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PreprintCC BY 4.0arXiv

Total Generalized Variation for Inverse Problems Involving Piecewise Linear Finite Elements

  • Moritz Kappes
  • Manuel Haas
  • Thomas Beiert
  • Simone Pezzuto
  • Alexander Effland
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Abstract

Higher-order regularization has shown improved results over first-order methods such as total variation. In this paper, convergence of a piecewise linear finite element discretization is shown for the second-order total generalized variation functional. Additionally, this convergence result is extended to a spatiotemporal setting after the spatiotemporal function space is comprehensively introduced. In both cases, discrete minimizers converge to the continuous minimizer with rate h1/4, matching the best known rate for finite element discretizations of total variation in the same general setting. Numerical experiments confirm the convergence rates and indicate faster convergence in practice. Comparisons with related spatial image reconstruction algorithms show comparable reconstruction results. The spatiotemporal case is discussed using the example of the inverse problem in electrocardiographic imaging, where the proposed functional improves the state of the art.

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Cite

BibTeXRIS
BibTeX
@misc{kappes2026total,
  title = {{Total Generalized Variation for Inverse Problems Involving Piecewise Linear Finite Elements}},
  author = {Moritz Kappes and Manuel Haas and Thomas Beiert and Simone Pezzuto and Alexander Effland},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12778},
  eprint = {2609.12778},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12778},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Total Generalized Variation for Inverse Problems Involving Piecewise Linear Finite Elements
AU  - Moritz Kappes
AU  - Manuel Haas
AU  - Thomas Beiert
AU  - Simone Pezzuto
AU  - Alexander Effland
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12778
UR  - https://arxiv.org/abs/2609.12778
AB  - Higher-order regularization has shown improved results over first-order methods such as total variation. In this paper, convergence of a piecewise linear finite element discretization is shown for the second-order total generalized variation functional. Additionally, this convergence result is extended to a spatiotemporal setting after the spatiotemporal function space is comprehensively introduced. In both cases, discrete minimizers converge to the continuous minimizer with rate h^1/4, matching the best known rate for finite element discretizations of total variation in the same general setting. Numerical experiments confirm the convergence rates and indicate faster convergence in practice. Comparisons with related spatial image reconstruction algorithms show comparable reconstruction results. The spatiotemporal case is discussed using the example of the inverse problem in electrocardiographic imaging, where the proposed functional improves the state of the art.
ER  -