×

Model reduction for Fisher’s equation with an error bound. (English) Zbl 1482.65108

Summary: This work considers a model-order reduction (MOR) for Fisher’s equation, which is generally used to describe many physical systems, such as chemical reactions, flame propagation, neurophysiology, nuclear reactors, and tissue engineering. Due to the nonlinearity in this type of system, solving the resulting discretized model for accurate solution could be time-consuming as the dimension gets large. Model-order reduction can be applied to improve the process of solving this large discretized model. In this work, a projection-based method called Proper Orthogonal Decomposition (POD) will be used first to project the state variables of the system on a low dimensional subspace, which will result in the decrease of unknowns in the systems. However, the computational complexity of the discretized nonlinear term still depends on the original large dimension. Discrete Empirical Interpolation Method (DEIM) is therefore used to eliminate this inefficiency. This POD-DEIM approach is applied on Fisher’s equation with discontinuous initial conditions. An apriori error bound is derived for the approximations from POD-DEIM reduced system for the semi-implicit numerical scheme. The usefulness of this approach is illustrated through the parametric study of the varying boundary conditions. This work also investigates the effect of adding the snapshot difference quotients to construct basis sets used in POD and POD-DEIM reduced systems. The numerical results show that this POD-DEIM can substantially decrease the computational time while providing accurate numerical solution.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
93B11 System structure simplification
PDFBibTeX XMLCite
Full Text: Link

References:

[1] D. Xiao A, F. Fang A, A. G. Buchan A, C. C. Pain A, I. M. Navon C, J. Du D, and G. Hu B. Non-linear model reduction for the navier-stokes equations using the residual deim method. · Zbl 1349.76288
[2] FISHER R. A. The wave of advance of advantageous genes.Annals of Eugenics, 7(4):355-369, 1937. · JFM 63.1111.04
[3] I. Petrovsky A. Kolomogoroff and 1 N. Piscounoff, Moscow Univ. Bull. Math. 1. 1937.
[4] Kamel Al-Khaled. Numerical study of fisher’s reactiondiffusion equation by the sinc collocation method.Journal of Computational and Applied Mathematics, 137(2):245 - 255, 2001. · Zbl 0992.65108
[5] S R Arridge, J P Kaipio, V Kolehmainen, M Schweiger, E Somersalo, T Tarvainen, and M Vauhkonen. Approximation errors and model reduction with an application in optical diffusion tomography.Inverse Problems, 22(1):175, 2006. · Zbl 1138.65042
[6] M Barrault. An empirical interpolation method: application to efficient reducedbasis discretization of partial differential equations.Comptes Rendus Mathematique, 339(9):667-672, 2004. · Zbl 1061.65118
[7] Maxime Barrault, Yvon Maday, Ngoc Cuong Nguyen, and Anthony T. Patera. An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations.Comptes Rendus Mathematique, 339(9):667 - 672, 2004. · Zbl 1061.65118
[8] Gal Berkooz, Philip Holmes, and John L. Lumley. The proper orthogonal decomposition in the analysis of turbulent flows.Annual Rev. Fluid Mech, pages 539-575, 1993.
[9] T. Bui-Thanh, M. Damodaran, and K. Willcox.Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition.AIAA Journal, 42:1505-1516, 2004.
[10] A. Cammilleri, F. Gueniat, J. Carlier, L. Pastur, E. Memin, F. Lusseyran, and G. Artana. Pod-spectral decomposition for fluid flow analysis and model reduction. Theoretical and Computational Fluid Dynamics, 27(6):787-815, 2013.
