2015/11/16 by Erdem Altuntac, Altuntac, Erdem
Mathematics · #FOS: Mathematics #Optimization and Control (math.OC) #math.OC
paper · pdf · doi:10.48550/arxiv.1511.05040
arxiv created 2015/11/16 · arxiv updated 2015/11/17
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 V into ℝ+, J : V → ℝ+. It is assumed that some given data fδ is defined on a compact domain G ⊂ ℝ+ and in the class of Hilbert space, fδ ∈ L2(G). Then general Tikhonov functional associated with some given linear, compact and injective forward operator T : V → L2(G) is formulated as Fα(φ, fδ) : V × L2(G) → ℝ+
(φ, fδ) ↦ Fα(φ, fδ) := (1)/(2)\VertTφ- fδ\VertL2(G)2 + αJ(φ) . Convergence of the regularized solution φα(δ) ∈ argminφ∈ V Fα(φ, fδ) to the true solution φ† 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 Fα. 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.