Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

By Christophe Prud'Homme, Dimitrios Rovas, Karen Veroy, L. Machiels, Yvon Maday, Anthony T. Patera, Gabriel Turinici. 2002

653 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Mechanics of Materials, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4177fd2bd74812e8b23b4be605c90d3a9572b437

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In Computational Mechanics

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Citations in the corpus

A static condensation reduced basis element method: Complex problems (2013, neutral): “Model order reduction methods, such as Proper Orthogonal Decomposition (POD) [1] or the Reduced Basis (RB) method [2, 3, 4], are thus an attractive alternative.”
A natural-norm Successive Constraint Method for inf-sup lower bounds (2010, methodology): “Introduction The certified reduced basis method is well developed for both coercive and non-coercive elliptic affinely parametrized partial differential equations [11, 15, 17].”
Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations (2002, background): “On renvoit aussi à [11] pour plus de détails sur la mise en oeuvre et les applications”
A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models (2004, methodology): “Finally, there are some interesting error bounds in [ 4 ], and these results could be applied to automatically select the reduction order.”
Data-driven operator inference for nonintrusive projection-based model reduction (2016, methodology): “The reduced basis method [3, 8] derives a reduced basis with a greedy approach [9] based on computationally cheap a posteriori error estimators of intermediate reduced models.”
A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations (2002, background): “Regarding the mathematical framework, it has been extended to several problem categories: non-coercive partial differential equations, parabolic equations, generalized eigenvalue problems, non-symmetric and non-compliant cases, see [ 19 ].”
REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS (2011, background): “(Online) RB evaluation is typically several orders of magnitude less expensive than the classical finite element approach [49,53].”
Reduced basis method for linear elasticity problems with many parameters (2008, background): “Other important aspects of the RB formulation are (i) the availability of an error bound theory to develop a rapid a posteriori error estimation based on residual calculations, see for example [25, 28], and (ii) the studies on a-priori convergence and error estimation, see [16, 24].”
Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders (2018, methodology): “Common choices for computing the reduced-basis matrix Φ when the velocity is linear in its first argument include balanced truncation [52], rational interpolation [4, 31], the reduced-basis method [59, 64], Rayleigh–Ritz eigenvectors [20]; common choices when the velocity is nonlinear include POD [3”
Structure‐preserving, stability, and accuracy properties of the energy‐conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models (2015, methodology): “In the original form of the reduced-basis method [5], a large number of snapshots is also pre-computed, but no data compression is explicitly performed.”
Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations (2007, methodology): “In the context of affine parameter dependence, in which the operator is expressible as the sum of products of parameter-dependent functions and parameter-independent operators, the Offline-Online idea is quite self-apparent and indeed has been re-invented often [15, 65, 70, 112]; however, applicatio”