On the assimilation and inversion of small data sets under chaotic regimes

Abstract

Assimilation and inverse problems are solved under a chaotic regime by combining a sequential data analysis, an iterative restriction of the search domain and a nonlinear optimization method. Our results are based on the Lorenz model which is run with known initial conditions and a parameter set which induces chaotic oscillations of the measured state variable. A small synthetic data set is formed which expands over several Lyapunov timescales. A cost function is then built from the model departures from the observed data and the corresponding predictions taken from any trial orbit of phase space. In the assimilation, this function reflects the complexity of the optimization problem (exponential growth of the number of local minima when increasing the number of data). The cost function also depends on geometric features of the dynamics, since the basin associated with any local minimum is stirred along the sheets of the local stable manifold. In the inversion, any search path in parameter space corresponds to a bifurcation sequence of the dynamics. The shape of the cost function is related to the orbit divergence in phase space, say to the local principal Lyapunov exponents of the trial and reference orbit. Paradoxically, under a chaotic regime, the sensitivity to initial conditions or versatility of the orbits when perturbing the control parameters allow us to dynamically probe the phase or parameter spaces with a high resolution, even with a small data set. In the presence of noise stirred basins of acceptable solutions can be efficiently delineated.