The lower bound of maximum predictable time can be formulated into a constrained nonlinear optimization problem, and the traditional solutions to this problem are the filtering method and the conditional nonlinear optimal perturbation (CNOP) method. Usually, the CNOP method is implemented with the help of a gradient descent algorithm based on the adjoint method, which is named the ADJ-CNOP. However, with the increasing improvement of actual prediction models, more and more physical processes are taken into consideration in models in the form of parameterization, thus giving rise to the on-off switch problem, which tremendously affects the effectiveness of the conventional gradient descent algorithm based on the adjoint method. In this study, we attempted to apply a genetic algorithm (GA) to the CNOP method, named GA-CNOP, to solve the predictability problems involving on-off switches. As the precision of the filtering method depends uniquely on the division of the constraint region, its results were taken as benchmarks, and a series of comparisons between the ADJ-CNOP and the GA-CNOP were performed for the modified Lorenz equation. Results show that the GA-CNOP can always determine the accurate lower bound of maximum predictable time, even in non-smooth cases, while the ADJ-CNOP, owing to the effect of on-off switches, often yields the incorrect lower bound of maximum predictable time. Therefore, in non-smooth cases, using GAs to solve predictability problems is more effective than using the conventional optimization algorithm based on gradients, as long as genetic operators in GAs are properly configured.Citation: Zheng, Q., Y. Dai, L. Zhang, J. X. Sha, and X. Q. Lu, 2012: On the application of a genetic algorithm to the predictability problems involving "on-off" switches.

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Abstract. In predictability problem research, the conditional nonlinear optimal perturbation (CNOP) describes the initial perturbation that satisfies the certain constraint condition and causes the largest prediction error at the prediction time. The CNOP method 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 on the prediction variable will lead to failure of the ADJ-CNOP method; (2) when the objective function has multiple extreme values, ADJ-CNOP has large probability to produce local CNOPs, hence make false estimation on the LBMPT. Furthermore, the particle swarm optimization (PSO) algorithm, one kind of intelligent algorithms, 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 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. Besides, to check the estimation accuracy of the LBMPT presented by the PSO-CNOP and the 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.