GNGTS 2013 - Atti del 32° Convegno Nazionale

The objective of the method is to generate an acceptable ensemble of “good fitting” models, rather than seeking a single optimal model. For this reason, a subsequent appraising stage is provided to extract information from the ensemble (Sambridge, 1999b). Several authors have proposed methods for analyzing an ensemble of data-acceptable models, primary using cluster analysis techniques (e.g. Kennett, 1978). In this algorithm, the appraising stage is developed in the framework of Bayesian Inference, that is, the input ensemble is reinterpolated to construct an approximate PPD, thus, such approximated PPD is importance sampled via a Gibbs sampler (Geman and Geman, 1984). Genetic Algorithms. Genetic Algorithms are search algorithms developed by Holland (1975) belonging to the larger class of evolutionary algorithms, and are based on the mechanics of natural selection and evolution to search through model space for optimal solutions. In a genetic algorithm, a population of strings (called chromosomes), which encodes candidate solutions (called individuals, or phenotypes) to an optimization problem, is evolved toward better solutions during the evolution process which starts from a population of randomly generated individuals. In each generation, the fitness (the error associated to each possible solution) of every individual is evaluated, then multiple individuals are stochastically selected from the current population on the basis of their fitness. Then they are modified (using crossover and mutation operators) to form a new population which is used in the next iteration. The algorithm terminates when either a maximum number of generations has been produced, or a satisfactory fitness level has been reached for the current population. Following the theory of the “punctuated equilibria” (Gould and Eldredge, 1977) it has been formulated a more sophisticated version of the GA, the so called Niched-Genetic Algorithms (N-GA), where the initial random population is divided in multiple subpopulation which are subjected to separated selection and evolution processes. Only for a fixed number of iterations the different populations can exchange some individuals to simulate the natural migration process. Therefore, a niching method must be able to form and maintain multiple diverse solutions and preserve them for the entire duration of the GA run. This method avoids the so called genetic drift (Horn, 1993), that is the loss of diversity inside a single population which can lead to convergence toward a local minimum in case of multimodal objective function (Sen and Stoffa, 1992). More details about GA can be found in Goldberg (1989) and Mitchell (1996). Inversion in case of Analytical Objective functionals. Convex Objective Functional. In this first test we compared the GA and the NA optimization methods on a very simple multidimensional parabolic objective function, which can be expressed by the following formula: where nd is the dimension of the model space and the n -dimensional model is m =[ m 1 , m 2 ,...., m nd ]. The accepted values of each variable range between -10 and +10. Since the misfit function is convex, we selected a single best models for each iteration for NA and a single population without niching for GA. Fig. 1a shows the number of model evaluations needed to ensure an optimal convergence toward the unique minimumwith an accuracy of 1e-6. The red circles and blue crosses represent the NA and GA, respectively, while the red and blue dotted lines represent a fourth degree polynomial fit for NA and GA respectively. For model space dimensions less than 16, the NA requires a smaller number of misfit evaluations. On the other hand, the curve fitting evidences the high number of model evaluations needed for convergence in case of high dimension model 61 GNGTS 2013 S essione 3.1