations. In the former the new approach serves to reduce online cost, $M \ll N$: the DRB model is restricted to a slice or subregion of a larger parameter domain associated with the intermediate RB model. In the latter the new approach enlarges the class of problems amenable to hp-RB treatment by a significant reduction in offline (precomputation) cost: in the development of the hp parameter domain partition and associated “local” (now derived) RB models the finite element truth is replaced by the intermediate RB model. We present numerical results to illustrate the new approach.

40. DBP Huynh, DJ Knezevic, and AT Patera, A Laplace Transform Certified Reduced Basis Method; Application to the Heat Equation and Wave Equation. CR Acad Sci Paris Series I, 349(7-8):401–405, 2011. full text available | doi:10.1016/j.crma.2011.02.003
We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary "modal" truncation; RB eigenfunction decomposition and contour integration for Offine-Online decomposition. We present numerical results to demonstrate the accuracy and efficiency of the approach.

39. DJ Knezevic and JW Peterson, A High-Performance Parallel Implementation of the Certified Reduced Basis Method. Computer Methods in Applied Mechanics and Engineering 200(13-16):1455–1466, 2011. full text available (revised December 2010) | doi: 10.1016/j.cma.2010.12.026
The Certified Reduced Basis method (herein RB method) is a popular approach for model reduction of parametrized partial differential equations. In this paper we introduce new techniques that are required to efficiently implement the Offine "Construction stage" of the RB method on high-performance parallel supercomputers. This enables us to generate certified RB models for large-scale three-dimensional problems that can be evaluated on standard workstations and other "thin" computing resources with speedup of many orders of magnitude compared to the corresponding full order model. We use our implementation to perform detailed numerical studies for two computationally expensive model problems: a natural convection fluid flow problem and a "many parameter" heat transfer problem. In the heat transfer problem, we exploit the computational efficiency of the RB method to perform a detailed study of "snapshot" selection in the Greedy algorithm, and we also examine statistics of the output sensitivity derivatives to obtain a "global" view of the relative importance of the parameters.

38. * JL Eftang, DJ Knezevic, and AT Patera, An hp Certified Reduced Basis Method for Parametrized Parabolic Partial Differential Equations. Mathematical and Computer Modelling of Dynamical Systems, 17(4):395–422, 2011. full text available | doi:10.1080/13873954.2011.547670
In this paper we introduce an hp certified reduced basis method for parabolic partial differential equations. We invoke a POD (in time) / Greedy (in parameter) sampling procedure first in the initial partition of the parameter domain (h-refinement) and subsequently in the construction of reduced basis approximation spaces restricted to each parameter subdomain (p-refinement). We show that proper balance between additional POD modes and additional parameter values in the initial subdivision process guarantees convergence of the approach. We present numerical results for two model problems: linear convection-diffusion and quadratically nonlinear Boussinesq natural convection. The new procedure is significantly faster (respectively, more costly) in the reduced basis Online (respectively, Offline) stage.

37. * DJ Knezevic, Reduced Basis Approximation and A Posteriori Error Estimates for a Multiscale Liquid Crystal Model. Mathematical and Computer Modelling of Dynamical Systems 17(4):443–461, 2011. full text available | doi: 10.1080/13873954.2011.547676
We present a reduced basis framework and associated a posteriori error estimates for the multiscale Stokes Fokker-Planck system that governs the flow of a dilute suspension of rod-like molecules immersed in a Newtonian solvent, relevant in liquid crystals modeling. The Fokker-Planck equation dictates the microscale behavior and must be solved at every quadrature point of the macroscale finite element mesh — this is a natural example of a many-query problem for which the certified reduced basis method is well suited. We focus on a Poiseuille flow problem to simplify the presentation of ideas, but we note that the methods developed in this paper generalize directly to more complicated problems. Numerical results demonstrate that our reduced basis approach leads to significant computational savings and also that our error estimator performs well for moderate parameter values.

