Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Application to transport and continuum mechanics. (English) Zbl 1304.65251
Summary: We consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations. The essential ingredients are (primal-dual) Galerkin projection onto a low-dimensional space associated with a smooth “parametric manifold”-dimension reduction; efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations-rapid convergence; a posteriori error estimation procedures-rigorous and sharp bounds for the linear-functional outputs of interest; and Offline-Online computational decomposition strategies-minimum marginal 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.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
Software:
rbMITReferences:
| [1] | rbMIT Software (2007) http://augustine.mit.edu/methodology/methodology_rbMIT_System.htm . MIT, Cambridge |
| [2] | Ainsworth M, Oden JT (1997) A posteriori error estimation in finite element analysis. Comput Methods Appl Mech Eng 142:1–88 · Zbl 0895.76040 · doi:10.1016/S0045-7825(96)01107-3 |
| [3] | Ainsworth M, Oden JT (2000) A posteriori error estimation in finite element analysis. Wiley-Interscience, New York · Zbl 1008.65076 |
| [4] | Almroth BO, Stern P, Brogan FA (1978) Automatic choice of global shape functions in structural analysis. AIAA J 16:525–528 · doi:10.2514/3.7539 |
| [5] | Anderson TL (2005) Fracture mechanics: fundamentals and application, 3rd edn. CRC Press, Boca Raton |
| [6] | Arpaci VS (1966) Conduction heat transfer. Addison-Wesley, Reading · Zbl 0144.46703 |
| [7] | Arpaci VS, Larsen PS (1984) Convection heat transfer. Prentice Hall, Englewood Cliffs |
| [8] | Atwell JA, King BB (2001) Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math Comput Model 33(1–3):1–19 · Zbl 0964.93032 · doi:10.1016/S0895-7177(00)00225-9 |
| [9] | Babuška I (1971) Error-bounds for finite element method. Numer Math 16:322–333 · Zbl 0214.42001 · doi:10.1007/BF02165003 |
| [10] | Babuška I, Osborn J (1991) Eigenvalue problems. In: Handbook of numerical analysis, vol II. Elsevier, Amsterdam, pp 641–787 · Zbl 0875.65087 |
| [11] | Babuška I, Rheinboldt W (1978) A posteriori error estimates for the finite element method. Int J Numer Methods Eng 12:1597–1615 · Zbl 0396.65068 · doi:10.1002/nme.1620121010 |
| [12] | Babuška I, Rheinboldt W (1978) Error estimates for adaptive finite element computations. SIAM J Numer Anal 15:736–754 · Zbl 0398.65069 · doi:10.1137/0715049 |
| [13] | Babuška I, Strouboulis T (2001) The finite element method and its reliability. Numerical mathematics and scientific computation. Clarendon, Oxford |
| [14] | Bai ZJ (2002) Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl Numer Math 43(1–2):9–44 · Zbl 1012.65136 · doi:10.1016/S0168-9274(02)00116-2 |
| [15] | Balmes E (1996) Parametric families of reduced finite element models: Theory and applications. Mech Syst Signal Process 10(4):381–394 · doi:10.1006/mssp.1996.0027 |
| [16] | Balsa-Canto E, Alonso A, Banga J (2004) Reduced-order models for nonlinear distributed process systems and their application in dynamic optimization. Ind Eng Chem Res 43(13):3353–3363 · doi:10.1021/ie049946y |
| [17] | Banks HT, Kunisch K (1989) Estimation techniques for distributed parameter systems. Systems & control: foundations & applications. Birkhäuser, Boston · Zbl 0695.93020 |
| [18] | 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 · Zbl 1061.65118 |
| [19] | Barrett A, Reddien G (1995) On the reduced basis method. Z Angew Math Mech 75(7):543–549 · Zbl 0832.65047 |
