Abstract
In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.
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rbMIT Software: http://augustine.mit.edu/methodology/ methodology rbMIT System.htm. © MIT, Cambridge, MA (2007)
Ainsworth, M., Oden, J.T.:A posteriori error estimation in finite element analysis. Comp. Meth. Appl. Mech. Engrg.142, 1–88 (1997)
Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley-Interscience (2000)
Almroth, B.O., Stern, P., Brogan, F.A.: Automatic choice of global shape functions in structural analysis. AIAA Journal16, 525–528 (1978)
Anderson, T.L.: Fracture Mechanics: Fundamentals and Application, third edn. CRC (2005)
Arpaci, V.S.: Conduction heat transfer. Addison-Wesley (1966)
Arpaci, V.S., Larsen, P.S.: Convection heat transfer. Prentice Hall (1984)
Atwell, J.A., King, B.B.: Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Mathematical and Computer Modelling33(1-3), 1–19 (2001)
Babuška, I.: Error-bounds for finite element method. Numerische Mathematik16, 322–333 (1971)
Babuška, I., Osborn, J.: Eigenvalue problems. In: Handbook of Numerical Analysis, vol. II, pp. 641-787. Elsevier (1991)
Babuška, I., Rheinboldt, W.:A posteriori error estimates for the finite element method. Int. J. Numer. Meth. Eng.12, 1597–1615 (1978)
Babuška, I., Rheinboldt, W.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal.15, 736–754 (1978)
Babuška, I., Strouboulis, T.: The Finite Element Method and its Reliability. Numerical Mathematics and Scientific Computation. Clarendon Press, Oxford,UK (2001)
Bai, Z.J.: Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Applied Numerical Mathematics43(1-2), 9–44 (2002)
Balmes, E.: Parametric families of reduced finite element models: Theory and applications. Mechanical Systems and Signal Processing10(4), 381–394 (1996)
Balsa-Canto, E., Alonso, A., Banga, J.: Reduced-order models for nonlinear distributed process systems and their application in dynamic optimization. Industrial & Engineering Chemistry Research43(13), 3353–3363 (2004)
Banks, H.T., Kunisch, K.: Estimation Techniques for Distributed Parameter Systems. Systems & Control: Foundations & Applications. Birkhäuser (1989)
Barrault, M., Nguyen, N.C., Maday, Y., Patera, A.T.: An “empirical interpolation” method: Application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris, Série I.339, 667–672 (2004)
Barrett, A., Reddien, G.: On the reduced basis method. Z. Angew. Math. Mech.75(7), 543–549 (1995)
Barsom, J.M., Rolfe, S.T.: Fracture and Fatigue Control in Structures. American Society for Testing and Metals (1999)
Bashir, O., Willcox, K., Ghattas, O.: Hessian-based model reduction for large-scale systems with initial condition inputs. Int. J. for Num. Meth. in Engineering (2007). Submitted
Bathe, K.J.: Finite Element Procedures. Prentice Hall (1996)
Becker, R., Rannacher, R.: A feedback approach to error control in finite element method: Basic analysis and examples. East - West J. Numer. Math.4, 237–264 (1996)
Benner, P., Mehrmann, V., (Eds.), D.S.: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering. Springer, Heildeberg (2003)
Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis of Periodic Structures. North-Holland, Amsterdam (1978)
Boyaval, S.: Application of reduced basis approximation and a posteriori error estimation to homogenization theory. SIAM Multiscale Modeling and Simulation (2007). Submitted
Braess, D.: Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, UK (2001)
Brezzi, F.: On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers. R.A.I.R.O., Anal. Numér.2, 129–151 (1974)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods,Springer Series in Computational Mathematics, vol. 15. Springer Verlag (1991)
Brezzi, F., Rappaz, J., Raviart, P.: Finite dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions. Numerische Mathematik36, 1–25 (1980)
Bui-Thanh, T., Damodaran, M., Willcox, K.: Proper orthogonal decomposition extensions for parametric applications in transonic aerodynamics (AIAA Paper 2003-4213). In: Proceedings of the 15th AIAA Computational Fluid Dynamics Conference (2003)
