Rapid Reliable Solution
of the Parametrized Partial Differential Equations
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Methodology: Reduced Basis Methods and A Posteriori Error Estimators

We include here selected recent publications of our group relevant to reduced basis approximation and a posteriori error estimation. The * to the left 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. Back to new list »

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 $ N \rightarrow \infty$), 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.
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 $S_N$ and hence spaces $W_N$ — 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.

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 $N$) — 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, ${\cal N}$ — for non-polynomial nonlinearities. Our "empirical interpolation" approach recovers ${\cal N}$-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, REJECTED. 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. 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).

22. 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.

23. * G Rozza, NC Nguyen, AT Patera, and S Deparis, Reduced Basis Methods and A Posteriori Error Estimators for Heat Transfer Problems. Proceedings of HT2009, 2009 ASME Summer Heat Transfer Conference, July 19–23, 2009, San Francisco, California, paper number HT2009-88211. full text available
This paper focuses on the parametric study of steady and unsteady forced and natural convection problems by the certified reduced basis method. These problems are characterized by an input-output relationship in which given an input parameter vector — material properties, boundary conditions and sources, and geometry — we would like to compute certain outputs of engineering interest — heat fluxes and average temperatures. The certified reduced basis method provides both (i) a very inexpensive yet accurate reduced basis output prediction, and (ii) a rigorous bound for the error in the reduced basis prediction relative to an underlying expensive high-fidelity finite element discretization. The feasibility and efficiency of the method is demonstrated for three natural convection model problems: a scalar steady forced convection problem in a rectangular channel characterized by two parameters — Péclet number and the aspect ratio of the channel — and an output — the average temperature over the domain; a steady natural convection problem in a laterally heated cavity characterized by three parameters — Grashof and Prandtl numbers, and the aspect ratio of the cavity — and an output — the inverse of the Nusselt number; and an unsteady natural convection problem in a laterally heated cavity characterized by two parameters — Grashof and Prandtl numbers — and a time-dependent output — the average of the horizontal velocity over a specified area of the cavity.
ERRATA We have discovered that the exponential stability factors reflected in Section 3.2.2 of this paper for the unsteady Boussinesq equations are in error — and in the wrong direction: the error bounds reported are thus not rigorous and in particular too optimistic. The theory of Section 2.2 is correct; however, it is now clear that the theory as presented is practically restricted to much smaller times or lower Grashof number than claimed in the paper. We will report elsewhere on improvements to the theory and associated (new) computational results. Note all other sections of the paper are free of (known) errors.
Correct Navier-Stokes/Boussinesq results may be found in Paper 34.

24. * 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.

25. * CN 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.
ERRATA There are no (known) errors in this paper; however, two other papers and associated results (Paper 26 and Paper 28 below) cited in this Calcolo paper do contain errors — see errata below.
Correct Navier-Stokes/Boussinesq results may be found in Paper 34.
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26. * 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 Biegler, Biros, Ghattas, Heinkenschloss, Keyes, Mallick, Tenorio, van Bloemen Waanders, & Willcox (Eds), 2009, Computational Methods for Large Scale Inverse Problems and Uncertainty Quantification, John Wiley & Sons, UK (submitted). full text available REVISED/RESUBMITTED (see Paper 31 below)
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.
In this paper we also summarize the extension of our RB approach to nonlinear parabolic equations, in particular — building on the Burgers work described in detail in Paper 22 (submitted to Calcolo) above — the unsteady incompressible Navier-Stokes equations. We consider a periodic square-in-channel “eddy-promoter" flow with Reynolds number as parameter. The (POD-Greedy) RB approximation accurately captures both the subcritical and supercritical Hopf-bifurcation dynamics. Furthermore, the rigorous RB a posteriori error bounds remain practicable even for modest times and modest Reynolds numbers — thanks in large part to accurate (conservative) formulation and estimation of the requisite stability factor by the Successive Constraint Method (see Papers 18 and 21 above).
ERRATA We have discovered that the exponential stability factors reported in the paper in Section 3 (for Navier-Stokes) are in error — and in the wrong direction: the error bounds reported are thus not rigorous and in particular too optimistic. The theory is correct; however, it is now clear that the theory as presented is practically restricted to much smaller times or lower Reynolds number than claimed in the paper. We will report elsewhere on improvements to the theory and associated (new) computational results. There is also a small error in the L2 error bound, equation (18) (first line), and associated numerical results. A corrected version of the paper is found below as Paper 31: we have removed Section 3; we have also corrected the small error in the L2 error bound (and associated numerical results).
Correct Navier-Stokes/Boussinesq results may be found in Paper 34.