| [20] | Barsom JM, Rolfe ST (1999) Fracture and fatigue control in structures. American society for testing and metals. Butterworth, Stoneham |
| [21] | Bashir O, Willcox K, Ghattas O, var Bloemen Waanders B, Hill J (2008) Hessian-based model reduction for large-scale systems with initial condition inputs. Int J Numer Methods Eng 73(6):844–868 · Zbl 1195.76311 · doi:10.1002/nme.2100 |
| [22] | Bathe KJ (1996) Finite element procedures. Prentice Hall, Englewood Cliffs |
| [23] | Becker R, Rannacher R (1996) A feedback approach to error control in finite element method: Basic analysis and examples. East-West J Numer Math 4:237–264 · Zbl 0868.65076 |
| [24] | Benner P, Mehrmann V, Sorensen D (eds) (2003) Dimension reduction of large-scale systems. Lecture notes in computational science and engineering. Springer, Heidelberg |
| [25] | Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis of periodic structures. North-Holland, Amsterdam · Zbl 0404.35001 |
| [26] | Boyaval S (2007) Application of reduced basis approximation and a posteriori error estimation to homogenization theory. Multiscale Model Simul (to appear) |
| [27] | Braess D (2001) Finite elements. Theory, fast solvers, and applications in solid mechanics. Cambridge University Press, Cambridge · Zbl 0976.65099 |
| [28] | Brezzi F (1974) On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO Anal Numer 2:129–151 · Zbl 0338.90047 |
| [29] | Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer series in computational mathematics, vol 15. Springer, Berlin · Zbl 0788.73002 |
| [30] | Brezzi F, Rappaz J, Raviart P (1980) Finite dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions. Numer Math 36:1–25 · Zbl 0488.65021 · doi:10.1007/BF01395985 |
| [31] | Bui-Thanh T, Damodaran M, Willcox K (2003) Proper orthogonal decomposition extensions for parametric applications in transonic aerodynamics (AIAA Paper 2003-4213). In: Proceedings of the 15th AIAA computational fluid dynamics conference |
| [32] | Bui-Thanh T, Willcox K, Ghattas O (2008) Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J Sci Comput (to appear) · Zbl 1196.37127 |
| [33] | Bui-Thanh T, Willcox K, Ghattas O (2007) 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 · Zbl 1196.37127 |
| [34] | Caloz G, Rappaz J (1997) Numerical analysis for nonlinear and bifurcation problems. In: Ciarlet P, Lions J (eds) Handbook of numerical analysis. Techniques of scientific computing (Part 2), vol V. Elsevier, Amsterdam, pp 487–637 |
| [35] | Cancès E, Le Bris C, Maday Y, Turinici G (2002) Towards reduced basis approaches in ab initio electronic structure computations. J Sci Comput 17(1–4):461–469 · Zbl 0998.81122 · doi:10.1023/A:1015150025426 |
| [36] | Cancès E, Le Bris C, Nguyen NC, Maday Y, Patera AT, Pau GSH (2007) 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) · Zbl 1330.81022 |
| [37] | Cazemier W (1997) Proper orthogonal decomposition and low dimensional models for turbolent flows. University of Groningen, Groningen |
| [38] | Chen J, Kang SM (2001) 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 |
| [39] | Chen Y, White J (2000) A quadratic method for nonlinear model order reduction. In: Proceedings of the international conference on modeling and simulation of microsystems, pp 477–480 |
| [40] | Christensen E, Brøns M, Sørensen J (2000) Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J Sci Comput 21(4):1419–1434 · Zbl 0959.35018 · doi:10.1137/S1064827598333181 |
