# A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations

@article{Grepl2005APE, title={A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations}, author={Martin A. Grepl and Anthony T. Patera}, journal={Mathematical Modelling and Numerical Analysis}, year={2005}, volume={39}, pages={157-181} }

In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem - we shall "simply" treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe - Galerkin projection onto a space WN spanned by solutions of…

## 360 Citations

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## References

SHOWING 1-10 OF 47 REFERENCES

A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations

- Mathematics
- 2003

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic partial differential equations with affine parameter dependence. The essential components are (i)…

A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations : “convex inverse” bound conditioners

- Mathematics, Computer Science
- 2002

A new class of improved bound conditioners is introduced: the critical innovation is the direct approximation of the parametric dependence of the inverse of the operator (rather than the operator itself); this helps accommodate higher-order effectivity constructions while simultaneously preserving on-line efficiency.

Reduced--Basis Output Bound Methods for Parametrized Partial Differential Equations

- Computer Science, Mathematics
- 2002

The method is ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

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

- Computer Science, Mathematics
- 2002

The method is ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

Reduced-basis output bound methods for parabolic problems

- Computer Science
- 2006

Without assuming a time discretization, a reduced-basis procedure is presented to ‘efficiently’ compute accurate approximations to the solution of the parabolic problem and ‘relevant’ outputs of interest and an error estimation procedure is developed to 'a posteriori validate’ the accuracy of the output predictions.

Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations

- Computer Science
- 2007

Abstract
We present a technique for the rapid and reliable optimization of systems characterized by linear-functional outputs of coercive elliptic partial differential equations with affine (input)…

An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations

- Mathematics
- 2004

Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds

- Computer Science
- 2003

Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems

- Computer Science
- 2000

Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds

- Computer Science, Mathematics
- 2005

This paper extends the methodology to the parametrized steady incompressible Navier–Stokes equations and exploits affine parametric structure and offline–online computational decompositions to provide real‐time deployed response.