Regularization of Inverse Problems by Filtered Diagonal Frame Decomposition under general source

2025/07/31 by Trong, Dang Duc, Minh, Nguyen Dang, Thang, Luu Xuan +1
#Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2507.23651

Abstract

Let X and Y be Hilbert spaces, and K: dom K ⊂ X → Y a bounded linear operator. This paper addresses the inverse problem Kx = y, where exact data y is replaced by noisy data y^δ satisfying ‖y^δ- y‖_Y ≤ δ. Due to the ill-posedness of such problems, we employ regularization methods to stabilize solutions. While singular value decomposition (SVD) provides a classical approach, its computation can be costly and impractical for certain operators. We explore alternatives via Diagonal Frame Decomposition (DFD), generalizing SVD-based techniques, and introduce a regularized solution x^δ_α= ∑_{λ∈ Λ} κ_λg_α(κ_λ²) ⟨ y^δ, v_λ⟩ u_λ. Convergence rates and optimality are analyzed under a generalized source condition M_{φ, E} = \ x ∈ dom K : ∑_{λ∈ Λ} [φ(κ_λ²)]^-1 |⟨ x, u_λ⟩|² ≤ E² \. Key questions include constructing DFD systems, relating DFD and SVD singular values, and extending source conditions. We present theoretical results, including modulus of continuity bounds and convergence rates for a priori and a posteriori parameter choices, with applications to polynomial and exponentially ill-posed problems.

Citations

• Convergence of non-linear diagonal frame filtering for regularizing inverse problems
• Regularization of Inverse Problems by Filtered Diagonal Frame Decomposition
• Sparse regularization of inverse problems by operator-adapted frame thresholding
• Efficient regularization with wavelet sparsity constraints in photoacoustic tomography
• Continuity of solutions of a class of fractional equations
• Numerical Evaluation of Two and Three Parameter Mittag-Leffler Functions
• Total variation regularization for a backward time-fractional diffusion problem

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