| [41] | Ciarlet PG (2002) The finite element method for elliptic problems. Classics in applied mathematics, vol 40. SIAM, Philadelphia |
| [42] | Daniel L, Ong C, White J (2002) Geometrically parametrized interconnect performance models for interconnect synthesis. In: Proceedings of the 2002 international symposium on physical design. Assoc Comput Mach, New York, pp 202–207 |
| [43] | Dedè L (2008) Advanced numerical methods for the solution of optimal control problems described by pdes with environmental applications. PhD thesis, Politecnico di Milano |
| [44] | Demmel JW (1997) Applied numerical linear algebra. SIAM, Philadelphia · Zbl 0879.65017 |
| [45] | Deparis S (2008) Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural norm approach. SIAM J Numer Anal 46:2039 · Zbl 1177.35148 · doi:10.1137/060674181 |
| [46] | Farle O, Hill V, Nickel P, Dyczij-Edlinger R (2006) Multivariate finite element model order reduction for permittivity or permeability estimation. IEEE Trans Megn 42:623–626 · doi:10.1109/TMAG.2006.872492 |
| [47] | Fink JP, Rheinboldt WC (1983) On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z Angew Math Mech 63(1):21–28 · Zbl 0533.73071 · doi:10.1002/zamm.19830630105 |
| [48] | Fox RL, Miura H (1971) An approximate analysis technique for design calculations. AIAA J 9(1):177–179 · doi:10.2514/3.6141 |
| [49] | Ganapathysubramanian S, Zabaras N (2004) Design across length scales: a reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties. Comput Methods Appl Mech Eng 193:5017–5034 · Zbl 1112.74353 · doi:10.1016/j.cma.2004.04.004 |
| [50] | Girault V, Raviart P (1986) Finite element approximation of the Navier-Stokes equations. Springer, Berlin · Zbl 0585.65077 |
| [51] | Goberna MA, Lopez MA (1998) Linear semi-infinite optimization. Wiley, New York |
| [52] | Grepl M (2005) Reduced-basis approximations and a posteriori error estimation for parabolic partial differential equations. PhD thesis, Massachusetts Institute of Technology · Zbl 1079.65096 |
| [53] | Grepl MA, Maday Y, Nguyen NC, Patera AT (2007) Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Model Math Anal Numer · Zbl 1142.65078 |
| [54] | Grepl MA, Nguyen NC, Veroy K, Patera AT, Liu GR (2007) Certified rapid solution of partial differential equations for real-time parameter estimation and optimization. In: Biegler LT, Ghattas O, Heinkenschloss M, Keyes D, van Wandeers B (eds) Proceedings of the 2nd Sandia workshop of PDE-constrained optimization: Real-time PDE-constrained optimization. SIAM computational science and engineering book series. SIAM, Philadelphia, pp 197–216 |
| [55] | Grepl MA, Patera AT (2005) A Posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. Model Math Anal Numer 39(1):157–181 · Zbl 1079.65096 · doi:10.1051/m2an:2005006 |
| [56] | Gresho P, Sani R (1998) Incompressible flow and the finite element method: advection-diffusion and isothermal laminar flow. Wiley, New York · Zbl 0941.76002 |
| [57] | Gunzburger MD (1989) Finite element methods for viscous incompressible flows. Academic Press, San Diego · Zbl 0697.76031 |
| [58] | Gunzburger MD (2003) Perspectives in flow control and optimization. Advances in design and control. SIAM, Philadelphia |
| [59] | Gunzburger MD, Peterson J, Shadid JN (2007) Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Comput Methods Appl Mech 196:1030–1047 · Zbl 1121.65354 · doi:10.1016/j.cma.2006.08.004 |
