Parameter estimation with model order reduction for elliptic differential equations. (English) Zbl 1398.65277
Summary: In this paper a new method for parameter estimation for elliptic partial differential equations is introduced. Parameter estimation includes minimizing an objective function, which is a measure for the difference between the parameter-dependent solution of the differential equation and some given data. We assume, that the given data results in a good approximation of the state of the system. In order to evaluate the objective function the solution of a differential equation has to be computed and hence, a large system of linear equations has to be solved. Minimization methods involve many evaluations of the objective function and therefore, the differential equation has to be solved multiple times. Thus, the computing time for parameter estimation can be large. Model order reduction was developed in order to reduce the computational effort of solving these differential equations multiple times. We use the given approximation of the state of the system as reduced basis and omit computing any snapshots. Therefore, our approach decreases the effort of the offline phase drastically. Furthermore, the dimension of the reduced system is one and thus, is much smaller than the dimension of other approaches. However, the obtained reduced model is a good approximation only close to the given data. Hence, the reduced system can lead to large errors for parameter sets, which correspond to solutions far away from the given approximation of the state of the system. In order to prevent convergence of the parameter estimator to such a local minimizer we penalise the approximation error between the full and the reduced system.
MSC:
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
Keywords:
model order reduction; parameter estimation; parametrized partial differential equations; penalty function; reduced basis methods; residual; thermal blockReferences:
| [1] | Santoro R, Semerjian H, Emmerman P, et al. Optical tomography for flow field diagnostics. Int J Heat Mass Transfer. 1981;24(7):1139-1150. |
| [2] | Elsinga GE, Scarano F, Wieneke B, et al. Tomographic particle image velocimetry. Exp Fluids. 2006;41(6):933-947. |
| [3] | Kunisch K, White LW. Regularity properties in parameter estimation of diffusion coefficients in one dimensional elliptic boundary value problems. Appl Anal. 1986;21(1-2):71-87. · Zbl 0585.49008 |
| [4] | Liu J. A multiresolution method for distributed parameter estimation. SIAM J Sci Comput. 1993;14(2):389-405. · Zbl 0773.65059 |
| [5] | Knowles I. Parameter identification for elliptic problems. J Comput Appl Math. 2001;131(1):175-194. · Zbl 0983.65121 |
| [6] | Rannacher R, Vexler B. A priori error estimates for the finite element discretization of elliptic parameter identification problems with pointwise measurements. SIAM J Control Optim. 2005;44(5):1844-1863. · Zbl 1113.65102 |
| [7] | Van den Bos A. Parameter estimation for scientists and engineers. New Jersey (NJ): John Wiley & Sons; 2007. · Zbl 1129.62001 |
| [8] | Patera AT, Rozza G. Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Cambridge: MIT; 2006. |
| [9] | Haasdonk B. Reduced basis methods for parametrized PDEs-A tutorial introduction for stationary and instationary problems. Stuttgart: Reduced order modelling; 2014. (Luminy Book series). |
| [10] | Holmes PJ, Lumley JL, Berkooz G, et al. Low-dimensional models of coherent structures in turbulence. Phys Rep. 1997;287(4):337-384. |
| [11] | Homescu C, Petzold LR, Serban R. Error estimation for reduced-order models of dynamical systems. SIAM J Numer Anal. 2005;43(4):1693-1714. · Zbl 1096.65076 |
| [12] | Kahlbacher M, Volkwein S. Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems. Discussiones Math Differ Inclusions Control Optim. 2007;27(1):95-117. · Zbl 1156.35020 |
| [13] | Haasdonk B, Ohlberger M. Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition. Math Comput Modell Dyn Syst. 2011;17(2):145-161. · Zbl 1230.37110 |
| [14] | Volkwein S. Proper orthogonal decomposition: theory and reduced-order modelling; University of Konstanz; 2013. p. 4. (Lecture notes; Vol. 4). · Zbl 1005.49029 |
| [15] | Quarteroni A, Manzoni A, Negri F. Reduced basis methods for partial differential equations: an introduction. Vol. 92. Heidelberg: Springer; 2015. · Zbl 1337.65113 |
| [16] | Siade A. Model reduction and parameter estimation in groundwater modeling [PhD thesis]. University of California; 2012. |
