A Variant Constrained Genetic Algorithm for Solving Conditional Nonlinear Optimal Perturbations

A variant constrained genetic algorithm （VCGA） for effective tracking of conditional nonlinear optimal perturbations （CNOPs） is presented. Compared with traditional constraint handling methods, the treatment of the constraint condition in VCGA is relatively easy to implement. Moreover, it does not require adjustments to indefinite pararneters. Using a hybrid crossover operator and the newly developed multi-ply mutation operator, VCGA improves the performance of GAs. To demonstrate the capability of VCGA to catch CNOPS in non-smooth cases, a partial differential equation, which has ＂on off＂ switches in its forcing term, is employed as the nonlinear model. To search global CNOPs of the nonlinear model, numerical experiments using VCGA, the traditional gradient descent algorithm based on the adjoint method （ADJ）, and a GA using tournament selection operation and the niching technique （GA-DEB） were performed. The results with various initial reference states showed that, in smooth cases, all three optimization methods are able to catch global CNOPs. Nevertheless, in non-smooth situations, a large proportion of CNOPs captured by the ADJ are local. Compared with ADJ, the performance of GA-DEB shows considerable improvement, but it is far below VCGA. Further, the impacts of population sizes on both VCGA and GA-DEB were investigated. The results were used to estimate the computation time of ~CGA and GA-DEB in obtaining CNOPs. The computational costs for VCGA, GA-DEB and ADJ to catch CNOPs of the nonlinear model are also compared.

On the Application of a Genetic Algorithm to the Predictability Problems Involving "On-Off" Switches

The lower bound of maximum predictable time can be formulated into a constrained nonlinear opti- mization 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 ad- joint 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.

An effective genetic algorithm to VDA with discontinuous “on-off” switches

With more and more improvements of atmosphere or ocean models,a growing number of physical processes in the form of parameterization are incorporated into the models,which,on the one hand,makes the models capable of describing the at-mospheric or oceanic movement more precisely,and on the other hand,introduces non-smoothness in the form of "on-off" switches into the models."On-off" switches enhance the nonlinearity of the models and finally result in the loss of the effec-tiveness of variational data assimilation(VDA) based on the conventional adjoint method(ADJ).This study,in virtue of the optimization ability of a genetic algorithm(GA) for non-smooth problems,presents a new GA(referred to as GA NEW) to solve the problems of the VDA with discontinuous "on-off" processes.In the GA-NEW,adaptive selection and mutation oper-ators,blend crossover operator,and elitist strategy are combined in application.In order to verify the effectiveness and feasi-bility of the GA NEW in VDA,an idealized model of partial differential equation with discontinuous "on-off" switches in the forcing term is adopted as the governing equation.By comparison with the ADJ,it is shown that the GA NEW in VDA is more effective and can yield better assimilation retrievals.In addition,VDA experiments demonstrate that the performance of a GA is greatly related to the configuration of genetic operators(selection,crossover and mutation operators) and much better results may be attained with more proper genetic operations.Furthermore,the robustness of the GA NEW to observational noise,model errors and observation density is investigated,and the results show that the GA NEW has stronger robustness than the ADJ with respect to all the three observation noises,model errors,and sparse observation.