| [60] | Haasdonk B, Ohlberger M (2008) Reduced basis method for finite volume approximations of parametrized evolution equations. Math Model Numer Anal 42(2):277–302 · Zbl 1388.76177 · doi:10.1051/m2an:2008001 |
| [61] | Huynh DBP (2007) Reduced-basis approximation and application to fracture and inverse problems. PhD thesis, Singapore-MIT Alliance, National University of Singapore |
| [62] | Huynh DBP, Patera AT (2007) Reduced-basis approximation and a posteriori error estimation for stress intensity factors. Int J Numer Methods Eng 72(10):1219–1259 · Zbl 1194.74413 · doi:10.1002/nme.2090 |
| [63] | Huynh DBP, Peraire J, Patera AT, Liu GR (2007) Reduced basis approximation and a posteriori error estimation for stress intensity factors: Application to failure analysis. In: Singapore-MIT alliance symposium |
| [64] | Huynh DBP, Rozza G, Sen S, Patera AT (2007) A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C R Acad Sci Paris Ser I 345:473–478 · Zbl 1127.65086 |
| [65] | Isaacson E, Keller HB (1994) Computation of eigenvalues and eigenvectors, analysis of numerical methods. Dover, New York |
| [66] | Ito K, Ravindran SS (1998) A reduced basis method for control problems governed by PDEs. In: Desch W, Kappel F, Kunisch K (eds) Control and estimation of distributed parameter systems. Birkhäuser, Boston, pp 153–168 · Zbl 0908.93025 |
| [67] | Ito K, Ravindran SS (1998) A reduced-order method for simulation and control of fluid flows. J Comput Phys 143(2):403–425 · Zbl 0936.76031 · doi:10.1006/jcph.1998.5943 |
| [68] | Ito K, Ravindran SS (2001) Reduced basis method for optimal control of unsteady viscous flows. Int J Comput Fluid Dyn 15(2):97–113 · Zbl 1036.76011 · doi:10.1080/10618560108970021 |
| [69] | Ito K, Schroeter JD (2001) Reduced order feedback synthesis for viscous incompressible flows. Math Comput Model 33(1–3):173–192 · Zbl 0967.93013 · doi:10.1016/S0895-7177(00)00237-5 |
| [70] | Jabbar M, Azeman A (2004) Fast optimization of electromagnetic-problems: the reduced-basis finite element approach. IEEE Trans Magn 40(4):2161–2163 · doi:10.1109/TMAG.2004.829184 |
| [71] | Johnson CR (1989) A Gershgorin-type lower bound for the smallest singular value. Linear Algebra Appl 112:1–7 · Zbl 0723.15013 · doi:10.1016/0024-3795(89)90583-1 |
| [72] | Karhunen K (1946) Zur spektraltheorie stochastischer prozesse. Ann Acad Sci Fenn 37 |
| [73] | Kunisch K, Volkwein S (2002) Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J Numer Anal 40(2):492–515 · Zbl 1075.65118 · doi:10.1137/S0036142900382612 |
| [74] | Kwang ATY (2006) 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 |
| [75] | Le Bris C (2006) Private communication. MIT |
| [76] | Lee MYL (1991) Estimation of the error in the reduced-basis method solution of differential algebraic equations. SIAM J Numer Anal 28:512–528 · Zbl 0737.65058 · doi:10.1137/0728028 |
| [77] | LeGresley PA, Alonso JJ (2000) Airfoil design optimization using reduced order models based on proper orthogonal decomposition. In: Fluids 2000 conference and exhibit. Denver, CO (2000). Paper 2000-2545 |
| [78] | Loeve MM (1955) Probability theory. Van Nostrand, Princeton · Zbl 0066.10903 |
| [79] | Løvgren AE, Maday Y, Rønquist EM (2006) A reduced basis element method for complex flow systems. In: Wesseling P, Oñate E, Periaux J (eds) ECCOMAS CFD 2006 proceedings. TU Delft, Delft · Zbl 1129.76036 |
| [80] | Løvgren AE, Maday Y, Rønquist EM (2006) A reduced basis element method for the steady Stokes problem. Math Model Numer Anal 40(3):529–552 · Zbl 1129.76036 · doi:10.1051/m2an:2006021 |