[11] Kevin Carlberg, Charbel Farhat, Julien Cortial, and David Amsallem. The{GNAT} method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows.Journal of Computational Physics, 242(0):623 - 647, 2013. · Zbl 1299.76180
[12] Dominique Chapelle, Asven Gariah, and Jacques Sainte-Marie. Galerkin approximation with Proper Orthogonal Decomposition: new error estimates and illustrative examples.ESAIM: Mathematical Modelling and Numerical Analysis, 46(4):731-757, 2012. · Zbl 1273.65125
[13] Saifon Chaturantabut and Danny C. Sorensen. Discrete Empirical Interpolation for nonlinear model reduction. InConference on Decision and Control, pages 4316-4321, 2009.
[14] Saifon Chaturantabut and Danny C. Sorensen. Nonlinear model reduction via discrete empirical interpolation.SIAM J. Sci. Comput., 32(5):2737-2764, September 2010. · Zbl 1217.65169
[15] Saifon Chaturantabut and Danny C. Sorensen.Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media. Mathematical and Computer Modelling of Dynamical Systems, 17:337-353, 2011. · Zbl 1302.76127
[16] Saifon Chaturantabut and Danny C. Sorensen. A state space error estimate for pod-deim nonlinear model reduction.SIAM J. Numer. Anal., 50(1):46-63, January 2012. · Zbl 1237.93035
[17] Laurent Cordier, Manuel Girault, and Daniel Petit. Reduced order modeling by modal identification method and pod-galerkin approach of the heated circular cylinder wake in mixed convection.Journal of Physics: Conference Series, 395(1):012102, 2012.
[18] R. S¸tef˘anescu and I. M. Navon. POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model.Journal of Computational Physics, 237:95-114, March 2013. · Zbl 1286.76106
[19] J. Duffin, O. Sobczyk, A.P. Crawley, J. Poublanc, D.J. Mikulis, and J.A. Fisher. The dynamics of cerebrovascular reactivity shown with transfer function analysis. NeuroImage, 114(0):207 - 216, 2015.
[20] Freddy Dumortier, Nikola Popovi, and Tasso J Kaper. The critical wave speed for the fisherkolmogorovpetrowskiipiscounov equation with cut-off.Nonlinearity, 20(4):855, 2007. · Zbl 1139.35056
[21] Kristine Embree. Fisher-KPP equation nonlinear traveling waves, 2001. [Online; accessed 30 August 2001].
[22] F. Fang, C.C. Pain, I.M. Navon, A.H. Elsheikh, J. Du, and D. Xiao. Non-linear petrovgalerkin methods for reduced order hyperbolic equations and discontinuous finite element methods.Journal of Computational Physics, 234(0):540 - 559, 2013. · Zbl 1284.65132
[23] Zhengkun Feng and Azzeddine Soulaimani. Reduced order modelling based on pod method for 3d nonlinear aeroelasticity. InThe 18th IASTED International Conference on Modelling and Simulation, MS ’07, pages 489-494, Anaheim, CA, USA, 2007. ACTA Press.
[24] Kimberly J. Fink and Laura Ray. Individualization of head related transfer functions using principal component analysis.Applied Acoustics, 87(0):162 - 173, 2015.
[25] R. A. FISHER. The wave of advance of advantageous genes.Annals of Eugenics, 7(4):355-369, 1937. · JFM 63.1111.04
[26] Brian A. Freno and Paul G.A. Cizmas. A proper orthogonal decomposition method for nonlinear flows with deforming meshes.International Journal of Heat and Fluid Flow, 50(0):145 - 159, 2014.
[27] Sudipta Ghosh and Nilanjan Senroy. Balanced truncation based reduced order modeling of wind farm.International Journal of Electrical Power and Energy Systems, 53(0):649 - 655, 2013.
[28] G.H. Golub and C.F. Van Loan.Matrix Computations. · Zbl 0559.65011
[29] M.D. Gunzburger, L.S. Hou, and W. Zhu. Fully discrete finite element approximations of the forced fisher equation.Journal of Mathematical Analysis and Applications, 313(2):419 - 440, 2006. · Zbl 1095.65094
[30] Roi Gurka, Alexander Liberzon, and Gad Hetsroni.{POD}of vorticity fields: A method for spatial characterization of coherent structures.International Journal of Heat and Fluid Flow, 27(3):416 - 423, 2006.
