Parameter identification in noisy extended systems: a hydrodynamic case

Abstract This paper is concerned with the robustness of parameter identification methods with respect to the noise levels typically found in experiments. More precisely, we focus on the case of an extended nonlinear system: a system of coupled local maps akin to a discretized complex Ginzburg-Landau equation, modeling a wake experiment. After a brief description of this hydrodynamic experiment as well as of the associated cost function and synthetic data generation, we introduce two inversion methods: a one-time-step approach, and a more sophisticated n -time-step optimization procedure, solved by a backpropagation method. The one-time-step approach reduces to a small linear system for the unknown parameters, while the n -time-step approach involves a backpropagation equation for a set of Lagrage multipliers. The sensitivity of each method with respect to noise is then discussed: while the n -time-step method is very robust even with large amounts of noise, the one-time-step approach is shown to be affected by small noise levels.

[1]  Philip E. Gill,et al.  Practical optimization , 1981 .

[2]  Daniel H. Rothman,et al.  Automatic estimation of large residual statics corrections , 1986 .

[3]  Achi Brandt,et al.  Multi-level approaches to discrete-state and stochastic problems , 1986 .

[4]  Nadine Aubry,et al.  The dynamics of coherent structures in the wall region of a turbulent boundary layer , 1988, Journal of Fluid Mechanics.

[5]  P. L. Gal,et al.  Complex bi-orthogonal decomposition of a chain of coupled wakes , 1992 .

[6]  St'ephane Zaleski,et al.  Identification of parameters in amplitude equations describing coupled wakes , 1996, chao-dyn/9601008.

[7]  Schreiber,et al.  Noise reduction in chaotic time-series data: A survey of common methods. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  Y. Sunahara,et al.  CHAPTER 2 – IDENTIFICATION OF DISTRIBUTED-PARAMETER SYSTEMS , 1982 .

[9]  A. Tarantola,et al.  Two‐dimensional nonlinear inversion of seismic waveforms: Numerical results , 1986 .

[10]  Stephen A. Billings,et al.  Identification of models for chaotic systems from noisy data: implications for performance and nonlinear filtering , 1995 .

[11]  C. Bunks,et al.  Multiscale seismic waveform inversion , 1995 .

[12]  Andreas S. Weigend,et al.  The Future of Time Series: Learning and Understanding , 1993 .

[13]  Gottfried Mayer-Kress,et al.  Dimension Densities for Turbulent Systems with Spatially Decaying Correlation Functions , 1987, Complex Syst..

[14]  M. Provansal,et al.  Bénard-von Kármán instability: transient and forced regimes , 1987, Journal of Fluid Mechanics.

[15]  Brown,et al.  Modeling and synchronizing chaotic systems from time-series data. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  L. Keefe,et al.  Two nonlinear control schemes contrasted on a hydrodynamiclike model , 1993 .

[17]  L. Tsimring,et al.  The analysis of observed chaotic data in physical systems , 1993 .

[18]  Hung Man Tong,et al.  Threshold models in non-linear time series analysis. Lecture notes in statistics, No.21 , 1983 .

[19]  J. D. Farmer,et al.  Optimal shadowing and noise reduction , 1991 .

[20]  J. Max.  Méthodes et techniques de traitement du signal et application aux mesures physiques. Tome 2 , 1981 .

[21]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[22]  M. B. Priestley,et al.  Non-linear and non-stationary time series analysis , 1990 .