| [81] | Løvgren AE, Maday Y, Rønquist EM (2006) A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems. Model Identif Control 27(2):79–94 · doi:10.4173/mic.2006.2.1 |
| [82] | Løvgren AE, Maday Y, Rønquist EM (2007) The reduced basis element method for fluid flows. In: Analysis and simulation of fluid dynamics. Advances in mathematical fluid mechanics. Birkauser, Boston, pp 129–154 · Zbl 1291.76247 |
| [83] | Ly H, Tran H (2001) Modeling and control of physical processes using proper orthogonal decomposition. Math Comput Model 33:223–236 · Zbl 0966.93018 · doi:10.1016/S0895-7177(00)00240-5 |
| [84] | Machiels L, Maday Y, Oliveira IB, Patera A, Rovas D (2000) Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C R Acad Sci Paris, Sér I 331(2):153–158 · Zbl 0960.65063 |
| [85] | Maday Y, Patera A, 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 · Zbl 1014.65115 · doi:10.1023/A:1015145924517 |
| [86] | Maday Y, Patera AT, Rovas DV (2002) A blackbox reduced-basis output bound method for noncoercive linear problems. In: Cioranescu D, Lions JL (eds) Nonlinear partial differential equations and their applications, Collége de France Seminar, vol XIV. Elsevier, Amsterdam, pp 533–569 · Zbl 1006.65124 |
| [87] | Maday Y, Patera AT, Turinici G (2002) 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 335(3):289–294 · Zbl 1009.65066 |
| [88] | Meyer CD (2000) Matrix analysis and applied linear algebra. SIAM, Philadelphia |
| [89] | Meyer M, Matthies HG (2003) Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods. Comput Mech 31(1–2):179–191 · Zbl 1038.74559 · doi:10.1007/s00466-002-0404-1 |
| [90] | Mortenson ME (1990) Computer graphics handbook. Industrial Press, New York |
| [91] | Murakami Y (2001) Stress intensity factors handbook. Elsevier, Amsterdam |
| [92] | Nagy DA (1979) Modal representation of geometrically nonlinear behaviour by the finite element method. Comput Struct 10:683–688 · Zbl 0406.73071 · doi:10.1016/0045-7949(79)90012-9 |
| [93] | Newman AJ (1996) Model reduction via the Karhunen-Loeve expansion part i: an exposition. Technical report, Institute for System Research University of Maryland, pp 96–322 |
| [94] | Newman JN (1977) Marine hydrodynamics. MIT Press, Cambridge |
| [95] | Nguyen NC (2005) Reduced-basis approximation and a posteriori error bounds for nonaffine and nonlinear partial differential equations: Application to inverse analysis. PhD thesis, Singapore-MIT Alliance, National University of Singapore |
| [96] | Nguyen NC, Patera AT (2007) Efficient and reliable parameter estimation in heat conduction using Bayesian inference and a reduced basis method (in preparation) |
| [97] | 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 |
| [98] | Noor AK (1981) Recent advances in reduction methods for nonlinear problems. Comput Struct 13:31–44 · Zbl 0455.73080 · doi:10.1016/0045-7949(81)90106-1 |
| [99] | Noor AK (1982) On making large nonlinear problems small. Comput Methods Appl Mech Eng 34:955–985 · doi:10.1016/0045-7825(82)90096-2 |
| [100] | Noor AK, Balch CD, Shibut MA (1984) Reduction methods for non-linear steady-state thermal analysis. Int J Numer Methods Eng 20:1323–1348 · Zbl 0557.65076 · doi:10.1002/nme.1620200711 |
| [101] | Noor AK, Peters JM (1980) Reduced basis technique for nonlinear analysis of structures. AIAA J 18(4):455–462 · doi:10.2514/3.50778 |
| [102] | Noor AK, Peters JM (1981) Bifurcation and post-buckling analysis of laminated composite plates via reduced basis techniques. Comput Methods Appl Mech Eng 29:271–295 · Zbl 0474.73100 · doi:10.1016/0045-7825(81)90046-3 |
