Linearized primal-dual methods for linear inverse problems with total variation regularization and finite element discretization

Abstract

Linear inverse problems with total variation regularization can be reformulated as saddle-point problems; the primal and dual variables of such a saddle-point reformulation can be discretized in piecewise affine and constant finite element spaces, respectively. Thus, the well-developed primal-dual approach (a.k.a. the inexact Uzawa method) is conceptually applicable to such a regularized and discretized model. When the primal-dual approach is applied, the resulting subproblems may be highly nontrivial and it is necessary to discuss how to tackle them and thus make the primal-dual approach implementable. In this paper, we suggest linearizing the data-fidelity quadratic term of the hard subproblems so as to obtain easier ones. A linearized primal-dual method is thus proposed. Inspired by the fact that the linearized primal-dual method can be explained as an application of the proximal point algorithm, a relaxed version of the linearized primal-dual method, which can often accelerate the convergence numerically with the same order of computation, is also proposed. The global convergence and worst-case convergence rate measured by the iteration complexity are established for the new algorithms. Their efficiency is verified by some numerical results.

A primal-dual algorithm framework for convex saddle-point optimization

“…Recently, Zhang, Zhu, and Wang [ 10 ] proposed a simple primal-dual method for total-variation image restoration problems and showed that their iterative scheme has the convergence rate in the ergodic sense. When we had finished this paper, we found the algorithm proposed in [ 11 ], where convergence analysis was similar to our proposed frame. However, the algorithm proposed in [ 11 ] is actually a particular case of our unified framework when the precondition matrix in our frame is fixed.…”

“…When we had finished this paper, we found the algorithm proposed in [ 11 ], where convergence analysis was similar to our proposed frame. However, the algorithm proposed in [ 11 ] is actually a particular case of our unified framework when the precondition matrix in our frame is fixed.…”

“…Also, the authors gave an algorithm based on Bregman operator splitting (BOS) when A is not diagonalizable. Recently, Tian and Yuan [ 11 ] proposed a linearized primal-dual method for linear inverse problems with total-variation regularization and showed that this variant yields significant computational benefits. Next, we show that different P in ( 7 ) can induce the following well-known methods: the linearized primal-dual method, SIU, BOS, and split Bregman methods and some new primal-dual algorithms with the correction step ( 8 ).…”

“…The linearized primal-dual method in [ 11 ] can be directly induced by setting and We can also easily show that the positive definiteness of the matrices P and Q in ( 34 ) is guaranteed if , , . In this situation, the scheme ( 7 ) can be written as follows: and the scheme ( 8 ) can be expressed as The idea is also similar to that of [ 22 , 23 ], which uses the symmetric positive semi-definite matrix instead of the identity matrix in the proximal term.…”

Finite element approximation of source term identification with TV-regularization

“…is proposed in [49,Algorithm 1], where the parameters τ, θ are chosen suitably. Furthermore, we note that the solution p h n+1 to the sub-problem (1.13) is given by the explicit form (see [8,49])…”