Abordagens robustas para problemas inversos baseados nas estatísticas generalizadas de Rényi, Tsallis e de Kaniadakis
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 —...
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| Formato: | Dissertação |
| Lenguaje: | pt_BR |
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Universidade Federal do Rio Grande do Norte
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| Acceso en línea: | https://repositorio.ufrn.br/handle/123456789/47117 |
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| Sumario: | 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. |
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