| [103] | Noor AK, Peters JM (1981) Tracing post-limit-point paths with reduced basis technique. Comput Methods Appl Mech Eng 28:217–240 · Zbl 0466.73090 · doi:10.1016/0045-7825(81)90106-7 |
| [104] | Noor AK, Peters JM (1983) Multiple-parameter reduced basis technique for bifurcation and post-buckling analysis of composite plates. Int J Numer Methods Eng 19:1783–1803 · Zbl 0557.73070 · doi:10.1002/nme.1620191206 |
| [105] | Noor AK, Peters JM (1983) Recent advances in reduction methods for instability analysis of structures. Comput Struct 16:67–80 · Zbl 0498.73094 · doi:10.1016/0045-7949(83)90148-7 |
| [106] | Noor AK, Peters JM, Andersen CM (1984) Mixed models and reduction techniques for large-rotation nonlinear problems. Comput Methods Appl Mech Eng 44:67–89 · doi:10.1016/0045-7825(84)90120-8 |
| [107] | Oliveira I, Patera AT (2007) Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations. Optim Eng 8:43–65 · Zbl 1171.65404 · doi:10.1007/s11081-007-9002-6 |
| [108] | Paraschivoiu M, Peraire J, Maday Y, Patera AT (1998) Fast bounds for outputs of partial differential equations. In: Borgaard J, Burns J, Cliff E, Schreck S (eds) Computational methods for optimal design and control. Birkhäuser, Boston, pp 323–360 · Zbl 0924.65057 |
| [109] | Parks DM (1974) A stiffness derivative finite element technique for determination of crack tip stress intensity factors. Int J Fract 10(4):487–502 · doi:10.1007/BF00155252 |
| [110] | Parlett BN (1998) The symmetric eigenvalue problem. SIAM, Philadelphia |
| [111] | Patera AT, Rønquist EM (2007) Reduced basis approximations and a posteriori error estimation for a Boltzmann model. Comput Methods Appl Mech Eng 196:2925–2942 · Zbl 1178.76236 · doi:10.1016/j.cma.2007.02.008 |
| [112] | Patera AT, Rozza G (2008) Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Copyright MIT (2006–2007). MIT Pappalardo monographs in mechanical engineering (to appear) |
| [113] | Pau GSH (2007) Reduced-basis method for quantum models of periodic solids. PhD thesis, Massachusetts Institute of Technology |
| [114] | Peterson JS (1989) The reduced basis method for incompressible viscous flow calculations. SIAM J Sci Stat Comput 10(4):777–786 · Zbl 0672.76034 · doi:10.1137/0910047 |
| [115] | Phillips JR (2000) Projection frameworks for model reduction of weakly nonlinear systems. In: Proceeding of the 37th ACM/IEEE Design Automation Conference, pp 184–189 |
| [116] | Phillips JR (2003) Projection-based approaches for model reduction of weakly nonlinear systems, time-varying systems. IEEE Trans Comput Aided Des Integr Circuits Syst 22:171–187 · doi:10.1109/TCAD.2002.806605 |
| [117] | Pierce N, Giles MB (2000) Adjoint recovery of superconvergent functionals from PDE approximations. SIAM Rev 42(2):247–264 · Zbl 0948.65119 · doi:10.1137/S0036144598349423 |
| [118] | Pironneau O (2006) Calibration of barrier options. In: Fitzgibbon W, Hoppe R, Periaux J, Pironneau O, Vassilevski Y (eds) Advances in numerical mathematics. Moscow/Houston, Russian Academy of Sciences/University of Houston, pp 183–192 |
| [119] | Porsching TA (1985) Estimation of the error in the reduced basis method solution of nonlinear equations. Math Comput 45(172):487–496 · Zbl 0586.65040 · doi:10.1090/S0025-5718-1985-0804937-0 |
| [120] | Porsching TA, Lee MYL (1987) The reduced-basis method for initial value problems. SIAM J Numer Anal 24:1277–1287 · Zbl 0639.65039 · doi:10.1137/0724083 |
