Modal reduction of PDE models by means of snapshot archetypes. (English) Zbl 1040.35089

Summary: A new method for constructing low-dimensional reduced models of dissipative partial differential equations is proposed. The original PDE, \(u_t=F(u)\), is projected onto a linear subspace spanned by the so-called snapshot archetypes, that are selected spatial profiles of \(u(x,t)\). The selection rule of the snapshot archetypes characterizes the method. Two different selection methods are proposed. We provide an “energetic” criterion for the minimum number of archetypes needed for an accurate approximation of the asymptotic dynamics. This approach is tested for several PDE systems such as the Kuramoto-Sivashinsky equation, the Arneodo-Elezgaray reaction-diffusion model, and the self-ignition dynamics of a coal stockpile. The latter two systems exhibit a rich bifurcative structure and are suitable for checking the robustness of the snapshot archetype reduced models with respect to parameter variations.


35Q53 KdV equations (Korteweg-de Vries equations)
35J20 Variational methods for second-order elliptic equations
35A22 Transform methods (e.g., integral transforms) applied to PDEs


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[1] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Springer, Berlin, 1997. · Zbl 0871.35001
[2] Robinson, J.C., Arbitrarily accurate approximate inertial manifolds of fixed dimensions, Phys. lett. A, 230, 301, (1997) · Zbl 1052.34509
[3] P. Holmes, J.L. Lumley, G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, 1996. · Zbl 0890.76001
[4] P.D. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes, Birkhäuser, Boston, 2001. · Zbl 1018.93001
[5] J.L. Lumley, Stochastic Tools in Turbulence, Academic Press, New York, 1970. · Zbl 0273.76035
[6] P. Constantin, C. Foias, B. Nicolaenko, R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer, Berlin, 1989. · Zbl 0683.58002
[7] H. Haken, Advanced Synergetics, Springer, Berlin, 1983. · Zbl 0521.93002
[8] Jolly, M.S.; Kevrekidis, I.G.; Titi, E.S., Approximate inertial manifolds for the kuramoto – sivashinsky equation: analysis and computations, Physica D, 44, 38, (1990) · Zbl 0704.58030
[9] Foias, C.; Jolly, M.S.; Kevrekidis, I.G.; Sell, G.R.; Titi, S., On the computation of inertial manifold, Phys. lett. A, 131, 433, (1988)
[10] Graham, M.D.; Kevrekidis, I.G., Alternative approaches to the karhunen – loeve decomposition for model reduction and data analysis, Comp. chem. eng., 20, 495, (1996)
[11] F. Takens, Detecting strange attractors in turbulence, in: A. Dold, B. Eckmann (Eds.), Dynamical Systems and Turbulence, Lecture Notes in Mathematics 898, Springer, Berlin, 1981, pp. 366-381. · Zbl 0513.58032
[12] Robinson, J.C., A concise proof of the “geometric” construction of inertial manifolds, Phys. lett. A, 200, 415, (1995) · Zbl 1020.34509
[13] Tarman, I.H.; Sirovich, L., Extension to karhunen – loeve based approximations of complicated phenomena, Comput. meth. appl. mech. eng., 155, 359, (1998) · Zbl 0959.76072
[14] Kirby, M., Minimal dynamical systems for PDE using Sobolev eigenfunctions, Physica D, 57, 466, (1991)
[15] Kwasniok, F., The reduction of complex dynamical systems using principal interaction patterns, Physica D, 92, 28, (1996) · Zbl 0900.76481
[16] Cutler, A.; Breiman, L., Archetypal analysis, Technometrics, 36, 338, (1994) · Zbl 0804.62002
[17] Stone, E.; Cutler, A., Archetypal analysis of spatio-temporal dynamics, Physica D, 90, 209, (1996) · Zbl 0900.76189
[18] Kuramoto, Y.; Tsuzuki, T., Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. theor. phys., 55, 365, (1976)
[19] Sivashinsky, G.I., Nonlinear analysis of hydrodynamic instability in laminar flames. I. derivation of basic equations, Acta abstr., 4, 1177, (1977) · Zbl 0427.76047
[20] Elezgaray, J.; Arneodo, A., Crisis-induced intermittent bursting in reaction – diffusion chemical systems, Phys. rev. lett., 68, 714, (1992)
[21] Adrover, A.; Continillo, G.; Crescitelli, S.; Giona, M.; Russo, L., Wavelet-like collocation method for finite-dimensional reduction of distributed systems, Comp. chem. eng., 24, 2687, (2000)
[22] R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology: Functional and Variational Methods, vol. II, Springer, Berlin, 2000. · Zbl 0944.47001
[23] M.F. Barnsley, Fractals Everywhere, Academic Press, Boston, 1988.
[24] A.N. Kolmogorov, S.V. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover, New York, 1975. · Zbl 0235.46001
[25] Sirovich, L., Chaotic dynamics of coherent structures, Physica D, 37, 126, (1989)
[26] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin, 1983. · Zbl 0515.34001
[27] Eckmann, J.-P.; Ruelle, D., Ergodic theory of chaos and strange attractors, Rev. mod. phys., 57, 617, (1985) · Zbl 0989.37516
[28] H. Bai-lin, Elementary Symbolic Dynamics and Chaos in Dissipative Systems, World Scientific, Singapore, 1989. · Zbl 0724.58001
[29] Continillo, G.; Faraoni, V.; Maffettone, P.L.; Crescitelli, S., Non-linear dynamics of a self-ignited reaction – diffusion system, Chem. eng. sci., 56, S1071-S1076, (2000)
[30] E.J. Döedel, J.P. Kernevez, AUTO: A Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, Report, Applied Mathematics, California Institute of Technology, 1986.
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