Regularization of inverse problems by filtered diagonal frame decomposition under general source

This paper addresses the ill-posed inverse problem Kx=y in real or complex Hilbert settings, where data y is contaminated by noise. We propose regularization methods utilizing diagonal frame decomposition (DFD) as a generalization of singular value decomposition (SVD)-based techniques to achieve stable solutions. Our approach introduces a regularization solution through filter-based methods, and we establish comprehensive theoretical results on convergence rates and optimality under a generalized source condition. These findings are applied to the fractional backward problem, specifically examining DFD system construction, relationships between DFD and SVD singular values, and extending existing source conditions for optimal regularization in polynomially and exponentially ill-posed scenarios.

• Publication year

  2025

• Venue

  Inverse Problems

• Publication date

  2025-07-31

• Fields of study

  Mathematics, Physics, Computer Science

• Identifiers
  DOI 10.1088/1361-6420/ae1998arXiv 2507.23651
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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