| [121] | Prud’homme C, Rovas D, Veroy K, Maday Y, Patera A, Turinici G (2002) Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bounds methods. J Fluids Eng 124(1):70–80 · doi:10.1115/1.1448332 |
| [122] | Prud’homme C, Rovas D, Veroy K, Patera AT (2002) A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. Model Math Anal Numer 36(5):747–771 · Zbl 1024.65104 · doi:10.1051/m2an:2002035 |
| [123] | Quarteroni A, Rozza G (2007) Numerical solution of parametrized Navier-Stokes equations by reduced basis method. Numer Methods Partial Differ Equ 23:923–948 · Zbl 1178.76238 · doi:10.1002/num.20249 |
| [124] | Quarteroni A, Rozza G, Quaini A (2006) Reduced basis method for optimal control af advection-diffusion processes. In: Fitzgibbon W, Hoppe R, Periaux J, Pironneau O, Vassilevski Y (eds) Advances in numerical mathematics. Russian Academy of Sciences/University of Houston, Moscow/Houston, pp 193–216 |
| [125] | Quarteroni A, Sacco R, Saleri F (2000) Numerical mathematics. Texts in applied mathematics, vol 37. Springer, New York |
| [126] | Quarteroni A, Valli A (1997) Numerical approximation of partial differential equations, 2nd edn. Springer, Berlin · Zbl 1151.65339 |
| [127] | Ravindran SS (2000) Reduced-order adaptive controllers for fluid flows using pod. J Sci Comput 15(4):457–478 · Zbl 1048.76016 · doi:10.1023/A:1011184714898 |
| [128] | Ravindran SS (2000) A reduced order approach to optimal control of fluids flow using proper orthogonal decomposition. Int J Numer Methods Fluids 34(5):425–448 · Zbl 1005.76020 · doi:10.1002/1097-0363(20001115)34:5<425::AID-FLD67>3.0.CO;2-W |
| [129] | Ravindran SS (2002) Adaptive reduced-order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J Sci Comput 23(6):1924–1942 · Zbl 1026.76015 · doi:10.1137/S1064827500374716 |
| [130] | Rewienski M, White J (2003) A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. IEEE Trans Comput-Aided Des Integr Circuits Syst 22:155–170 · doi:10.1109/TCAD.2002.806601 |
| [131] | Rheinboldt WC (1981) Numerical analysis of continuation methods for nonlinear structural problems. Comput Struct 13(1–3):103–113 · Zbl 0465.65030 · doi:10.1016/0045-7949(81)90114-0 |
| [132] | Rheinboldt WC (1993) On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Anal Theory, Methods Appl 21(11):849–858 · Zbl 0802.65068 · doi:10.1016/0362-546X(93)90050-3 |
| [133] | Rovas D, Machiels L, Maday Y (2005) Reduced basis output bounds methods for parabolic problems. IMA J Appl Math · Zbl 1101.65099 |
| [134] | Rovas DV (2002) Reduced-basis output bound methods for parametrized partial differential equations. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA |
| [135] | Rozza G (2005) Real-time reduced basis techniques for arterial bypass geometries. In: Bathe K (ed) Proceedings of the third MIT conference on computational fluid and solid mechanics, June 14–17, 2005. Computational fluid and solid mechanics. Elsevier, Amsterdam, pp 1283–1287 |
| [136] | Rozza G (2005) Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity. Appl Numer Math 55(4):403–424 · Zbl 1112.65122 · doi:10.1016/j.apnum.2004.11.004 |
| [137] | Rozza G (2005) Shape design by optimal flow control and reduced basis techniques: Applications to bypass configurations in haemodynamics. PhD thesis, EPFL, Ecole Polytechnique Federale de Lausanne |
| [138] | Rozza G (2008) Reduced basis method for Stokes equations in domains with non-affine parametric dependence. Comput Vis Sci 11(4). doi: 10.1007/s00791-006-0044-7 |
