Optimization and Engineering

, Volume 8, Issue 1, pp 43–65 | Cite as

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

Article

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) parameter dependence. The critical ingredients are: reduced-basis approximation to effect significant reduction in state-space dimensionality; a posteriori error bounds to provide rigorous error estimation and control; “offline/online” computational decompositions to permit rapid evaluation of output bounds, output bound gradients, and output bound Hessians in the limit of many queries; and reformulation of the approximate optimization statement to ensure (true) feasibility and control of suboptimality. To illustrate the method we consider the design of a three-dimensional thermal fin: Given volume and power objective-function weights, and root temperature “not-to-exceed” limits, the optimal geometry and heat transfer coefficient can be determined—with guaranteed feasibility—in only 2.3 seconds on a 500 MHz Pentium machine; note the latter includes only the online component of the calculations. Our method permits not only interactive optimal design at conception and manufacturing, but also real-time reliable adaptive optimal design in operation.

Keywords

Engineering optimization Reduced-basis approximation A posteriori error estimation Output bounds Uncertainty control Parametrized partial differential equations Real-time computing 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alexandrov N, Dennis JE Jr (1995) Multilevel algorithms for nonlinear optimization. In: Optimal design and control (Blacksburg, VA, 1994). Progr. systems control theory, vol 19. Birkhäuser Boston, Boston, pp 1–22 Google Scholar
  2. Alexandrov N, Hussaini MY (1997) Multidisciplinary design optimization: state-of-the-art. SIAM, Philadelphia Google Scholar
  3. Almroth BO, Stern P, Brogan FA (1978) Automatic choice of global shape functions in structural analysis. AIAA J 16:525–528 Google Scholar
  4. Barrault M, Nguyen NC, Maday Y, Patera AT (2004) An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations. C R Acad Sci Paris Sér I 339:667–672 MATHMathSciNetGoogle Scholar
  5. Barrett A, Reddien G (1995) On the reduced basis method. Z Angew Math Mech 75(7):543–549 MATHMathSciNetGoogle Scholar
  6. Biros G, Ghattas O (2000) A Lagrange–Newton–Krylov–Schur method for PDE-constrained optimization. SIAG/OPT Views News 11(1) Google Scholar
  7. Byrd RH, Hribar ME, Nocedal J (1999) An interior point algorithm for large scale nonlinear programming. SIAM J Optim 9(4):877–900 MATHCrossRefMathSciNetGoogle Scholar
  8. Cliff EM, Heinkenschloss M, Shenoy AR (1998) Adjoint-based methods in aerodynamic design-optimization. In: Computational methods for optimal design and control (Arlington, VA, 1997). Progr. systems control theory, vol 24. Birkhäuser Boston, Boston, pp 91–112 Google Scholar
  9. Coleman TF, Li Y (1990) Large scale numerical optimization. SIAM, Philadelphia MATHGoogle Scholar
  10. Dennis JE, Torczon V (1997) Managing approximation models in optimization. In: Multidisciplinary design optimization (Hampton, VA, 1995). SIAM, Philadelphia, pp 330–347 Google Scholar
  11. Fink JP, Rheinboldt WC (1983) On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z Angew Math Mech 63:21–28 MATHMathSciNetGoogle Scholar
  12. Forsgren A, Gill PE (1998) Primal-dual interior methods for nonconvex nonlinear programming. SIAM J Optim 8(4):1132–1152 MATHCrossRefMathSciNetGoogle Scholar
  13. Forsgren A, Gill P, Wright M (2002) Interior methods for nonlinear optimization. SIAM Rev 44(4):535–597 CrossRefMathSciNetGoogle Scholar
  14. Fox R, Miura H (1971) An approximate analysis technique for design calculations. AIAA J 9(1):177–179 CrossRefGoogle Scholar
  15. Ghattas O, Orozco CE (1997) A parallel reduced Hessian SQP method for shape optimization. In: Multidisciplinary design optimization (Hampton, VA, 1995). SIAM, Philadelphia, pp 133–152 Google Scholar