36. * DBP Huynh, DJ Knezevic, JW Peterson, and AT Patera, High-Fidelity Real-Time Simulation on Deployed Platforms. Computers and Fluids, 43(1):74–81, 2011. full text available | doi:10.1016/j.compfluid.2010.07.007
We present a certified reduced basis method for high-fidelity real-time solution of parametrized partial differential equations on deployed platforms. Applications include in situ parameter estimation, adaptive design and control, interactive synthesis and visualization, and individuated product specification. We emphasize a new hierarchical architecture particularly well suited to the reduced basis computational paradigm: the expensive Offline stage is conducted pre-deployment on a parallel supercomputer (in our examples, the TeraGrid machine Ranger); the inexpensive Online stage is conducted "in the field" on ubiquitous thin/inexpensive platforms such as laptops, tablets, smartphones (in our examples, the Nexus One Android-based phone), or embedded chips. We illustrate our approach with three examples: a two-dimensional Helmholtz acoustics "horn" problem; a three-dimensional transient heat conduction "Swiss Cheese" problem; and a three-dimensional unsteady incompressible Navier-Stokes low-Reynolds-number "eddy-promoter" problem.

35. * JL Eftang, MA Grepl, and AT Patera, A Posteriori Error Bounds for the Empirical Interpolation Method. CR Acad Sci Paris Series I, 348(9–10): 575–579, 2010. full text (revised March 2010) | doi:10.1016/j.crma.2010.03.004
We present rigorous a posteriori error bounds for the Empirical Interpolation Method (EIM). The essential ingredients are (i) analytical upper bounds for the parametric derivatives of the function to be approximated, (ii) the EIM “Lebesgue constant,” and (iii) information concerning the EIM approximation error at a finite set of points in parameter space. The bound is computed “offline” and is valid over the entire parameter domain; it is thus readily employed in (say) the “online” reduced basis context. We present numerical results that confirm the validity of our approach.
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34. * JL Eftang, AT Patera, and EM Rønquist, An “hp” Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations. SIAM Journal on Scientific Computing, 32(6):3170–3200 (2010). full text (revised July 2010) | doi: 10.1137/090780122
We present a new “hp” parameter multi-domain certified reduced basis method for rapid and reliable online evaluation of functional outputs associated with parametrized elliptic partial differential equations. We propose, and provide theoretical justification for, a new procedure for adaptive partition (“h”-refinement) of the parameter domain into smaller parameter subdomains: we pursue a hierarchical splitting of the parameter (sub)domains based on proximity to judiciously chosen parameter anchor points within each subdomain. Subsequently, we construct individual standard RB approximation spaces (“p”-refinement) over each subdomain. Greedy parameter sampling procedures and a posteriori error estimation play important roles in both the “h”-type and “p”-type stages of the new algorithm. We present illustrative numerical results for a convection-diffusion problem: the new “hp”-approach is considerably faster (respectively, more costly) than the standard “p”-type reduced basis method in the online (respectively, offline) stage.

33. Y Maday, NC Nguyen, AT Patera, and SH Pau, A General Multipurpose Interpolation Procedure: The Magic Points. Communications on Pure and Applied Analysis (CPAA), 8(1):383–404, 2009. doi: 10.3934/cpaa.2009.8.383
Lagrangian interpolation is a classical way to approximate general functions by finite sums of well chosen, pre-defined, linearly independent interpolating functions; it is much simpler to implement than determining the best fits with respect to some Banach (or even Hilbert) norms. In addition, only partial knowledge is required (here values on some set of points). The problem of defining the best sample of points is nevertheless rather complex and is in general open. In this paper we propose a way to derive such sets of points. We do not claim that the points resulting from the construction explained here are optimal in any sense. Nevertheless, the resulting interpolation method is proven to work under certain hypothesis, the process is very general and simple to implement, and compared to situations where the best behavior is known, it is relatively competitive.

