2014/06/04 by Erdem Altuntac, Altuntac, Erdem
Mathematics · Engineering · Computer Science · #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.1406.1227
The problem of minimization of the least squares functional with a smooth, lower semi-continuous, convex regularizer J(⋅) is considered to be solved. Over some compact and convex subset Ω of the Hilbert space H, the regularizer is implicitly defined as J(⋅) : Ck(Ω, H) → ℝ+ where k ∈ \1,2\. So the cost functional associated with some given linear, compact and injective forward operator T :Ω⊂ H → H, Fα(⋅ , fδ) := (1)/(2) \Vert T( ⋅ ) - fδ\VertH2 + αJ(⋅) , where fδ is the given perturbed data with its perturbation amount δ in it. Convergence of the regularized optimum solution φα(δ) ∈ argmin Fα(φ, fδ) to the true solution φ† is analysed depending on the smoothness degree of the regularizer, i.e. the cases k ∈ \1,2\ in J(⋅) : Ck(Ω, H) → ℝ+. In both cases, we define such a regularization parameter that is in cooperation with the condition α(δ, fδ) ∈ \ αgt; 0 \vert \VertTφαδ - fδ\Vert ≤ τδ\ , for some fixed τ≥ 1. In the case of k = 2, we are able to evaluate the discrepancy \VertTφα(δ) - fδ\Vert≤ τδ with the Hessian Lipschitz constant LH of the functional Fα(⋅ , fδ).