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Reduced-basis output bounds for approximately parametrized elliptic coercive partial differential equations

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Computing and Visualization in Science

Abstract

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

In our earlier work, we develop a rigorous a posteriori error bound framework for the case in which the parametrization of the partial differential equation is exact; in this paper, we address the situation in which our mathematical model is not “complete.” In particular, we permit error in the data that define our partial differential operator: this error may be introduced, for example, by imperfect specification, measurement, calculation, or parametric expansion of a coefficient function. We develop both accurate predictions for the outputs of interest and associated rigorous a posteriori error bounds; and the latter incorporate both numerical discretization and “model truncation” effects. Numerical results are presented for a particular instantiation in which the model error originates in the (approximately) prescribed velocity field associated with a three-dimensional convection-diffusion problem.

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References

  1. Akgun, M.A., Garcelon, J.H., Haftka, R.T.: Fast exact linear and non-linear structural reanalysis and the Sherman–Morrison–Woodbury formulas. International Journal for Numerical Methods in Engineering. 50(7), 1587–1606 (2001)

    Article  Google Scholar 

  2. Allgower, E., Georg, K.: Simplicial and continuation methods for approximating fixed-points and solutions to systems of equations. SIAM Review. 22(1), 28–85 (1980)

    Article  MathSciNet  Google Scholar 

  3. Almroth, B.O., Stern, P., Brogan, F.A.: Automatic choice of global shape functions in structural analysis. AIAA Journal. 16, 525–528 (1978)

    Article  Google Scholar 

  4. Balmes, E.: Parametric families of reduced finite element models: Theory and applications. Mechanical Systems and Signal Processing. 10(4), 381–394 (1996)

    Article  Google Scholar 

  5. Barrett, A., Reddien, G.: On the reduced basis method. Z. Angew. Math. Mech. 75(7), 543–549 (1995)

    Article  MathSciNet  Google Scholar 

  6. Chan, T.F., Wan, W.L.: Analysis of projection methods for solving linear systems with multiple right-hand sides. SIAM Journal on Scientific Computing. 18(6), 1698,1721 (1997)

    Article  MathSciNet  Google Scholar 

  7. Farhat, C., Crivelli, L., Roux, F.X.: Extending substructure based iterative solvers to multiple load and repeated analyses. Computer Methods in Applied Mechanics and Engineering. 117(1–2), 195–209 (1994)

  8. Fink, J.P., Rheinboldt, W.C.: On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech. 63, 21–28 (1983)

    Article  MathSciNet  Google Scholar 

  9. Machiels, L., Maday, Y., Oliveira, I.B., Patera, A.T., Rovas, D.V.: Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris, Série I. 331(2), 153–158 (2000)

    Article  MathSciNet  Google Scholar 

  10. Maday, Y.: Private communication

  11. Maday, Y., Patera, A.T., Rovas, D.V.: A blackbox reduced-basis output bound method for noncoercive linear problems. In D.Cioranescu and J.-L. Lions (eds.), Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar Volume XIV. Elsevier Science B.V. 2002, pp. 533–569, Amsterdam: North Holland

  12. Maday, Y., Patera, A.T., Turinici, G.: Global a priori convergence theory for reduced-basis approximation of single-parameter symmetric coercive elliptic partial differential equations. C. R. Acad. Sci. Paris, Série I. 335, 1–6 (2002)

    Article  Google Scholar 

  13. Noor, A.K., Peters, J.M.: Reduced basis technique for nonlinear analysis of structures. AIAA Journal. 18(4), 455–462 (1980)

    Article  Google Scholar 

  14. Peterson, J.S.: The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10(4), 777–786 (1989)

    Article  Google Scholar 

  15. Porsching, T.A.: Estimation of the error in the reduced basis method solution of nonlinear equations. Mathematics of Computation. 45(172), 487–496 (1985)

    Article  MathSciNet  Google Scholar 

  16. Prud’homme, C., Rovas, D., Veroy, K., Maday, Y., Patera, A.T., Turinici, G.: Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. Journal of Fluids Engineering. 124(1), 70–80 (2002)

    Article  Google Scholar 

  17. Quarteroni, A.,Valli, A.: Numerical Approximation of Partial Differential Equations. 2nd edn. New York: Springer-Verlag 1997

  18. Rheinboldt, W.C.: Numerical analysis of continuation methods for nonlinear structural problems. Computers and Structures. 13(1–3), 103–113 (1981)

  19. Rheinboldt, W.C.: On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Analysis, Theory, Methods and Applications. 21(11), 849–858 (1993)

    Article  Google Scholar 

  20. Solodukhov, Y.: Reduced-Basis Methods Applied to Locally Non-Affine Problems. PhD thesis, Massachusetts Institute of Technology, 2004, In progress

  21. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Englewood Cliffs, NJ: Prentice-Hall 1973

  22. Veroy, K.: Reduced-Basis Methods Applied to Problems in Elasticity: Analysis and Applications. PhD thesis, Massachusetts Institute of Technology, April 2003

  23. Veroy, K., Rovas, D., Patera, A.T.: A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations: “convex inverse” bound conditioners. Control, Optimisation and Calculus of Variations. Special Volume: A tribute to J.-L.Lions. 8, 1007–1028 (2002)

    Article  MathSciNet  Google Scholar 

  24. Wendland, W.L.: Elliptic Systems in the Plane. London, San Francisco: Pitman 1979

  25. Yip, E.L.: A note on the stability of solving a rank-p modification of a linear system by the Sherman–Morrison–Woodbury formula. SIAM Journal on Scientific and Statistical Computating. 7(2), 507–513 (1986)

    Article  Google Scholar 

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  1. Department of Mechanical Engineering, Massachusetts Institute of Technology, Room 3-266, 77 Massachusetts Avenue, 02139-4307, Cambridge, MA, USA

    C. Prud’homme & A.T. Patera

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  1. C. Prud’homme
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  2. A.T. Patera
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Correspondence to C. Prud’homme.

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M.S. Espedal, A. Quarteroni, A. Sequeira

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Prud’homme, C., Patera , A. Reduced-basis output bounds for approximately parametrized elliptic coercive partial differential equations. Comput Visual Sci 6, 147–162 (2004). https://doi.org/10.1007/s00791-003-0119-7

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