32. S Boyaval, C Le Bris, T Lelièvre, Y Maday, NC Nguyen, and AT Patera, Reduced Basis Techniques for Stochastic Problems. Archives of Computational Methods in Engineering 17(4):435–454, 2010. full text available (revised March 2010) | doi: 10.1007/s11831-010-9056-z
In this review paper, we look at recent applications of a now classical general reduction technique, the Reduced Basis (RB) approach initiated in [Prud'homme, et al., JFE, 124(1):70–80, 2002], to the specific context of differential equations with random coefficients. First we give an overview of the RB approach. Next we look at two specific applications. The first is the application of the RB approach for the discretization of a simple second-order elliptic equation with a random boundary condition [S Boyaval, et al., CMAME, 198(41-44):3187–3206, 2009]; the second is the application of the RB approach to reduce the variance in the Monte-Carlo simulation of a stochastic differential equation [S Boyaval and T. Lelièvre, A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm. In P Zhang, editor, Communication in Mathematical Sciences, volume Special Issue “Mathematical Issues on Complex Fluids”, accepted for publication 2009]. Finally, we conclude with some general comments and a discussion of possible tracks for further research.

31. A Buffa, Y Maday, AT Patera, C Prud'homme, and G Turinici, A Priori Convergence of the Greedy Algorithm for the Parametrized Reduced Basis. Mathematical Modelling and Numerical Analysis 46(3):595–603, 2012. full text available | doi: 10.1051/m2an/2011056
The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the "reduced basis." The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that three greedy algorithms converge; the last algorithm, based on the use of an a posteriori estimator, is the approach actually employed in the calculations.

30. DJ Knezevic, NC Nguyen, and AT Patera, Reduced Basis Approximation and A Posteriori Error Estimation for the Parametrized Unsteady Boussinesq Equations. Mathematical Models and Methods in Applied Sciences 21(7):1415–1442, 2011. full text available | doi: 10.1142/S0218202511005441
In this paper we present reduced basis approximations and associated rigorous a posteriori error bounds for the parametrized unsteady Boussinesq equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold — to provide dimension reduction; an efficient POD-Greedy sampling method for identification of optimal and numerically stable approximations — to yield rapid convergence; accurate (Online) calculation of the solution-dependent stability factor by the Successive Constraint Method — to quantify the growth of perturbations/residuals in time; rigorous a posteriori bounds for the errors in the reduced basis approximation and associated outputs — to provide certainty in our predictions; and an Offline-Online computational decomposition strategy for our reduced basis approximation and associated error bound — to minimize marginal cost and hence achieve high performance in the real-time and many-query contexts. The method is applied to a transient natural convection problem in a two-dimensional “complex” enclosure — a square with a small rectangle cut-out — parametrized by Grashof number and orientation with respect to gravity. Numerical results indicate that the reduced basis approximation converges rapidly and that furthermore the (inexpensive) rigorous a posteriori error bounds remain practicable for parameter domains and final times of physical interest.
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29. JL Eftang, AT Patera, and EM Rønquist, An “hp” Certified Reduced Basis Method for Parametrized Parabolic Partial Differential Equations. JS Hesthaven and EM Rønquist (eds.), Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, 76:179–187, 2011. full text (revised January 2010) | doi: 0.1007/978-3-642-15337-2_15
We extend previous work on a parameter multi-element “hp” certified reduced basis method for elliptic equations to the case of parabolic equations. A POD (in time) / Greedy (in parameter) sampling procedure is invoked both in the partitioning of the parameter domain (“h”-refinement) and in the construction of individual reduced basis approximation spaces for each parameter subdomain (“p”-refinement). The critical new issue is proper balance between additional POD modes and additional parameter values in the initial subdivision process. We present numerical results to compare the computational cost of the new approach to the standard (“p”-type) reduced basis method.

28. JL Eftang and EM Rønquist, Evaluation of Flux Integral Outputs for the Reduced Basis Method. Mathematical Models and Methods in Applied Sciences, 20(3): 351–374, 2010. full text (revised October 2009) | doi: 10.1142/S021820251000426X
In this paper, we consider the evaluation of flux integral outputs from reduced basis solutions to second-order PDE’s. In order to evaluate such outputs, a lifting function v* must be chosen. In the standard finite element context, this choice is not relevant, whereas in the reduced basis context, as we show, it greatly affects the output error. We propose two “good” choices for v* and illustrate their effect on the output error by examining a numerical example. We also make clear the role of v* in a more general primal-dual reduced basis approximation framework.

