# A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations

Christophe Prud'homme; Dimitrios V. Rovas; Karen Veroy; Anthony T. Patera

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 36, Issue: 5, page 747-771
- ISSN: 0764-583X

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topPrud'homme, Christophe, et al. "A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations." ESAIM: Mathematical Modelling and Numerical Analysis 36.5 (2010): 747-771. <http://eudml.org/doc/194124>.

@article{Prudhomme2010,

abstract = {
We present in this article two components: these components can in fact serve various goals
independently, though we consider them here as an ensemble. The first component is a technique for
the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and
parabolic) partial differential equations with affine parameter dependence.
The essential features are (i) (provably) rapidly convergent global
reduced–basis approximations — Galerkin projection onto a space
WN 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. This component is ideally suited — considering
the operation count of the online stage — for the repeated and rapid evaluation required in the
context of parameter estimation, design, optimization, and
real–time control. The second component is a framework for distributed simulations. This framework
comprises a library providing the necessary abstractions/concepts for distributed simulations and a
small set of tools — namely SimTeXand SimLaB— allowing an easy manipulation of those
simulations. While the library is the backbone of the framework and is therefore general, the
various interfaces answer specific needs. We shall describe both components and present how they
interact.
},

author = {Prud'homme, Christophe, Rovas, Dimitrios V., Veroy, Karen, Patera, Anthony T.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Mathematical framework; reduced-basis methods; error bounds; computational framework; simulations repository; distributed and parallel computing; CORBA; C++.; mathematical framework; C++; elliptic equations; parabolic equations; convergence; Galerkin projection},

language = {eng},

month = {3},

number = {5},

pages = {747-771},

publisher = {EDP Sciences},

title = {A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations},

url = {http://eudml.org/doc/194124},

volume = {36},

year = {2010},

}

TY - JOUR

AU - Prud'homme, Christophe

AU - Rovas, Dimitrios V.

AU - Veroy, Karen

AU - Patera, Anthony T.

TI - A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 36

IS - 5

SP - 747

EP - 771

AB -
We present in this article two components: these components can in fact serve various goals
independently, though we consider them here as an ensemble. The first component is a technique for
the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and
parabolic) partial differential equations with affine parameter dependence.
The essential features are (i) (provably) rapidly convergent global
reduced–basis approximations — Galerkin projection onto a space
WN 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. This component is ideally suited — considering
the operation count of the online stage — for the repeated and rapid evaluation required in the
context of parameter estimation, design, optimization, and
real–time control. The second component is a framework for distributed simulations. This framework
comprises a library providing the necessary abstractions/concepts for distributed simulations and a
small set of tools — namely SimTeXand SimLaB— allowing an easy manipulation of those
simulations. While the library is the backbone of the framework and is therefore general, the
various interfaces answer specific needs. We shall describe both components and present how they
interact.

LA - eng

KW - Mathematical framework; reduced-basis methods; error bounds; computational framework; simulations repository; distributed and parallel computing; CORBA; C++.; mathematical framework; C++; elliptic equations; parabolic equations; convergence; Galerkin projection

UR - http://eudml.org/doc/194124

ER -

## References

top- M.A. Akgun, J.H. Garcelon and R.T. Haftka, Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas. Int. J. Numer. Methods Engrg.50 (2001) 1587-1606.
- E. Allgower and K. Georg, Simplicial and continuation methods for approximating fixed-points and solutions to systems of equations. SIAM Rev.22 (1980) 28-85.
- B.O. Almroth, P. Stern and F.A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA Journal16 (1978) 525-528.
- A. Barrett and G. Reddien, On the reduced basis method. Z. Angew. Math. Mech.75 (1995) 543-549.
- T.F. Chan and W.L. Wan, Analysis of projection methods for solving linear systems with multiple right-hand sides. SIAM J. Sci. Comput.18 (1997) 1698-1721.
- A.G. Evans, J.W. Hutchinson, N.A. Fleck, M.F. Ashby and H.N.G. Wadley, The topological design of multifunctional cellular metals. Prog. Mater. Sci.46 (2001) 309-327.
- C. Farhat, L. Crivelli and F.X. Roux, Extending substructure based iterative solvers to multiple load and repeated analyses. Comput. Methods Appl. Mech. Engrg.117 (1994) 195-209.
- J.P. Fink and W.C. Rheinboldt, On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech.63 (1983) 21-28.
- L. Machiels, J. Peraire and A.T. Patera, A posteriori finite element output bounds for the incompressible Navier-Stokes equations; Application to a natural convection problem. J. Comput. Phys.172 (2001) 401-425.
- Y. Maday, L. Machiels, A.T. Patera and D.V. Rovas, Blackbox reduced-basis output bound methods for shape optimization, in Proceedings 12th International Domain Decomposition Conference, Chiba, Japan (2000) 429-436.
- Y. Maday, A.T. Patera and J. Peraire, A general formulation for a posteriori bounds for output functionals of partial differential equations; Application to the eigenvalue problem. C. R. Acad. Sci. Paris Sér. I Math.328 (1999) 823-828.
- Y. Maday, A.T. Patera and G. Turinici, Global a priori convergence theory for reduced-basis approximation of single-parameter symmetric coercive elliptic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math.335 (2002) 1-6.
- A.K. Noor and J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA Journal18 (1980) 455-462.
- A.T. Patera and E.M. Rønquist, A general output bound result: Application to discretization and iteration error estimation and control. Math. Models Methods Appl. Sci.11 (2001) 685-712.
- A.T. Patera and E.M. Rønquist, A general output bound result: Application to discretization and iteration error estimation and control. Math. Models Methods Appl. Sci. (2000). MIT FML Report 98-12-1.
- J.S. Peterson, The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput.10 (1989) 777-786.
- T.A. Porsching, Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comp.45 (1985) 487-496.
- C. Prud'homme, A Framework for Reliable Real-Time Web-Based Distributed Simulations. MIT (to appear).
- C. Prud'homme, D. Rovas, K. Veroy, Y. Maday, A.T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bounds methods. J. Fluids Engrg.124 (2002) 70-80.
- W.C. Rheinboldt, Numerical analysis of continuation methods for nonlinear structural problems. Comput. Structures13 (1981) 103-113.
- W.C. Rheinboldt, On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Anal.21 (1993) 849-858.
- D. Rovas, Reduced-Basis Output Bound Methods for Partial Differential Equations. Ph.D. thesis, MIT (in progress).
- K. Veroy, Reduced Basis Methods Applied to Problems in Elasticity: Analysis and Applications. Ph.D. thesis, MIT (in progress).
- N. Wicks and J. W. Hutchinson, Optimal truss plates. Internat. J. Solids Structures38 (2001) 5165-5183.
- E.L. Yip, A note on the stability of solving a rank-p modification of a linear system by the Sherman-Morrison-Woodbury formula. SIAM J. Sci. Stat. Comput.7 (1986) 507-513.

## Citations in EuDML Documents

top- Toni Lassila, Andrea Manzoni, Gianluigi Rozza, On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition
- Toni Lassila, Andrea Manzoni, Gianluigi Rozza, On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition
- Toni Lassila, Andrea Manzoni, Gianluigi Rozza, On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition
- Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie, Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples
- Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie, Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples
- Dominique Chapelle, Asven Gariah, Jacques Sainte-Marie, Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples
- Bernard Haasdonk, Mario Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations

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