Overlapping domain decomposition and multigrid methods for inverse problems

Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients

“…In practical implementations, we need to invert the matrix C to get C À1 . The matrix C À1 can be replaced by domain decomposition or multigrid preconditioners for the Laplacian operator as in [40]. In all the examples, the mesh size is h ¼ 1=64.…”

Neumann--Neumann Domain Decomposition Preconditioners for Linear-Quadratic Elliptic Optimal Control Problems

“…Other DD methods for linear-quadratic elliptic optimal control problems are given in [1,2,3,4,5,6,42,36,37] and DD methods for a class of elliptic parameter identification problems are discussed in [19,40,49]. We refer to [37] and [44] for a brief comparison of the different approaches.…”

An Efficient Linear Solver for Nonlinear Parameter Identification Problems

“…Chen and Zou [5] and Keung and Zou [14] generalized the method to the case which allows the identifying coefficients to be discontinuous by using the regularization of bounded variations and they provided the rigorous theoretical justifications of the method and its finite element approximation. Independently, Chan and Tai [3,4,18] considered also the regularization of bounded variations and did numerous experiments on the performance of the augmented Lagrangian method for identifying highly discontinuous parameters.…”

“…at each iteration, where ε is the smoothing parameter introduced to smooth the BVnorm term in numerical implementations and c(q) is a linear function of q, which causes the indefiniteness of the system; see [3,4,5,14,18]. It seems there are very few iterative methods which are known to be globally convergent for solving such a troublesome system.…”