Variational Convergence Analysis With Smoothed-TV Interpretation

Abstract

The problem of minimizing the least squares functional with a Fr\'echet differentiable, lower semi-continuous, convex penalizer $J$ is considered to be solved. The penalizer maps the functions of Banach space $\mathcal{V}$ into $\mathbb{R}_{+},$ $ J : \mathcal{V} \rightarrow \mathbb{R}_{+}.$ It is assumed that some given data $f^{\delta}$ is defined on a compact domain $\mathcal{G} \subset \mathbb{R}_{+}$ and in the class of Hilbert space, $f^{\delta} \in \mathcal{L}^{2}(\mathcal{G}).$ Then general Tikhonov functional associated with some given linear, compact and injective forward operator $\mathcal{T} : \mathcal{V} \rightarrow \mathcal{L}^{2}(\mathcal{G})$ is formulated as \begin{eqnarray} F_{\alpha}(\varphi, f^{\delta}) : & \mathcal{V} \times \mathcal{L}^{2}(\mathcal{G}) & \rightarrow \mathbb{R}_{+} \nonumber\\ & (\varphi, f^{\delta}) & \mapsto F_{\alpha}(\varphi, f^{\delta}) := \frac{1}{2}\Vert\mathcal{T}\varphi - f^{\delta}\Vert_{\mathcal{L}^{2}(\mathcal{G})}^2 + \alpha J(\varphi) . \nonumber \end{eqnarray} Convergence of the regularized solution $\varphi_{\alpha(\delta)} \in \mathrm{argmin}_{\varphi \in \mathcal{V}} F_{\alpha}(\varphi, f^{\delta})$ to the true solution $\varphi^{\dagger}$ is analysed by means of Bregman divergence. First part of this aims to provide some general convergence analysis for generally strongly convex functional $J$ in the cost functional $F_{\alpha}$. In this part the key observation is that strong convexity of the penalty term $J$ with its convexity modulus implies norm convergence in the Bregman metric sense. In the second part, this general analysis will be interepreted for the smoothed-TV functional. The result of this work is applicable for any strongly convex functional.