[31] Michael Hinze and Stefan Volkwein. Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: Error estimates and suboptimal control. In Peter Benner, DannyC. Sorensen, and Volker Mehrmann, editors,Dimension Reduction of Large-Scale Systems, volume 45 ofLecture Notes in Computational Science and Engineering, pages 261-306. Springer Berlin Heidelberg, 2005. · Zbl 1079.65533
[32] Amit Hochman, Bradley N. Bond, and Jacob K. White. A stabilized discrete empirical interpolation method for model reduction of electrical, thermal, and microelectromechanical systems. InProceedings of the 48th Design Automation Conference, DAC ’11, pages 540-545, New York, NY, USA, 2011. ACM.
[33] Traian Iliescu and Zhu Wang.Are the snapshot difference quotients needed in the proper orthogonal decomposition?SIAM Journal on Scientific Computing, 36(3):A1221-A1250, 2014. · Zbl 1297.65092
[34] G.B. Jasmon and L.H.C.C. Lee. Distribution network reduction for voltage stability analysis and loadflow calculations.International Journal of Electrical Power and Energy Systems, 13(1):9 - 13, 1991.
[35] Eileen Kammann, Fredi Trltzsch, and Stefan Volkwein. A method of a-posteriori error estimation with application to proper orthogonal decomposition. 2012. · Zbl 1282.49021
[36] Wei Kang, Jia-Zhong Zhang, Sheng Ren, and Peng-Fei Lei.Nonlinear galerkin method for low-dimensional modeling of fluid dynamic system using{POD}modes. Communications in Nonlinear Science and Numerical Simulation, 22(13):943 - 952, 2015. · Zbl 1329.65223
[37] Anthony R. Kellems, Saifon Chaturantabut, and Steven J. Cox. Morphologically accurate reduced order modeling of spiking neurons.Journal of Computational Neuroscience, 28:477-494, 2010.
[38] K. Kunisch and S. Volkwein. Galerkin proper orthogonal decomposition methods for parabolic problems.Numerische Mathematik, 90(1):117-148, 2001. · Zbl 1005.65112
[39] K. Kunisch and S. Volkwein. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics.SIAM J. Numer. Anal., 40(2):492-515, February 2002. · Zbl 1075.65118
[40] F Lanata and A Del Grosso. Damage detection and localization for continuous static monitoring of structures using a proper orthogonal decomposition of signals.Smart Materials and Structures, 15(6):1811, 2006.
[41] Norman Lang, Jens Saak, and Tatjana Stykel. Towards practical implementations of balanced truncation for{LTV}systems.IFAC-PapersOnLine, 48(1):7 - 8, 2015.
[42] Alex Liberzon, Roi Gurka, Iztok Tiselj, and Gad Hetsroni. Spatial characterization of the numerically simulated vorticity fields of a flow in a flume.Theoretical and Computational Fluid Dynamics, 19(2):115-125, 2005. · Zbl 1148.76330
[43] J. L. Lumley. Stochastic Tools in Turbulence.Academic Press, New York, 1970. · Zbl 0273.76035
[44] J.L. Lumley. The structure of inhomogeneous turbulent flows.in Atmospheric Turbulence and Radio Wave Propagation (A. M. Yaglom and V. I. Tararsky, eds.), (Nauka, Moscow), 1967.
[45] Zhendong Luo, Jiang Zhu, Ruiwen Wang, and I.M. Navon. Proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical pacific ocean reduced gravity model.Computer Methods in Applied Mechanics and Engineering, 196(4144):4184 - 4195, 2007. · Zbl 1173.76348
[46] Hung V. Ly and Hien T. Tran. Proper orthogonal decomposition for flow calculations and optimal control in a horizontal cvd reactor. Technical report, 1998. · Zbl 1146.76631
[47] Vicenc Mendez, Sergei Fedotov, and Werner Horsthemke.Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities (Springer Series in Synergetics). Springer, 2010 edition, June 2010.
