Variational Convergence Analysis With Smoothed-TV Interpretation

Abstract

The problem of minimizing the least squares functional with a Fréchet differentiable, lower semi-continuous, convex penalizer $J$ is considered to be solved. The penalizer maps the functions of Banach space $\mathcal{V}$ into ${\mathbb{R}}_{+},$ $J:\mathcal{V}\to {\mathbb{R}}_{+}.$ 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 $\mathcal{T}:\mathcal{V}\to {\mathcal{L}}^2(\mathcal{G})$ is formulated as \begin{eqnarray} F_α(φ, f^δ) : & \mathcal{V} \times \mathcal{L}^{2}(\mathcal{G}) & \rightarrow \mathbb{R}_{+} \nonumber\\ & (φ, f^δ) & \mapsto F_α(φ, f^δ) := \frac{1}{2}\Vert\mathcal{T}φ- f^δ\Vert_{\mathcal{L}^{2}(\mathcal{G})}^2 + αJ(φ) . \nonumber \end{eqnarray} Convergence of the regularized solution ${\phi }_{\alpha (\delta )}\in {\mathrm{argmin}}_{\phi \in \mathcal{V}}{F}_{\alpha }(\phi ,f^{\delta })$ to the true solution ${\phi }^{†}$ 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.