An efficient reuced-order modeling approach for non-linear parametrized partial differential equations

International Journal for Numerical Methods in Engineering

Volumn 76, Issue 1, 2008, Pages 27-55

  Nguyen, N C ^a   Peraire, J ^a  

^a Massachusetts Institute of Technology Cambridge   (United States)

Author keywords

Coefficient function approximation; Convection diffusion equations; Elliptic equations; Parametrized PDEs; Reduced order approximation; Standard Galerkin reduced order model

Indexed keywords

AIRCRAFT ENGINES; COMPUTATIONAL COMPLEXITY; COMPUTATIONAL FLUID DYNAMICS; DIFFERENCE EQUATIONS; DIFFERENTIAL EQUATIONS; DIFFERENTIATION (CALCULUS); GALERKIN METHODS; IMAGE SEGMENTATION; PARTIAL DIFFERENTIAL EQUATIONS;

COEFFICIENT-FUNCTION APPROXIMATION; CONVECTION-DIFFUSION EQUATIONS; ELLIPTIC EQUATIONS; PARAMETRIZED PDES; REDUCED-ORDER APPROXIMATION; STANDARD GALERKIN REDUCED-ORDER MODEL;

LINEAR EQUATIONS;

EID: 53749102512     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2309     Document Type: Article

References (43)