27. * NC Nguyen, G Rozza, DBP Huynh, and AT Patera, Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Parabolic PDEs; Application to Real-Time Bayesian Parameter Estimation. In Large-Scale Inverse Problems and Quantification of Uncertainty, L Biegler, G Biros, O Ghattas, M Heinkenschloss, D Keyes, B Mallick, Y Marzouk, L Tenorio, B van Bloemen Waanders, and K. Willcox (eds), John Wiley & Sons, Ltd, UK. (to appear 2010). full text available
In this paper we first review our most recent RB techniques — approximation and a posteriori error estimation — for linear parabolic equations, with particular emphasis on combined POD (in time) - Greedy (in parameter) sampling procedures as first proposed by Haasdonk and Ohlberger [M2AN 42(2):277–302, 2008]; as a numerical example, we consider transient thermal analysis of a delamination crack in a two-material (Fiber-Reinforced-Polymer and concrete) system. We also consider the application of these RB approximations and error bounds — within the context of our particular crack delamination example — to Bayesian parameter estimation: we provide very rapid and certified RB parameter estimators that permit effectively real-time (reliable) response.
We observe numerically that the Bayesian upper bound for our parameter estimate increases linearly with the RB error bound when the RB error bound is sufficiently smaller than the experimental error, but exponentially when the RB error bound is significantly larger than the experimental error. Therefore, the RB error bound must be chosen smaller than the experimental error to obtain sharp Bayesian bounds.

26. * DBP Huynh, DJ Knezevic, Y Chen, JS Hesthaven, and AT Patera, A Natural-Norm Successive Constraint Method for Inf-Sup Lower Bounds. Computer Methods in Applied Mechanics and Engineering, 199:1963–1975, 2010. full text (revised February 2010) | doi: 10.1016/j.cma.2010.02.011
We present a new approach for the construction of lower bounds for the inf-sup stability constants required in a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations. We combine the "linearized" inf-sup statement of the natural-norm approach with the approximation procedure of the Successive Constraint Method (SCM): the former (natural-norm) provides an economical parameter expansion and local concavity in parameter — a small(er) optimization problem which enjoys intrinsic lower bound properties; the latter (SCM) provides a systematic optimization framework — a Linear Program (LP) relaxation which readily incorporates continuity and stability constraints. The natural-norm SCM requires a parameter domain decomposition: we propose a greedy algorithm for selection of the SCM control points as well as adaptive construction of the optimal subdomains. The efficacy of the natural-norm SCM is illustrated through numerical results for two types of non-coercive problems: the Helmholtz equation (for acoustics, elasticity, and electromagnetics), and the convection-diffusion equation.

25. * DJ Knezevic and AT Patera, A Certified Reduced Basis Method for the Fokker-Planck Equation of Dilute Polymeric Fluids: FENE Dumbbells in Extensional Flow. SIAM Journal on Scientific Computing, 32(2):793–817, 2010. full text (revised December 2009) | doi:10.1137/090759239
In this paper we present a reduced basis method for the parametrized Fokker-Planck equation associated with evolution of Finitely Extensible Nonlinear Elastic (FENE) dumbbells in a Newtonian solvent for a (prescribed) extensional macroscale flow. There are two new important ingredients: a projection-based POD-Greedy sampling procedure (proposed by Haasdonk and Ohlberger [M2AN 42(2):277–302, 2008]) for the stable identification of optimal reduced basis spaces; and a finite-time a posteriori bound for the error in the reduced basis prediction of the two outputs of interest — the optical anisotropy and the first normal stress difference. We present numerical results for stress-conformation hysteresis as a function of Weissenberg number and final time that demonstrate the rapid convergence of the reduced basis approximation and the effectiveness of the a posteriori error bounds.
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24. * S Boyaval, C Le Bris, Y Maday, NC Nguyen, and AT Patera, A Reduced Basis Approach for Variational Problems with Stochastic Parameters: Application to Heat Conduction with Variable Robin Coefficient. Computer Methods in Applied Mechanics and Engineering, 198(41-44):3187–3206, 2009. doi: 10.1016/j.cma.2009.05.019
In this work, a Reduced Basis (RB) approach is used to solve a large number of Boundary Value Problems (BVPs) parametrized by a stochastic input — expressed as a Karhunen-Loève expansion — in order to compute outputs that are smooth functionals of the random solution fields. The RB method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the RB approach developed previously for deterministic systems. However, the stochastic framework requires the development of new a posteriori estimates for "statistical" outputs — such as the first two moments of integrals of the random solution fields; these error bounds, in turn, permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many-query context.