[48] Ha Binh Minh, Carles Batlle, and Enric Fossas. A new estimation of the lower error bound in balanced truncation method.Automatica, 50(8):2196 - 2198, 2014. · Zbl 1297.93073
[49] Thien Duy Nguyen, John Craig Wells, Paritosh Mokhasi, and Dietmar Rempfer. Proper orthogonal decomposition-based estimations of the flow field from particle image velocimetry wall-gradient measurements in the backward-facing step flow. Measurement Science and Technology, 21(11):115406, 2010.
[50] Daniel Olmos and Bernie D. Shizgal. A pseudospectral method of solution of fisher’s equation.J. Comput. Appl. Math., 193(1):219-242, August 2006. · Zbl 1092.65088
[51] Casian Alexandru Pantea.Mathematical and Computational Analysis of Biochemical Reaction Networks. PhD thesis, University of Wisconsin at Madison, Madison, WI, USA, 2010. AAI3437121. · Zbl 1094.20012
[52] A. Placzek, D.-M. Tran, and R. Ohayon. A nonlinear pod-galerkin reduced-order model for compressible flows taking into account rigid body motions.Computer Methods in Applied Mechanics and Engineering, 200(4952):3497 - 3514, 2011. · Zbl 1239.76046
[53] Michael Presho, Anastasiya Protasov, and Eduardo Gildin. Localglobal model reduction of parameter-dependent, single-phase flow models via balanced truncation. Journal of Computational and Applied Mathematics, 271(0):163 - 179, 2014. · Zbl 1329.76187
[54] Y. Qiu and D.M. Sloan. Numerical solution of fisher’s equation using a moving mesh method.Journal of Computational Physics, 146(2):726 - 746, 1998. · Zbl 0927.65108
[55] Shodhan Rao, Arjan van der Schaft, Karen van Eunen, Barbara M. Bakker, and Bayu Jayawardhana. Model-order reduction of biochemical reaction networks.CoRR, abs/1212.2438, 2012. · Zbl 1285.05169
[56] Micha Rewieski and Jacob White. Model order reduction for nonlinear dynamical systems based on trajectory piecewise-linear approximations.Linear Algebra and its Applications, 415(23):426 - 454, 2006. Special Issue on Order Reduction of LargeScale Systems. · Zbl 1105.93020
[57] P.R. Runkle, M.A. Blommer, and G.H. Wakefield. A comparison of head related transfer function interpolation methods.InApplications of Signal Processing to Audio and Acoustics, 1995., IEEE ASSP Workshop on, pages 88-91, Oct 1995.
[58] S. QIN S. TANG and R. O. WEBER.NUMERICAL STUDIES ON 2DIMENSIONAL REACTION-DIFFUSION EQUATIONS.J. Austral. Math. Soc. Sen B, 35:223-243, 1993. · Zbl 0807.65103
[59] K. Kunisch S. Volkwein, M. Kahlbacher and F. Troltzsch.Proper Orthogonal Decomposition: Applications in Optimization and Control.
[60] Elisa Schenone.Reduced Order Models, Forward and Inverse Problems in Cardiac Electrophysiology. Theses, Universit´e Pierre et Marie Curie - Paris VI, November 2014.
[61] John R. Singler. Optimality of balanced proper orthogonal decomposition for data reconstruction ii: Further approximation results.Journal of Mathematical Analysis and Applications, 421(2):1006 - 1020, 2015. · Zbl 1302.41025
[62] Angela Slavova. Cellular neural network models of some equations from biology, physics and ecology.FUNCTIONAL DIFFERENTIAL EQUATIONS, 10:579-591, 2003. · Zbl 1040.92003
[63] D. Stan and J. L. V´azquez. The Fisher-KPP equation with nonlinear fractional diffusion.ArXiv e-prints, March 2013.
