On the stabilization of singular identification problem of an unknown discontinuous diffusion parameter in elliptic equation

Convergence analysis of an alternating direction method of multipliers for the identification of nonsmooth diffusion parameters with total variation

“…Theorem 3.3 presents a valuable way to characterize solutions of optimization problems through saddle points of the augmented Lagrangian, which is advantageous for numerical methods. Specifically, building upon the surjectivity of the error operators' derivatives demonstrated in theorem 3.2, we establish an equivalence between resolving the optimization problem (21) and identifying the saddle points of the augmented Lagrangian (see also [16]). This equivalence introduces a broad range of duality techniques that can be utilized from a numerical standpoint.…”

“…In recent years, several studies have explored convergence rates for TV regularization, such as a rigorous finite element method analysis proposed in [25] for conductivity identification in electrical impedance tomography with TV regularization. Another work in [24] introduced a new discrete variant of TV regularization using the finite element method and nodal quadrature formula, while an alternative approach based on the finite volume method was proposed in [16]. In [22,23], convergence rates were derived for identifying parameters in elliptic equations.…”

“…Several studies have tackled minimization problems similar to (11). For instance, in [13,16], the authors used duality techniques to approach the solutions. They introduced an augmented Lagrangian functional and sought the solutions as saddle points of this functional.…”