23. * NC Nguyen, G Rozza, & AT Patera, Reduced Basis Approximation and A Posteriori Error Estimation for the Time-Dependent Viscous Burgers' Equation. Calcolo, 46(3):157–185 2009. doi: 10.1007/s10092-009-0005-x
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 a posteriori error estimator, derived from standard analysis of the error-residual equation, comprises two key ingredients — both of which admit efficient Offline-Online treatment: the first is a sum over timesteps of the square of the dual norm of the residual; the second is an accurate upper bound (computed by the Successive Constraint Method) for the exponential-in-time stability factor. These error bounds serve both Offline for construction of the reduced basis space by a new POD-Greedy procedure and Online for verification of fidelity. The a posteriori error bounds are practicable for final times (measured in convective units) T ≈ O(1) and Reynolds numbers ν ⁻¹ ≫ 1; we present numerical results for a (stationary) steepening front for T = 2 and 1 ≤ ν ⁻¹ ≤ 200.
Note: Extension of the results of this paper to Navier-Stokes can be found in Paper 30.

22. * G Rozza, NC Nguyen, DBP Huynh, and AT Patera, Real-Time Reliable Simulation of Heat Transfer Phenomea. Proceedings of HT2009, 2009 ASME Summer Heat Transfer Conference, July 19–23, 2009, San Francisco, California, paper number HT2009-88212. full text available
This paper includes, in addition to a brief summary of the certified reduced basis methodology, a description of the rbMIT software package, a "worked problem" framework for educational applications of the rbMIT software, and finally several illustrative "worked problems" in heat transfer.

21. S Sen, Reduced-Basis Approximation and A Posteriori Error Estimation for Many-Parameter Heat Conduction Problems. Numerical Heat Transfer, Part B: Fundamentals 54(5):369–389, 2008. doi:10.1080/10407790802424204
Reduced-basis (RB) methods enable repeated and rapid evaluation of parametrized partial differential equation (PDE)-constrained input-output relationships required in the context of parameter estimation, design, optimization, and control. These methods have been successfully applied to problems with few parameters [O(3)]. Here we introduce efficient sampling algorithms that enable the efficient exploration of many parameters. We apply the RB methods to an illustrative heat conduction problem with P = 25 parameters, obtaining accurate and certified results in real time with significant computational savings relative to standard finite-element techniques.

20. * 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. Archives of Computational Methods in Engineering 15(3):229–275, 2008. 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).
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19. * 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.

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.
ERRATA Note we should consider the symmetric part in the formation of the $\hat{a}^{q}$ leading up to Equation(6).

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.

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.)

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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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.

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. Also G Rozza, Reduced basis method for Stokes equations in domains with non-affine parametric dependence, Comp Vis Science, 12(1):23–35, 2009, Rozza thesis, and AE Løvgren, Reduced basis modeling of hierarchical flow systems, PhD Thesis, Norwegian University of Science and Technology, Trondheim, Norway, 2005.

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.

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.

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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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.

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.

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.

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.

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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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 Rovas thesis) and for nonlinear problems (a monotonic cubic nonlinearity); additionally, we discuss a first quasi-rigorous treatment for the Burgers equation.
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.
ERRATA Note the Helmholtz deflation procedure described in this paper is not correct.

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.

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.

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.
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