Bui-Thanh, T., Willcox, K., Ghattas, O.: Model reduction for large-scale systems with high-dimensional parametric input space (AIAA Paper 2007-2049). In: Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conference (2007)
Caloz, G., Rappaz, J.: Numerical analysis for nonlinear and bifurcation problems. In: P. Ciarlet, J. Lions (eds.) Handbook of Numerical Analysis, Vol. V, Techniques of Scientific Computing (Part 2), pp. 487-637. Elsevier Science B.V. (1997)
Cancès, E., Le Bris, C., Maday, Y., Turinici, G.: Towards reduced basis approaches inab initio electronic structure computations. J. Sci. Comput.17(1-4), 461–469 (2002)
Cancès, E., Le Bris, C., Nguyen, N.C., Maday, Y., Patera, A.T., Pau, G.S.H.: Feasibility and competitiveness of a reduced basis approach for rapid electronic structure calculations in quantum chemistry. In: Proceedings of the Workshop for High-dimensional Partial Differential Equations in Science and Engineering (Montreal) (2007)
Cazemier, W.: Proper Orthogonal Decomposition and Low Dimensional Models for Turbolent Flows. University of Groningen (1997)
Chen, J., Kang, S.M.: Model-order reduction of nonlinear mems devices through arclength-based Karhunen-Loéve decomposition. In: Proceeding of the IEEE international Symposium on Circuits and Systems, vol. 2, pp. 457-460 (2001)
Chen, Y., White, J.: A quadratic method for nonlinear model order reduction. In: Proceedings of the international Conference on Modeling and Simulation of Microsystems, pp. 477-480 (2000)
Christensen, E., Brøns, M., Sørensen, J.: Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J. Scientific Computing21(4), 1419–1434 (2000)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, 40. SIAM (2002)
Daniel, L., Ong, C., White, J.: Geometrically parametrized interconnect performance models for interconnect synthesis. In: Proceedings of the 2002 International Symposium on Physical Design, ACM press, pp. 202-207 (2002)
Dedè, L.: Advanced numerical methods for the solution of optimal control problems described by pdes with environmental applications. Ph.D. thesis, Politecnico di Milano (In progress)
Demmel, J.W.: Applied Numerical Linear Algebra. SIAM (1997)
Deparis, S.: Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach. SIAM Journal of Numerical Analysis (2007). Submitted
Farle, O., Hill, V., Nickel, P., Dyczij-Edlinger, R.: Multivariate finite element model order reduction for permittivity or permeability estimation. IEEE Transactions on Megnetics42, 623–626 (2006)
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(1), 21–28 (1983)
Fox, R.L., Miura, H.: An approximate analysis technique for design calculations. AIAA Journal9(1), 177–179 (1971)
Ganapathysubramanian, S., Zabaras, N.: Design across length scales: a reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties. Computer Methods in Applied Mechanics and Engineering193, 5017–5034 (2004)
Girault, V., Raviart, P.: Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag (1986)
Goberna, M.A., Lopez, M.A.: Linear Semi-Infinite Optimization. J.Wiley, New York (1998)
Grepl, M.: Reduced-basis approximations and aposteriori error estimation for parabolic partial differential equations. Ph.D. thesis, Massachusetts Institute of Technology (2005)
Grepl, M.A., Maday, Y., Nguyen, N.C., Patera, A.T.: Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. M2AN (Math. Model. Numer. Anal.) (2007)
Grepl, M.A., Nguyen, N.C., Veroy, K., Patera, A.T., Liu, G.R.: Certified rapid solution of partial differential equations for real-time parameter estimation and optimization. In: L.T. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes, B. van B. Wandeers (eds.) Proceedings of the 2nd Sandia Workshop of PDE-Constrained Optimization: Real-Time PDE-Constrained Optimization, SIAM Computational Science and Engineering Book Series, pp. 197-216 (2007)
Grepl, M.A., Patera, A.T.:A Posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. M2AN (Math. Model. Numer. Anal.)39(1), 157–181 (2005)
Gresho, P., Sani, R.: Incompressible Flow and the Finite Element Method: Advection-Diffusion and Isothermal Laminar Flow. John Wiley & Sons (1998)
Gunzburger, M.D.: Finite Element Methods for Viscous Incompressible Flows. Academic Press (1989)
Gunzburger, M.D.: Perspectives in Flow Control and Optimization. Advances in Design and Control. SIAM (2003)
