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

Abstract

Let $X$ and $Y$ be Hilbert spaces, and $\mathbf{K}: \text{dom} \mathbf{K} \subset X \to Y$ a bounded linear operator. This paper addresses the inverse problem $\mathbf{K}x = y$, where exact data $y$ is replaced by noisy data $y^\delta$ satisfying $\|y^\delta - y\|_Y \leq \delta$. 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 $x^\delta_\alpha = \sum_{\lambda \in \Lambda} \kappa_\lambda g_\alpha(\kappa_\lambda^2) \langle y^\delta, v_\lambda \rangle \overline{u}_\lambda$. Convergence rates and optimality are analyzed under a generalized source condition $\mathbf{M}_{\varphi, E} = \{ x \in \text{dom} \mathbf{K} : \sum_{\lambda \in \Lambda} [\varphi(\kappa_\lambda^2)]^{-1} |\langle x, u_\lambda \rangle|^2 \leq E^2 \}$. 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.