| [139] | Rozza G, Veroy K (2007) On the stability of reduced basis method for Stokes equations in parametrized domains. Comput Methods Appl Mech Eng 196:1244–1260 · Zbl 1173.76352 · doi:10.1016/j.cma.2006.09.005 |
| [140] | Schiesser WE, Silebi CA (1997) Computational transport phenomena: numerical methods for the solution of transport problems. Cambridge University Press, Cambridge |
| [141] | Sen S (2007) Reduced-basis approximation and a posteriori error estimation for non-coercive elliptic problems: Application to acoustics. PhD thesis, Massachusetts Institute of Technology |
| [142] | Sen S, Veroy K, Huynh DBP, Deparis S, Nguyen NC, Patera AT (2006) ”Natural norm” a posteriori error estimators for reduced basis approximations. J Comput Phys 217:37–62 · Zbl 1100.65094 · doi:10.1016/j.jcp.2006.02.012 |
| [143] | Shi G, Shi CJR (2004) 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, New York, pp 774–779 |
| [144] | Sirisup S, Xiu D, Karniadakis G (2005) Equation-free/Galerkin-free POD-assisted computation of incompressible flows. J Comput Phys 207:617–642 · Zbl 1213.76146 · doi:10.1016/j.jcp.2005.01.024 |
| [145] | Sirovich L (1987) Turbulence and the dynamics of coherent structures, part 1: Coherent structures. Q Appl Math 45(3):561–571 · Zbl 0676.76047 |
| [146] | Strang G (2003) Introduction to linear algebra. Wellesley-Cambridge, Wellesley · Zbl 1042.15001 |
| [147] | Strang G, Fix GJ (1973) An analysis of the finite element method. Prentice-Hall, New York · Zbl 0356.65096 |
| [148] | Tonn T, Urban K (2006) A reduced-basis method for solving parameter-dependent convection-diffusion problems around rigid bodies. In: Wesseling P, Oñate E, Periaux J (eds) ECCOMAS CFD 2006 proceedings. TU Delft, Delft |
| [149] | Trefethen L, III DB (1997) Numerical linear algebra. SIAM, Philadelphia · Zbl 0874.65013 |
| [150] | Veroy K (2003) Reduced-basis methods applied to problems in elasticity: Analysis and applications. PhD thesis, Massachusetts Institute of Technology |
| [151] | 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 Methods Fluids 47:773–788 · Zbl 1134.76326 · doi:10.1002/fld.867 |
| [152] | Veroy K, Prud’homme C, Patera AT (2003) Reduced-basis approximation of the viscous Burgers equation: Rigorous a posteriori error bounds. C R Acad Sci Paris, Sér I 337(9):619–624 · Zbl 1036.65075 |
| [153] | 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. In: Proceedings of the 16th AIAA computational fluid dynamics conference. Paper 2003-3847 |
| [154] | Wang J, Zabaras N (2005) Using Bayesian statistics in the estimation of heat source in radiation. Int J Heat Mass Transfer 48:15–29 · Zbl 1122.80307 · doi:10.1016/j.ijheatmasstransfer.2004.08.009 |
| [155] | Weile DS, Michielssen E (2001) Analysis of frequency selective surfaces using two-parameter generalized rational Krylov model-order reduction. IEEE Trans Antennas Propag 49(11):1539–1549 · Zbl 1002.78532 · doi:10.1109/8.964089 |
| [156] | Weile DS, Michielssen E, Gallivan K (2001) Reduced-order modeling of multiscreen frequency-selective surfaces using Krylov-based rational interpolation. IEEE Trans Antennas Propag 49(5):801–813 · doi:10.1109/8.929635 |
| [157] | Willcox K, Peraire J (2002) Balanced model reduction via the proper orthogonal decomposition. AIAA J 40(11):2323–2330 · doi:10.2514/2.1570 |
| [158] | Zienkiewicz O, Taylor R (2000) Finite element method. The basis, vol 1. Butterworth-Heinemann, London · Zbl 0991.74002 |
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