Gunzburger, M.D., Peterson, J., Shadid, J.N.: Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Comp. Meth. Applied Mech.196, 1030–1047 (2007)
Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized evolution equations. Mathematical Modelling and Numerical Analysis (M2AN) (2006). Submitted
Huynh, D.B.P.: Reduced-basis approximation and application to fracture and inverse problems. Ph.D. thesis, Singapore-MIT Alliance, National University of Singapore (2007)
Huynh, D.B.P., Patera, A.T.: Reduced-basis approximation anda posteriori error estimation for stress intensity factors. Int. J. Num. Meth. Eng. (2007). In press (DOI: 10.1002/nme.2090)
Huynh, D.B.P., Peraire, J., Patera, A.T., Liu, G.R.: Reduced basis approximation and a posteriori error estimation for stress intensity factors: Application to failure analysis. In: Singapore-MIT Alliance Symposium (2007)
Huynh, D.B.P., Rozza, G., Sen, S., Patera, A.T.: A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Acad. Sci. Paris, Analyse Numérique (2007). Submitted
Isaacson, E., Keller, H.B.: Computation of Eigenvalues and Eigenvectors, Analysis of Numerical Methods. Dover Publications, New York (1994)
Ito, K., Ravindran, S.S.: A reduced basis method for control problems governed by PDEs. In: W. Desch, F. Kappel, K. Kunisch (eds.) Control and Estimation of Distributed Parameter Systems, pp. 153-168. Birkhäuser (1998)
Ito, K., Ravindran, S.S.: A reduced-order method for simulation and control of fluid flows. Journal of Computational Physics143(2), 403–425 (1998)
Ito, K., Ravindran, S.S.: Reduced basis method for optimal control of unsteady viscous flows. International Journal of Computational Fluid Dynamics15(2), 97–113 (2001)
Ito, K., Schroeter, J.D.: Reduced order feedback synthesis for viscous incompressible flows. Mathematical And Computer Modelling33(1-3), 173–192 (2001)
Jabbar, M., Azeman, A.: Fast optimization of electromagnetic-problems:the reduced-basis finite element approach. IEEE Transactions on Magnetics40(4), 2161–2163 (2004)
Johnson, C.R.: A Gershgorin-type lower bound for the smallest singular value. Linear Algebra and Appl112, 1–7 (1989)
Karhunen, K.: Zur spektraltheorie stochastischer prozesse. Annales Academiae Scientiarum Fennicae 37 (1946)
Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Num. Analysis40(2), 492–515 (2002)
Kwang, A.T.Y.: Reduced basis methods for 2nd order wave equation: Application to one dimensional seismic problem. Master’s thesis, Singapore-MIT Alliance, Computation for Design and Optimization (2006)
Le Bris, C.: Private Communication. MIT (2006)
Lee, M.Y.L.: Estimation of the error in the reducedbasis method solution of differential algebraic equations. SIAM Journal of Numerical Analysis28, 512–528 (1991)
LeGresley, P.A., Alonso, J.J.: Airfoil design optimization using reduced order models based on proper orthogonal decomposition. In: Fluids 2000 Conference and Exhibit, Denver, CO (2000). Paper 2000-2545
Loeve, M.M.: Probability Theory. Van Nostrand (1955)
Løvgren, A.E., Maday, Y., Rønquist, E.M.: A reduced basis element method for complex flow systems. In: ECCOMAS CFD 2006 Proceedings, P. Wesseling, E. Oñate, J. Periaux (Eds.) TU Delft, The Netherlands (2006)
Løvgren, A.E., Maday, Y., Rønquist, E.M.: A reduced basis element method for the steady Stokes problem. Mathematical Modelling and Numerical Analysis (M2AN)40(3), 529–552 (2006)
Løvgren, A.E., Maday, Y., Rønquist, E.M.: A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems. Modeling, Identification and Control27(2), 79–94 (2006)
82. Løvgren, A.E., Maday, Y., Rønquist, E.M.: The reduced basis element method for fluid flows. Journal of Mathematical Fluid Mechanics (2007). In press
Ly, H., Tran, H.: Modeling and control of physical processes using proper orthogonal decomposition. Mathematical and Computer Modelling33,223–2366 (2001)
Machiels, L., Maday, Y., Oliveira, I.B., Patera, A., Rovas, D.: Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris, Série I331(2), 153–158 (2000)
Maday, Y., Patera, A., Turinici, G.: A Priori convergence theory for reduced-basis approximations of singleparameter elliptic partial differential equations. Journal of Scientific Computing17(1-4), 437–446 (2002)