1. Almroth BO, Stem P, Brogan FA. Automatic choice of global shape functions in structural analysis. AIAA Journal 1978; 16:525-528.
2. Noor AK, Peters JM. Reduced basis technique for nonlinear analysis of structures. AIAA Journal 1980; 18(4): 455-462.
3. Fink JP, Rheinboldt WC. On the error behavior of the reduced basis technique for nonlinear finite element approximations. Zeitschrift fuer Angewandte Mathematik and Mechanik 1983; 63:21-28.
4. Porsching TA. Estimation of the error in the reduced basis method solution of nonlinear equations. Mathematics of Computation 1985; 45(172):487-496.
5. Gunzburger MD. Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms. Academic Press: Boston, MA, 1989.
6. Peterson JS. The reduced basis method for incompressible viscous flow calculations. SIAM Journal on Scientific and Statistical Computing 1989; 10(4):777-786.
7. Prud'homme C, Rovas D, Veroy K, Maday Y, Patera AT, Turinici G. Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods. Journal of Fluids Engineering 2002; 124(1):70-80.
8. Deane A, Kevrekidis I, Karniadakis G, Orszag S. Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders. Physics of Fluids A 1991; 3(10):2337-2354.
9. Berkooz G, Titi E. Galerkin projections and the proper orthogonal decomposition for equivariant equations. Physics Letters A 1993; 174:94-102.
10. Kunisch K, Volkwein S. Control of burgers' equation by a reduced order approach using proper orthogonal decomposition. Journal on Optimization Theory and Applications 1999; 102:345-371.
11. Ravindran SS. A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. International Journal for Numerical Methods in Fluids 2000; 34:425-448.
12. Atwell JA, King BB. Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Mathematical and Computer Modelling 2001; 33(1-3):1-19.
13. Willcox K, Peraire J. Balanced model reduction via the proper orthogonal decomposition. AIAA Journal 2002; 40(11):2323.
14. Meyer M, Matthies HG. Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods. Computational Mechanics 2003; 31(1-2):179-191.
15. Machiels L, Maday Y, Oliveira IB, Patera AT, Rovas D. Output hounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. Comptes Rendus de'l Academie des Sciences Paris, Séries I 2000; 331(2):153-158.
16. Maday Y, Patera AT, Turinici G. Global a priori convergence theory for reduced-basis approximation of single-parameter symmetric coercive elliptic partial differential equations. Comptes Rendus de'l Academie des Sciences Paris, Séries I 2002; 335(3):289-294.
17. Veroy K, Rovas D, Patera AT. 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 2002; 8:1007-1028. Special volume: a tribute to J.-L. Lions.
18. Grepl MA, Nguyen NC, Patera KAT, Liu GR. Certified rapid solution of parametrized partial differential equations for real-time applications. Proceedings of the 2nd Sandia Workshop of PDE-Constrained Optimization: Towards Real-time and On-line PDE-constrained Optimization, SIAM Computational Science and Engineering Book Series. SIAM: Philadelphia, PA, 2007; 197-212.
19. Grepl MA, Patera AT. Reduced-basis approximation for time-dependent parametrized partial differential equations. Mathematical Modelling and Numerical Analysis 2005; 39:157-181.
20. Veroy K, Prud'homme C, Rovas DV, Patera AT. A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations (AIAA Paper 2003-3847). Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, Orlando, Florida, 2003.
21. Nguyen NC, Veroy K, Patera AT. Certified real-time solution of parametrized partial differential equations. In Handbook of Materials Modelling, Yip S (ed.). Springer: Berlin, 2005; 1523-1559.
22. Veroy K, Patera AT. Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations; rigorous reduced-basis a posteriori error bounds. International Journal for Numerical Methods in Fluids 2004; 47:773-788.
23. Bui-Thanh T, Willcox K, Ghattas O, van Bloemen Waanders B. Goal-oriented, model-constrained optimization for reduction of large-scale systems. Journal of Computational Physics 2007; 224:880-896.
24. Hinze M, Volkwein S. Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control. Lecture Notes in Computational Science and Engineering, vol. 45. Springer: Berlin, 2005; 45:261-306.
25. Graham WR, Peraire J, Tang KY. Optimal control of vortex shedding using low order models - Part I: Open-loop model development. International Journal for Numerical Methods in Engineering 1999; 44:945-972.
26. Gunzburger M, Peterson J, Shadid J. Reduced-order modeling of time-dependent pdes with multiple parameters in the boundary data. Computer Methods in Applied Mechanics and Engineering 2007; 196:1030-1047.
27. Bai ZJ. Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Applied Numerical Mathematics 2002; 43(1-2):9-44.
28. Rewienski M, White J. A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. IEEE Transactions on Computer-Aided Design of Integrated Circuit and Systems 2003; 22:155-170.
29. Scherpen JMA. Balancing for nonlinear systems. Systems and Control Letters 1993; 21:143-153.
30. Chen Y, White J. A quadratic method for nonlinear model order reduction. Proceedings of the International Conference on Modelling and Simulation of Microsystems, San Diego, CA, 2000; 477-480.
31. Phillips J. Projection-based approaches for model reduction of weakly nonlinear systems, time-varying systems. IEEE Transactions on Computer-Aided Design of Integrated Circuit and Systems 2003; 22:171-187.
32. Barrault M, Maday Y, Nguyen NC, Patera AT. An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus de'l Academie des Sciences Paris, Séries I 2004; 339:667-672.
33. Grepl MA, Maday Y, Nguyen NC, Patera AT. Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Mathematical Modelling and Numerical Analysis 2007; 41:575-605.
34. Nguyen NC, Patera AT, Peraire J. A best points interpolation method for efficient approximation of parametrized functions. International Journal of Numerical Methods in Engineering, accepted.
35. Maday Y, Nguyen NC, Patera AT, Pau GSH. A general, multipurpose interpolation procedure: the magic points. Constructive Approximation, submitted.
36. Karhunen K. Zur spektraltheorie stochastischer prozesse. Annales Academiae Scientiarum Fennicae 1946; 37:3-7.
37. Loeve MM. Probability Theory, Van Nostrand: Princeton, NJ, 1955.
38. Sirovich L. Turbulence and the dynamics of coherent structures. Part 1: Coherent structures. Quarterly of Applied Mathematics 1987; 45(3):561-571.
39. Marquardt DW. An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics 1963; 11(1):431-444.
40. Rozza G, Huynh D, Patera A. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics. Archives of Computational Methods in Engineering, submitted.
41. Nguyen NC. Reduced-basis approximations and a posteriori error bounds for nonaffine and nonlinear partial differential equations: application to inverse analysis. Ph.D. Thesis, Singapore-MIT Alliance, National University of Singapore, Singapore 2005.
42. Cancès E, LeBris C, Maday Y, Nguyen NC, Patera AT, Pau GSH. Feasibility and competitiveness of a reduced basis approach for rapid electronic structure calculations in quantum chemistry. Proceedings of the Montreal Workshop for High-dimensional Partial Differential Equations in Science and Engineering, vol. 41. CRM Proceedings Series. American Mathematical Society: Providence, RI, 2007; 15-47.
43. Grepl M. Reduced-basis approximations for time-dependent partial differential equations: application to optimal control. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, May 2005.