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In the present work, we discuss variational regularization for ill-posed nonlinear problems with focus on an oversmoothing penalty term. This means in our model that the searched-for solution of the considered nonlinear operator equation does not belong to the domain of definition of the penalty functional. In the past years, such variational regularization has been investigated comprehensively in Hilbert scales. Our present study tries to continue and to extend those investigations to Banach scales. This new study includes convergence rates result for a priori choices of the regularization parameter, both for Hölder-type smoothness and low order-type smoothness. The necessary tools for low order smoothness in the Banach space setting are provided.

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Variational Regularization with Oversmoothing Penalty Term in Banach Spaces

  • 章节

  • 发表日期：

    2025年

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  • Robert Plato,
  • Bernd Hofmann

摘要

In the present work, we discuss variational regularization for ill-posed nonlinear problems with focus on an oversmoothing penalty term. This means in our model that the searched-for solution of the considered nonlinear operator equation does not belong to the domain of definition of the penalty functional. In the past years, such variational regularization has been investigated comprehensively in Hilbert scales. Our present study tries to continue and to extend those investigations to Banach scales. This new study includes convergence rates result for a priori choices of the regularization parameter, both for Hölder-type smoothness and low order-type smoothness. The necessary tools for low order smoothness in the Banach space setting are provided.