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Puzzle in inverse problems: Tsallis noise and Tsallis norm

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  • Adson Alexandre Quirino da Silveira

    (Universidade Federal do Rio Grande do Norte)

  • Renato Ferreira Souza

    (Universidade Federal do Rio Grande do Norte
    Senai Cimatec)

  • Jonathas da Silva Maciel

    (Senai Cimatec)

  • Jessica Lia Santos da Costa

    (Universidade Federal de Minas Gerais)

  • Daniel Teixeira dos Santos

    (Universidade Federal do Rio Grande do Norte
    Senai Cimatec)

  • João Medeiros Araujo

    (Universidade Federal do Rio Grande do Norte)

  • Sérgio Luiz E. F. da Silva

    (Politecnico di Torino
    Universidade Federal Fluminense)

  • Gilberto Corso

    (Universidade Federal do Rio Grande do Norte)

Abstract

Inverse problems are challenging in several ways, and we cite the non-linearity and the presence of non-Gaussian noise. Least squared is the standard method to construct a equivalent functional for optimization, which is equivalent to the L2 norm of the misfit. Alternative norms in the optimization process are an useful strategy in the inverse problem solution. Generalized statistics can be present at two sides of the inverse problem: in the noise that pollutes the data and in the norm used in the optimization algorithm. With help of a seismic problem, we polluted the signal with a q-exponential noise (using an exponent $$q_{\text {noise}}$$ q noise ) and subsequently inverted the problem using a norm associated to a q-exponential (with an exponent $$q_{\text {inv}}$$ q inv ). The same procedure was also applied to the simpler problem of a linear fitting. We tested the hypothesis of a relation between the exponents $$q_{noise}$$ q noise and $$q_{\text {inv}}$$ q inv . The overall pattern observed is the following: inversion error are smaller for low $$q_{\text {noise}}$$ q noise and high $$q_{\text {inv}}$$ q inv . In contrast, the worst inversion is found for high polluting noise (far from Gaussian noise) and for inversion with low $$q_{\text {inv}}$$ q inv (close to the Gaussian case). Graphic abstract

Suggested Citation

  • Adson Alexandre Quirino da Silveira & Renato Ferreira Souza & Jonathas da Silva Maciel & Jessica Lia Santos da Costa & Daniel Teixeira dos Santos & João Medeiros Araujo & Sérgio Luiz E. F. da Silva & , 2023. "Puzzle in inverse problems: Tsallis noise and Tsallis norm," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 96(3), pages 1-7, March.
  • Handle: RePEc:spr:eurphb:v:96:y:2023:i:3:d:10.1140_epjb_s10051-023-00496-0
    DOI: 10.1140/epjb/s10051-023-00496-0
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    References listed on IDEAS

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    1. da Silva, Sérgio Luiz Eduardo Ferreira & dos Santos Lima, Gustavo Zampier & de Araújo, João Medeiros & Corso, Gilberto, 2021. "Extensive and nonextensive statistics in seismic inversion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 563(C).
    2. da Silva, Sérgio Luiz Eduardo Ferreira & da Costa, Carlos A.N. & Carvalho, Pedro Tiago C. & de Araújo, João Medeiros & dos Santos Lucena, Liacir & Corso, Gilberto, 2020. "Robust full-waveform inversion using q-statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 548(C).
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