| [17] | Rehm AM, Scribner EY, Fathallah-Shaykh HM. Proper orthogonal decomposition for parameter estimation in oscillating biological networks. J Comput Appl Math. 2014;258:135-150. · Zbl 1291.92054 |
| [18] | Ly HV, Tran HT. Modeling and control of physical processes using proper orthogonal decomposition. Math Comput Model. 2001;33(1):223-236. · Zbl 0966.93018 |
| [19] | LeGresley PA, Alonso JJ. Investigation of non-linear projection for POD based reduced order models for aerodynamics. AIAA Paper. 2001;926:2001. |
| [20] | Ravindran S. Adaptive reduced-order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J Sci Comput. 2002;23(6):1924-1942. · Zbl 1026.76015 |
| [21] | Druskin V, Zaslavsky M. On combining model reduction and Gauss-Newton algorithms for inverse partial differential equation problems. Inverse Prob. 2007;23(4):1599. · Zbl 1127.65030 |
| [22] | Cui T, Marzouk YM, Willcox KE. Data-driven model reduction for the Bayesian solution of inverse problems. Int J Numer Methods Eng. 2015;102(5):966-990. · Zbl 1352.65445 |
| [23] | Garmatter D, Haasdonk B, Harrach B. A reduced basis Landweber method for nonlinear inverse problems. Inverse Prob. 2016;32(3):035001. · Zbl 1382.65362 |
| [24] | Müller T, Timmer J. Parameter identification techniques for partial differential equations. Int J Bifurcation Chaos. 2004;14(6):2053-2060. · Zbl 1062.35177 |
| [25] | Voit EO, Almeida J. Decoupling dynamical systems for pathway identification from metabolic profiles. Bioinformatics. 2004;20(11):1670-1681. |
| [26] | Chou I-C, Martens H, Voit EO. Parameter estimation in biochemical systems models with alternating regression. Theor Biol Med Model. 2006;3(1):1-11. |
| [27] | Grepl MA, Patera AT. A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: Math Model Numer Anal. 2005;39(1):157-181. · Zbl 1079.65096 |
| [28] | Rovas D, Machiels L, Maday Y. Reduced-basis output bound methods for parabolic problems. IMA J Numer Anal. 2006;26(3):423-445. · Zbl 1101.65099 |
| [29] | Eftang JL, Knezevic DJ, Patera AT. An hp certified reduced basis method for parametrized parabolic partial differential equations. Math Comput Modell Dyn Syst. 2011;17(4):395-422. · Zbl 1302.65223 |
| [30] | Barrault M, Maday Y, Nguyen NC, et al. An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C R Math. 2004;339(9):667-672. · Zbl 1061.65118 |
| [31] | Chaturantabut S, Sorensen DC. Discrete empirical interpolation for nonlinear model reduction. In: Proceedings of the 48th IEEE Conference on Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. IEEE; 2009. p. 4316-4321. |
| [32] | Chaturantabut S, Sorensen DC. Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput. 2010;32(5):2737-2764. · Zbl 1217.65169 |
| [33] | Engl HW, Hanke M, Neubauer A. Regularization of inverse problems. Vol. 375. Dordrecht: Springer Science & Business Media; 1996. · Zbl 0859.65054 |
| [34] | Rathinam M, Petzold LR. A new look at proper orthogonal decomposition. SIAM J Numer Anal. 2003;41(5):1893-1925. · Zbl 1053.65106 |
| [35] | Hinze M, Volkwein S. Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In: Benner P, Mehrmann V, Sorensen DC, editors. Dimension reduction of large-scale systems. Berlin: Springer Verlag; 2005. p. 261-306. · Zbl 1079.65533 |
| [36] | Sportisse B, Djouad R. Use of proper orthogonal decomposition for the reduction of atmospheric chemical kinetics. J Geophys Res: Atmos. 2007;112(D6):1-9. |
| [37] | Binev P, Cohen A, Dahmen W, et al. Convergence rates for greedy algorithms in reduced basis methods. SIAM J Math Anal. 2011;43(3):1457-1472. · Zbl 1229.65193 |
| [38] | Buffa A, Maday Y, Patera AT, et al. A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: Math Modell Numer Anal. 2012;46(3):595-603. · Zbl 1272.65084 |
| [39] | Evans L. Partial differential equations. Providence (RI): American Mathematical Society; 2010. (Graduate studies in mathematics). · Zbl 1194.35001 |
| [40] | Braess D. Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie [Finite elements: theory, fast solvers and applications in elasticity theory]. Berlin: Springer; 2007. · Zbl 1180.65146 |
| [41] | Lagarias JC, Reeds JA, Wright MH, et al. Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J Optim. 1998;9(1):112-147. · Zbl 1005.90056 |
| [42] | Martí R. Multi-start methods. In: Glover F, Kochenberger GA, editors. Handbook of metaheuristics. Boston: Springer; 2003. p. 1212-1240. |
| [43] | Bernardi C. Optimal finite-element interpolation on curved domains. SIAM J Numer Anal. 1989;26(5):1212-1240. · Zbl 0678.65003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
