Abordagens robustas para problemas inversos baseados nas estatísticas generalizadas de Rényi, Tsallis e de Kaniadakis

Abstract

In general, inverse problems can be faced as a task of optimizing a functional that promotes the fit between the experimental data and the data calculated from a physical model. Commonly, the objective function known as "least squares function" — which is based on Gaussian statistics — is used for this task, however this approach presents serious difficulties in a context in which the noises dont obey the Gaussian statistics. The type of non-Gaussian noise that we investigated in this work are the outliers, which are characterized as discrepant measures that contaminate the sample and make it difficult to interpretation of the experimental data. In this dissertation we approach the generalization of the inverse problem through the generalization of Gaussian statistics in the context of Rényi, Tsallis and Kaniadakis statistics. In this sense, we discuss the error distributions in the non-Gaussian context and the generalized objective functions that derive from these statistics and evaluate their robustness through the so-called Influence Function (objective function gradient). We exemplify the robustness of generalized methodologies using numerical experiments. In particular, we use the inverse problem generalization in a seismic inversion problem with high contamination from outliers. Our results show that the generalized inverse problem is resistant to outliers. Furthermore, we identified that the best data inversion performance occurs when the entropic index of each generalized statistic is associated with objective functions proportional to the inverse of the error amplitude. We argue that at such a limit the three approaches are resistant to outliers and are also equivalent. Furthermore, this approach suggests a lower computational cost for the inversion process due to the reduction of numerical simulations to be performed and the rapid convergence of the optimization process.