Maday, Y., Patera, A.T., Rovas, D.V.: A blackbox reduced-basis output bound method for noncoercive linear problems. In: D. Cioranescu, J.L. Lions (eds.) Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar Volume XIV, pp. 533-569. Elsevier Science B.V. (2002)
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 I335(3), 289–294 (2002)
Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM (2000)
Meyer, M., Matthies, H.G.: Efficient model reduction in non-linear dynamics using the Karhunen-Léve expansion and dual-weighted-residual methods. Computational Mechanics31(1-2), 179–191 (2003)
Mortenson, M.E.: Computer graphics handbook. Industrial Press (1990)
Murakami, Y.: Stress Intensity Factors Handbook. Elsevier (2001)
Nagy, D.A.: Modal representation of geometrically nonlinear behaviour by the finite element method. Computers and Structures10, 683–688 (1979)
Newman, A.J.: Model reduction via the Karhunen-Loeve expansion part i: an exposition. Technical Report Institute for System Research University of Maryland (96-322) (1996)
Newman, J.N.: Marine Hydrodynamics. MIT Press, Cambridge, MA (1977)
Nguyen, N.C.: Reduced-basis approximation 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 (2005)
Nguyen, N.C., Patera, A.T.: Efficient and reliable parameter estimation in heat conduction using Bayesian inference and a reduced basis method (2007). In preparation
Nguyen, N.C., Veroy, K., Patera, A.T.: Certified realtime solution of parametrized partial differential equations. In: S. Yip (ed.) Handbook of Materials Modeling, pp. 1523-1558. Springer (2005)
Noor, A.K.: Recent advances in reduction methods for nonlinear problems. Comput. Struct.13, 31–44 (1981)
Noor, A.K.: On making large nonlinear problems small. Comp. Meth. Appl. Mech. Engrg.34, 955–985 (1982)
Noor, A.K., Balch, C.D., Shibut, M.A.: Reduction methods for non-linear steady-state thermal analysis. Int. J. Num. Meth. Engrg.20, 1323–1348 (1984)
Noor, A.K., Peters, J.M.: Reduced basis technique for nonlinear analysis of structures. AIAA Journal18(4), 455–462 (1980)
Noor, A.K., Peters, J.M.: Bifurcation and post-buckling analysis of laminated composite plates via reduced basis techniques. Comp. Meth. Appl. Mech. Engrg.29, 271–295 (1981)
Noor, A.K., Peters, J.M.: Tracing post-limit-point paths with reduced basis technique. Comp. Meth. Appl. Mech. Engrg.28, 217–240 (1981)
Noor, A.K., Peters, J.M.: Multiple-parameter reduced basis technique for bifurcation and post-buckling analysis of composite plates. Int. J. Num. Meth. Engrg.19, 1783–1803 (1983)
Noor, A.K., Peters, J.M.: Recent advances in reduction methods for instability analysis of structures. Comput. Struct.16, 67–80 (1983)
Noor, A.K., Peters, J.M., Andersen, C.M.: Mixed models and reduction techniques for large-rotation nonlinear problems. Comp. Meth. Appl. Mech. Engrg.44, 67–89 (1984)
Oliveira, I., Patera, A.T.: Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations. Optim. Eng.8, 43–65 (2007)
Paraschivoiu, M., Peraire, J., Maday, Y., Patera, A.T.: Fast bounds for outputs of partial differential equations. In: J. Borgaard, J. Burns, E. Cliff, S. Schreck (eds.) Computational methods for optimal design and control, pp. 323-360. Birkhäuser (1998)
Parks, D.M.: A stiffness derivative finite element technique for determination of crack tip stress intensity factors. International Journal of Fracture10(4), 487–502 (1974)
Parlett, B.N.: The Symmetric Eigenvalue Problem. Society for Industrial and Applied Mathematics, Philadelphia (1998)
Patera, A.T., Rønquist, E.M.: Reduced basis approximations and a posteriori error estimation for a Boltzmann model. Computer Methods in Applied Mechanics and Engineering196, 2925–2942 (2007)
Patera, A.T., Rozza, G.: Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. Copyright MIT (2006-2007). To appear in MIT Pappalardo Monographs in Mechanical Engineering
Pau, G.S.H.: Reduced-basis method for quantum models of periodic solids. Ph.D. thesis, Massachusetts Institute of Technology. (2007)
Peterson, J.S.: The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput.10(4), 777–786 (1989)
