GNGTS 2013 - Atti del 32° Convegno Nazionale

space. This is a crucial limit of the NA method, in fact, we stopped the NA inversions at nd=20 due to the very high computational cost of the inversion procedure. Fig. 1b shows the computational time of the two different optimization methods tested on a esa-core Intel(R) Xeon(R) CPU E5645 @2.40GHz, 48 GB RAM. In such a simple inversion, no forward model computation is requested, therefore the computation time only depends on the internal algorithm operations. This figure shows a trend very similar to that previously observed in Fig. 1a. One can note a better performance of the NA method below nd =9 while the GA appears to overtake the NA for high-dimension model spaces. In case of NA, the most time consuming operation is the computation of the Voronoi cells which is linearly dependent on the space dimension and is proportional to the square of the number of models: (Sambridge, 1999a): where t is the computational time and ne is the number of evaluated models. Multiminima Objective Functional. In this second test we compared the GA and the NA optimization methods on a more challenging multidimensional objective functional which consists in a so-called multidimensional egg-box, that is, a product of sinusoids which we weighted with a n -dimensional gaussian centered on the true model, in order to enhance a single minimum with respect to the others. The analytical misfit can be expressed by the following formula: where l is the wavelength of the sinusoid, σ is the width of the gaussian, m true,i is the i-th component of the true model. We chose l =8, σ=2, and we located m true on one of the minima of the multidimensional sinusoid, at position ( l/4,l/4 ). As in the first test each parameter ranges between -10 and +10, while we convergence to 1e-2. When performing the GA inversions we used the same values for the control parameters independently of the dimension of the parameter space, namely a number of individuals of 100, 10 subpopulation, selection rate of 80% and a mutation rate of 1%. In addition, we set the migration process to take place every ten generations. Fig. 1 – Convex misfit functional: (a) comparison of number of models computed to reach convergence for GA (blue) and NA (red); (b) comparison of runtime occurred to reach convergence for GA (blue) and NA (red). Each experimental dataset in the graphs was fitted with a polynomial of 4 th degree. 62 GNGTS 2013 S essione 3.1