In predictability problem research, the conditional nonlinear
optimal perturbation (CNOP) describes the initial perturbation that satisfies
a certain constraint condition and causes the largest prediction error at the
prediction time. The CNOP has been successfully applied in estimation of the
lower bound of maximum predictable time (LBMPT). Generally, CNOPs are
calculated by a gradient descent algorithm based on the adjoint model, which
is called ADJ-CNOP. This study, through the two-dimensional Ikeda model,
investigates the impacts of the nonlinearity on ADJ-CNOP and the
corresponding precision problems when using ADJ-CNOP to estimate the LBMPT.
Our conclusions are that (1) when the initial perturbation is large
or the prediction time is long, the strong nonlinearity of the dynamical
model in the prediction variable will lead to failure of the ADJ-CNOP method,
and (2) when the objective function has multiple extreme values, ADJ-CNOP has
a large probability of producing local CNOPs, hence making a false estimation
of the LBMPT. Furthermore, the particle swarm optimization (PSO) algorithm,
one kind of intelligent algorithm, is introduced to solve this problem. The
method using PSO to compute CNOP is called PSO-CNOP. The results of numerical
experiments show that even with a large initial perturbation and long
prediction time, or when the objective function has multiple extreme values,
PSO-CNOP can always obtain the global CNOP. Since the PSO algorithm is a
heuristic search algorithm based on the population, it can overcome the
impact of nonlinearity and the disturbance from multiple extremes of the
objective function. In addition, to check the estimation accuracy of the
LBMPT presented by PSO-CNOP and ADJ-CNOP, we partition the constraint domain
of initial perturbations into sufficiently fine grid meshes and take the
LBMPT obtained by the filtering method as a benchmark. The result shows that
the estimation presented by PSO-CNOP is closer to the true value than the one
by ADJ-CNOP with the forecast time increasing.