Phillips, J.R.: Projection frameworks for model reduction of weakly nonlinear systems. In: Proceeding of the 37th ACM/IEEE Design Automation Conference, pp. 184-189 (2000)
Phillips, J.R.: Projection-based approaches for model reduction of weakly nonlinear systems, time-varying systems. In: IEEE Transactions On Computer-Aided Design of Integrated Circuit and Systems, vol. 22, pp. 171-187 (2003)
Pierce, N., Giles, M.B.: Adjoint recovery of superconvergent functionals from PDE approximations. SIAM Review42(2),247–2644 (2000)
Pironneau, O.: Calibration of barrier options. In: W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau, Y. Vassilevski (eds.) Advances in Numerical Mathematics, pp. 183–192. Moscow, Institute of Numerical Mathematics, Russian Academy of Sciences and Houston, Department of Mathematics, University of Houston (2006)
Porsching, T.A.: Estimation of the error in the reduced basis method solution of nonlinear equations. Mathematics of Computation45(172), 487–496 (1985)
Porsching, T.A., Lee, M.Y.L.: The reduced-basis method for initial value problems. SIAM Journal of Numerical Analysis24, 1277–1287 (1987)
Prud’homme, C., Rovas, D., Veroy, K., Maday, Y., Patera, A., Turinici, G.: Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bounds methods. Journal of Fluids Engineering124(1), 70–80 (2002)
Prud’homme, C., Rovas, D., Veroy, K., Patera, A.T.: A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. M2AN Math. Model. Numer. Anal.36(5), 747–771 (2002)
Quarteroni, A., Rozza, G.: Numerical solution of parametrized Navier-Stokes equations by reduced basis method. Num. Meth. PDEs23, 923–948 (2007)
Quarteroni, A., Rozza, G., Quaini, A.: Reduced basis method for optimal control af advection-diffusion processes. In: W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau, Y. Vassilevski (eds.) Advances in Numerical Mathematics, pp. 193-216. Moscow, Institute of Numerical Mathematics, Russian Academy of Sciences and Houston, Department of Mathematics, University of Houston (2006)
Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, Texts in Applied Mathematics, vol. 37. Springer, New York (2000)
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, 2nd edn. Springer (1997)
Ravindran, S.S.: Reduced-order adaptive controllers for fluid flows using pod. J. of Scientific Computing15(4), 457–478 (2000)
Ravindran, S.S.: A reduced order approach to optimal control of fluids flow using proper orthogonal decomposition. Int. J. of Numerical Methods in Fluids34(5), 425–448 (2000)
Ravindran, S.S.: Adaptive reduced-order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J. Sci. Comput.23(6), 1924–1942 (2002)
Rewienski, M., White, J.: A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. In: IEEE Transactions On Computer-Aided Design of Integrated Circuit and Systems, vol. 22, pp. 155-170 (2003)
Rheinboldt, W.C.: Numerical analysis of continuation methods for nonlinear structural problems. Computers and Structures13(1-3), 103–113 (1981)
Rheinboldt, W.C.: On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Analysis, Theory, Methods and Applications21(11), 849–858 (1993)
Rovas, D., Machiels, L., Maday, Y.: Reduced basis output bounds methods for parabolic problems. IMA J. Appl. Math. (2005)
Rovas, D.V.: Reduced-basis output bound methods for parametrized partial differential equations. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (2002)
Rozza, G.: Real-time reduced basis techniques for arterial bypass geometries. In: K. Bathe (ed.) Computational Fluid and Solid Mechanics, pp. 1283-1287. Elsevier (2005). Proceedings of the Third M.I.T. Conference on Computational Fluid and Solid Mechanics, June 14-17, 2005
Rozza, G.: Shape design by optimal flow control and reduced basis techniques: Applications to bypass configurations in haemodynamics. Ph.D. thesis, EPFL, Ecole Polytechnique Federale de Lausanne (2005)
Rozza, G.: Reduced basis method for Stokes equations in domains with non-affine parametric dependence. Comp. Vis. Science (2007). In press (DOI: 10.1007/s00791-006-0044-7)
Rozza, G.: Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity. Appl. Numer. Math.55(4), 403–424 (2007)
Rozza, G., Veroy, K.: On the stability of reduced basis method for Stokes equations in parametrized domains. Comp. Meth. Appl. Mech. and Eng.196, 1244–1260 (2007)
