Zadeh's extension principle for continuous functions of non-interactive variables: a parallel optimization approach

There is a growing interest in the use of fuzzy intervals in many engineering applications. However, a direct implementation of Zadeh's extension principle, which forms the basis for computing with fuzzy intervals, is still computationally too demanding for practical use. In the case of a continuous function and fuzzy intervals that describe non-interactive variables as inputs, the output is a fuzzy interval as well and can be determined for each alpha-cut separately. The problem, thus, reduces to finding the endpoints of these a-cuts, which amounts to a number of interwoven optimization problems. In the case of a non-monotone continuous function, however, these optimization problems are non-trivial. In this paper, different optimization algorithms are applied for that purpose: Gradient Descent based on Sequential Quadratic Programming, Simplex-Simulated Annealing, Particle Swarm Optimization, and Particle Swarm Optimization combined with Gradient Descent. In addition, two approaches are followed to determine a suitable number of a-cuts: either a fixed, predetermined number is used, or an initially (very) small number is chosen that is subsequently increased according to a linearity criterion. Both a non-parallel and a parallel implementation are designed. The parallel version is restricted to work with Particle Swarm Optimization and employs communication to optimize its (internal) performance by exploiting the dependence between the various optimization problems. Different configurations are evaluated on a set of benchmark functions in terms of the mean area under the output fuzzy interval and the number of function evaluations. Particle Swarm Optimization combined with Gradient Descent starting from a small number of a-cuts leads to the most accurate fuzzy intervals at the cost of a relatively large number of function evaluations

Calibration of a water and energy balance model: Recursive parameter estimation versus particle swarm optimization

It is well known that one of the major problems in the application of land surface models is the determination of the various model parameters. In most cases, only one or a limited number of variables are used to estimate these parameters. This study evaluates the use of two fundamentally different global optimization methods, multistart weight-adaptive recursive parameter estimation (MWARPE) and particle swarm optimization (PSO), for the estimation of hydrologic model parameters on the basis of data for multiple variables. MWARPE iteratively uses the linear recursive filter equations in a Monte Carlo setting and therefore does not rely on the explicit minimization of an objective function. However, a major drawback of the MWARPE method is the high dimensionality, determined by the number of observations, of the matrix to be inverted. On the other hand, PSO is a stochastic optimization method based on the collective strength of a population of individuals with flocking or herding behavior, as observed in a wide number of biological systems. In situ observations of net radiation; latent, sensible, and ground heat fluxes; and the soil moisture profile are used to determine the parameters of a simplified water and energy balance model. Both optimization methods are analyzed in terms of model performance and computational efficiency. Comparable results, expressed in terms of the root mean square error values, were obtained for both methods. However, it was found that MWARPE tends to slightly overfit the data