Improvement on the Nelder-Mead Simplex Algorithm and Its Application to Meteorology

Owing to computers providing great advantage in iterative calculation, the non-derivative algorithms have great potential applications to scientific research. Nelder-Mead Simplex (NMS) algorithm is a technique for minimizing an objective function in a multiple-dimensional space without differentiation, proposed by Nelder and Mead (1965). As a simple non-derivative technique, NMS algorithm is widely introduced in many computational books and used in numerical computations, and Matlab implements this algorithm for instance. Unfortunately, the method has some disadvantages such as slow converging and low precision, which need to be improved.Meteorological problems, usually nonlinear, are very complicated to solve, which require nonlinear fittings, solving the relationships between different meteorological variables, determining parameters in empirical formulas, solving the system of nonlinear equations and so forth. Conventionally, nonlinear problems are usually transformed into linear ones to solve, but this is not always practical. Fortunately the non-derivative algorithm could do this work, and an introduction to the application of the NMS algorithm to meteorological computations could be beneficial for meteorological community.An improvement on the NMS algorithm is proposed. To avoid the problems existing in the original NM simplex algorithm, a constraint is introduced to obtain next iterative point rather than finding the points of reflection, expansion, contraction and shrink, similar to that in the Powell's Method or in the Steepest Gradient algorithm. However, the searching direction is still along the downhill direction, i.e., the direction along the segment line between the vertex with the greatest functional value and the gravity center point. By introduction of the constraint, the multi-variable function will be transformed into a function with one variable, i.e., the constraint, which can be solved by the algorithms to calculate the minimum of a function with one variable, such as the Golden Section Search, Fibonacci Search and so forth. This approach will still keep the calculation without differentiation. After the improvement, the computation is greatly simplified, and its convergence will be accelerated.Some possible applications of the NMS algorithm to meteorology are also introduced, and it's also described how to implement the algorithm in fitting parameters in empirical formulas and solving the system of nonlinear equations.To testify this improvement, fitting experiments to some parameters in land surface process are made using the modified algorithm. Relationships between zero plane displacement and leaf area index (I_LA) and between aerodynamic resistance and I_LA at ground surface are calculated using the Least Square Method and the NMS algorithm to determine parameters. Experimental results show that the proposed algorithm have very high precision when fitting nonlinear formulas, therefore it can be used in computations for solving nonlinear issues or the system of nonlinear equations.