27. * 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.

28. * NC Nguyen and AT Patera, Reduced Basis Approximation and A Posteriori Error Estimation for the Parametrized Unsteady Boussinesq Equations: Application to Natural Convection in a Laterally Heated Cavity. Journal of Computational Physics (withdrawn August 2009). full text available RETRACTED
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 weighted 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 for high performance in the real-time (e.g., parameter-estimation, control) and many-query (e.g., design optimization, stability map) contexts. The method is applied to study stable steady-state solutions, the onset of oscillatory instability, and supercritical unsteady regimes for a natural convection "response to disturbance" flow in a laterally heated rectangular cavity over a range of Grashof number and Prandtl number. Numerical results indicate that the reduced basis approximation converges rapidly and that furthermore the (inexpensive) rigorous a posteriori error bounds remain practicable even for moderate final times and moderate parameter domains.
Our development here is based upon our previous work on the unsteady viscous Burger equation and the unsteady incompressible Navier-Stokes equations. However, we extend these initial efforts in several important ways. First, we address the additional formulation, computation, and theoretical difficulties associated with the coupled multi-field Boussinesq equations — proper scaling is crucial to ultimate reduced basis performance. Second, we consider a multi-parameter, Gr and Pr (and of course time), which places additional stress on the weighted POD-Greedy sampling procedure. Third, we consider more relevant outputs which permit us to quantitatively address stability and bifurcation issues. Finally, we include many details not presented in the previous work as well as more extensive physical results.
ERRATA We have discovered that the exponential stability factors reported in the paper are in error — and in the wrong direction: the error bounds reported are thus not rigorous and in particular too optimistic. The theory is correct; however, it is now clear that the theory as presented is practically restricted to much smaller times or lower Grashof number than claimed in the paper. We will report elsewhere on improvements to the theory and associated (new) computational results.
Correct Navier-Stokes/Boussinesq results may be found in Paper 34.

29. * 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 (in press). full text available (revised 16 October 2009)
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.

30. * 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 (submitted 2009). full text available
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.
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31. * 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 Biegler, Biros, Ghattas, Heinkenschloss, Keyes, Mallick, Tenorio, van Bloemen Waanders, & Willcox (Eds), 2009, Computational Methods for Large Scale Inverse Problems and Uncertainty Quantification, John Wiley & Sons, 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.

32. JL Eftang and EM Rønquist, Evaluation of Flux Integral Outputs for the Reduced Basis Method. Mathematical Models and Methods in Applied Sciences (submitted 2008). full text available(revised October 2009)
Abstract: 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.

33. JL Eftang, AT Patera, and EM Rønquist, An “hp” Certified Reduced Basis Method for Parametrized Parabolic Partial Differential Equations. 2009 ICOSAHOM Proceedings (to appear 2010). full text available (revised January 2010)
Abstract: 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.

34. 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 (submitted November 2009). full text available
Abstract: 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.

35. 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 (submitted November 2009). full text available
Abstract: In this paper we prove a priori convergence of the greedy algorithm for identification of the reduced basis space. It is shown that if a sufficiently good space exists — providing sufficiently rapid exponential convergence — then the greedy algorithm will find a (somewhat less) good space — still providing exponential convergence, albeit less rapid.
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36. 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 (submitted December 2009). full text available
Abstract: 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.

37. 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
Abstract: 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.

38. * JL Eftang, AT Patera, and EM Rønquist, An “hp” Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations. SIAM Journal on Scientific Computing (submitted December 2009). full text available
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 a new procedure and attendant theoretical foundations for adaptive partition of the parameter domain into parameter subdomains (“h”-refinement); subsequently, we construct individual standard reduced basis approximation spaces for each subdomain (“p”-refinement). Greedy parameter sampling procedures and a posteriori error estimation are the main ingredients 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.

39. * JL Eftang, MA Grepl, and AT Patera, A Posteriori Error Bounds for the Empirical Interpolation Method. CR Acad Sci Paris Series I (submitted February 2010). full text available
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.

40. * DBP Huynh, DJ Knezevic, JW Peterson, and AT Patera, High-Fidelity Real-Time Simulation on Deployed Platforms. Computers and Fluids (submitted April 2010). full text available
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.

 

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