[64] Razvan Stefanescu, Adrian Sandu, and Ionel Michael Navon. POD/DEIM strategies for reduced data assimilation systems.CoRR, abs/1402.5992, 2014.
[65] Kyle Chand William Arrighi Tanya Kostova, Geoffrey Oxberry. Error bounds and analysis of proper orthogonal decomposition model reduction methods using snapshots from the solution and the time derivatives. · Zbl 1376.65099
[66] Jonathan H. Tu and Clarence W. Rowley. An improved algorithm for balanced {POD}through an analytic treatment of impulse response tails.Journal of Computational Physics, 231(16):5317 - 5333, 2012. · Zbl 1426.76573
[67] Alexander Vendl and Heike Fabender. Projection-based model order reduction for steady aerodynamics. In Norbert Kroll, Rolf Radespiel, Jan Willem Burg, and Kaare Srensen, editors,Computational Flight Testing, volume 123 ofNotes on Numerical Fluid Mechanics and Multidisciplinary Design, pages 151-166. Springer Berlin Heidelberg, 2013.
[68] S. Volkwein, editor.Model Reduction using proper Orthogonal Decomposition. 2008. · Zbl 1191.49040
[69] T. Vo, R. Pulch, E.J.W. ter Maten, and A. El Guennouni. Trajectory piecewise linear approach for nonlinear differential-algebraic equations in circuit simulation. In Gabriela Ciuprina and Daniel Ioan, editors,Scientific Computing in Electrical Engineering, volume 11 ofMathematics in Industry, pages 167-173. Springer Berlin Heidelberg, 2007. · Zbl 1170.78447
[70] A-xia Wang and Yi-Chen Ma. An error estimate of the proper orthogonal decomposition in model reduction and data compression.Numerical Methods for Partial Differential Equations, 25(4):972-989, 2009. · Zbl 1419.76509
[71] K. Willcox and J. Peraire. Balanced model reduction via the proper orthogonal decomposition.AIAA Journal, pages 2323-2330, 2002.
[72] John Williamson. Note on a principal axis transformation for non-hermitian matrices. Bull. Amer. Math. Soc., 45:920-922, 12 1939. · Zbl 0022.29804
[73] Daniel Wirtz, Danny C. Sorensen, and Bernard Haasdonk. A-posteriori error estimation for deim reduced nonlinear dynamical systems.Submitted to SIAM Journal on Scientific Computing, 2012. · Zbl 1312.65127
[74] D. Xiao, F. Fang, J. Du, C.C. Pain, I.M. Navon, A.G. Buchan, A.H. ElSheikh, and G. Hu. Non-linear petrovgalerkin methods for reduced order modelling of the navierstokes equations using a mixed finite element pair.Computer Methods in Applied Mechanics and Engineering, 255(0):147 - 157, 2013. · Zbl 1297.76107
[75] Xunnian Yang and Jianmin Zheng.Curvature tensor computation by piecewise surface interpolation.Computer-Aided Design, 45(12):1639 - 1650, 2013.
[76] Xiaoji Ye, Peng Li, Min Zhao, R. Panda, and Jiang Hu. Scalable analysis of meshbased clock distribution networks using application-specific reduced order modeling. Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on, 29(9):1342-1353, Sept 2010.
[77] Wojciech Zalewski and Piotr Antoni Gryglaszewski. Mathematical model of heat and mass transfer processes in evaporative fluid coolers.Chemical Engineering and Processing: Process Intensification, 36(4):271 - 280, 1997.
[78] Huiyan Zhang, Ligang Wu, Peng Shi, and Yuxin Zhao. Balanced truncation approach to model reduction of markovian jump time-varying delay systems.Journal of the Franklin Institute, (0):-, 2015 · Zbl 1395.93137
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.