  16. Gill PE, Murray W, Saunders MA (2002) SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM J Optim 12(4):979–1006 MATHCrossRefMathSciNetGoogle Scholar
  17. Gould NIM (2003) Some reflections on the current state of active-set and interior-point methods for constrained optimization. SIAG/OPT Views News 14(1) Google Scholar
  18. Haftka R, Gürdal Z, Kamat M (1990) Elements of structural optimization. Kluwer Academic, Dordrecht MATHGoogle Scholar
  19. Hager WW, Hearn DW, Pardalos PM (1994) Large scale optimization: state of the art. Kluwer Academic, Dordrecht MATHGoogle Scholar
  20. Ito K, Ravindran SS (1998) A reduced-order method for simulation and control of fluid flows. J Comput Phys 143(2):403–425 MATHCrossRefMathSciNetGoogle Scholar
  21. Lewis RM (1998) Numerical computation of sensitivities and the adjoint approach. In: Computational methods for optimal design and control (Arlington, VA, 1997). Progr. systems control theory, vol 24. Birkhäuser Boston, Boston, pp 285–302 Google Scholar
  22. Machiels L, Maday Y, Oliveira IB, Patera AT, Rovas DV (2000) Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C R Acad Sci Paris Sér I 331(2):153–158 MATHMathSciNetGoogle Scholar
  23. Maday Y, Machiels L, Patera AT, Rovas DV (2000) Blackbox reduced-basis output bound methods for shape optimization. In: Proceedings 12th international domain decomposition conference. Chiba, pp 429–436 Google Scholar
  24. Maday Y, Patera AT, Turinici G (2002) A priori convergence theory for reduced–basis approximations of single-parameter elliptic partial differential equations. J Sci Comput 17(1–4):437–446 MATHCrossRefMathSciNetGoogle Scholar
  25. Nguyen NC, Veroy K, Patera AT (2005) Certified real-time solution of parametrized partial differential equations. In: Yip S (ed) Handbook of materials modeling. Springer, Berlin, pp 1523–1558 Google Scholar
  26. Noor AK, Peters JM (1980) Reduced basis technique for nonlinear analysis of structures. AIAA J 18(4):455–462 Google Scholar
  27. Oliveira IB (2007) An affine, directionally-scaled trust region SQP-IPM algorithm (in preparation) Google Scholar
  28. Patera AT, Rovas D, Machiels L (2000) Reduced–basis output–bound methods for elliptic partial differential equations. SIAG/OPT Views News 11(1) Google Scholar
  29. Porsching TA (1985) Estimation of the error in the reduced basis method solution of nonlinear equations. Math Comput 45(172):487–496 MATHCrossRefMathSciNetGoogle Scholar
  30. Prud’homme C, Rovas D, Veroy K, Maday Y, Patera AT, Turinici G (2002a) Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods. J Fluids Eng 124(1):70–80 CrossRefGoogle Scholar
  31. Prud’homme C, Rovas DV, Veroy K, Patera AT (2002b) A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. M2AN (Math Model Numer Anal) 36(5):747–771 MATHCrossRefMathSciNetGoogle Scholar
  32. Ravindran SS (1998) Numerical approximation of optimal flow control problems by SQP method. In: Optimal control of viscous flow. SIAM, Philadelphia, pp 181–198 Google Scholar
  33. Veroy K, Patera AT (2005) Certified real-time solution of the parametrized steady incompressible Navier–Stokes equations; Rigorous reduced-basis a posteriori error bounds. Int J Numer Meth Fluids 47:773–788 MATHCrossRefMathSciNetGoogle Scholar
  34. Veroy K, Rovas D, Patera AT (2002) A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations: “convex inverse” bound conditioners. Control Optim Calc Var 8:1007–1028. Special Volume: A tribute to J.-L. Lions MATHCrossRefMathSciNetGoogle Scholar
  35. Veroy K, Prud’homme C, Rovas DV, Patera AT (2003) A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations (AIAA Paper 2003-3847). In: Proceedings of the 16th AIAA computational fluid dynamics conference Google Scholar
  36. Waanders B, Bartlett R, Long K, Boggs P, Salinger A (2002) Large scale non-linear programming for PDE constrained optimization. Technical report SAND2002-3198, Sandia National Labs Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyDept. of Mechanical EngineeringCambridgeUSA
  2. 2.Operations Research R and DCaryUSA

Personalised recommendations