A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations
TL;DR: It is shown that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the exact solution uniformly in parameter space, thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges.
Abstract: We consider “Lagrangian” reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity-coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.
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TL;DR: A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs).
Abstract: A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.
1,695 citations
Cites background from "A Priori Convergence Theory for Red..."
...Its success is limited to the problems of linear elliptic parabolic PDEs with affine parameters or low-order polynomial nonlinearities [32, 25, 26, 41, 29]....
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TL;DR: Barrault et al. as discussed by the authors presented an efficient reduced-basis discretization procedure for partial differential equations with nonaffine parameter dependence, replacing non-affine coefficient functions with a collateral reducedbasis expansion, which then permits an affine offline-online computational decomposition.
1,265 citations
TL;DR: (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations are considered.
Abstract: In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.
1,090 citations
Cites background or methods from "A Priori Convergence Theory for Red..."
...2 for many parameters we investigate the critical role of parametric smoothness [47,85] in convergence theory and practice....
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...We note the analysis presented here in fact is relevant to a large class of single parameter coercive problems [85,87,112]....
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...(In fact, the parameters need only be quasi-uniform in lnμ [85,87]....
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...We present from [85,87,112] an a priori theory for RB approximations associated with the specific non-hierarchical equi-ln spaces (75)....
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TL;DR: The method is ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
Abstract: We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
588 citations
TL;DR: In this paper, the authors extended the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems in volving.
Abstract: In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems in- volving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numeri- cal results are presented to assess our approach.
532 citations
References
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TL;DR: The method is ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
Abstract: We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
588 citations
TL;DR: In this paper, a reduced basis technique and a computational algorithm are presented for predicting the nonlinear static response of structures, where a total Lagrangian formulation is used and the structure is discretized by using displacement finite element models.
Abstract: A reduced basis technique and a computational' algorithm are presented for predicting the nonlinear static response of structures. A total Lagrangian formulation is used and the structure is discretized by using displacement finite element models. The nodal displacement vector is expressed as a linear combination of a small number of basis vectors and a Rayleigh-Ritz technique is used to approximate the finite element equations by a reduced system of nonlinear equations. The Rayleigh-Ritz approximation functions (basis vectors) are chosen to be those commonly used in the static perturbation technique namely, a nonlinear solution and a number of its path derivatives. A procedure is outlined for automatically selecting the load (or displacement) step size and monitoring the solution accuracy. The high accuracy and effectiveness of the proposed approach is demonstrated by means of numerical examples.
414 citations
"A Priori Convergence Theory for Red..." refers methods in this paper
...One particular approach is the reduced-basis method, first introduced in the late 1970s for nonlinear structural analysis [1, 9], and subsequently developed more broadly in the 1980s and 1990s [3, 5, 10, 13, 2]....
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276 citations
"A Priori Convergence Theory for Red..." refers methods in this paper
...One particular approach is the reduced-basis method, first introduced in the late 1970s for nonlinear structural analysis [1, 9], and subsequently developed more broadly in the 1980s and 1990s [3, 5, 10, 13, 2]....
[...]