Schiesser, W.E., Silebi, C.A.: Computational Transport Phenomena: Numerical Methods for the Solution of Transport Problems. Cambridge Univeristy Press (1997)
Sen, S.: Reduced-basis approximation and a posteriori error estimation for non-coercive eliptic problems: Application to acoustics. Ph.D. thesis, Massachusetts Institute of Technology. (2007)
Sen, S., Veroy, K., Huynh, D.B.P., Deparis, S., Nguyen, N.C., Patera, A.T.: ‘Natural norm’ a posteriori error estimators for reduced basis approximations. Journal of Computational Physics217, 37–62 (2006)
Shi, G., Shi, C.J.R.: Parametric model order reduction for interconnect analysis. In: Proceedings of the 2004 Conference on Asia South Pacific design automation: electronic design and solution fair, IEEE press, pp. 774-779 (2004)
Sirisup, S., Xiu, D., Karniadakis, G.: Equationfree/ Galerkin-free POD-assisted computation of incompressible flows. Journal of Computational Physics207, 617–642 (2005)
Sirovich, L.: Turbulence and the dynamics of coherent structures, part 1: Coherent structures. Quarterly of Applied Mathematics45(3), 561–571 (1987)
Strang, G.: Introduction to linear algebra. Wellesley-Cambridge Press (2003)
Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall (1973)
Tonn, T., Urban, K.: A reduced-basis method for solving parameter-dependent convection-diffusion problems around rigid bodies. In: ECCOMAS CFD 2006 Proceedings, P. Wesseling, E. Oñate, J. Periaux (Eds.) TU Delft, The Netherlands (2006)
Trefethen, L., III, D.B.: Numerical Linear Algebra. SIAM (1997)
Veroy, K.: Reduced-basis methods applied to problems in elasticity: Analysis and applications. Ph.D. thesis, Massachusetts Institute of Technology (2003)
Veroy, K., Patera, A.T.: Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations; Rigorous reduced-basisa posteriori error bounds. International Journal for Numerical Methods in Fluids47, 773–788 (2005)
Veroy, K., Prud’homme, C., Patera, A.T.: Reducedbasis approximation of the viscous Burgers equation: Rigorous a posteriori error bounds. C. R. Acad. Sci. Paris, Série I337(9), 619–624 (2003)
Veroy, K., Prud’homme, C., Rovas, D.V., Patera, A.T.: A Posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: Proceedings of the 16th AIAA Computational Fluid Dynamics Conference (2003). Paper 2003-3847
Wang, J., Zabaras, N.: Using Bayesian statistics in the estimation of heat source in radiation. International Journal of Heat and Mass Transfer48, 15–29 (2005)
Weile, D.S., Michielssen, E.: Analysis of frequency selective surfaces using two-parameter generalized rational Krylov model-order reduction. IEEE Transactions on Antennas and Propagation49(11), 1539–1549 (2001)
Weile, D.S., Michielssen, E., Gallivan, K.: Reducedorder modeling of multiscreen frequency-selective surfaces using Krylov-based rational interpolation. IEEE Transactions on Antennas and Propagation49(5), 801–813 (2001)
Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. AIAA Journal40(11), 2323–2330 (2002)
Zienkiewicz, O., Taylor, R.: Finite Element Method: Volume 1. The Basis. Butterworth-Heinemann, London (2000)
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This work was supported by DARPA/AFOSR Grants FA9550-05-1-0114 and FA-9550-07-1-0425,the Singapore-MIT Alliance,the Pappalardo MIT Mechanical Engineering Graduate Monograph Fund,and the Progetto Roberto Rocca Politecnico di Milano-MIT.We acknowledge many helpful discussions with Professor Yvon Maday of University Paris6.
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Rozza, G., Huynh, D.B.P. & Patera, A.T. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. ARCO 15, 1–47 (2007). https://doi.org/10.1007/BF03024948
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DOI: https://doi.org/10.1007/BF03024948
Keywords
- Partial differential
- equations
- parameter
- variation
- affine geometry description
- Galerkin approximation
- a posteriori error estimation,reduced basis
- reduced order model,sampling strategies
- POD
- greedy techniques
- offline-online procedures
- marginal cost
- coercivity lower bound
- successive constraint method
- real-time computation
- many-query