. This paper considers problems of distributed parameter estimation from data measurements on solutions of partial differential equations (PDEs). A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e., the model). This functional consists of a data fitting term, involving the solution of a finite volume or finite element discretization of the forward differential equation, and a Tikhonov-type regularization term, involving the discretization of a mix of model derivatives. We develop a multigrid method for the resulting constrained optimization problem. The method directly addresses the discretized PDE system which defines a critical point of the Lagrangian. The discretization is cell-based. This system is strongly coupled when the regularization parameter is small. Moreover, the compactness of the discretization scheme does not necessarily follow from compact discretizations of the forward model and of the regularization term. We therefore employ a Marquardt-type modification on coarser grids. Alternatively, fewer grids are used and a preconditioned Krylov-space method is utilized on the coarsest grid. A collective point relaxation method (weighted Jacobi or a Gauss-Seidel variant) is used for smoothing. We demonstrate the efficiency of our method on a classical model problem from hydrology. 1.

In this paper, we consider a multigrid application in digital image processing. Here, the problem is to find a map, which transforms an image T into another image R such that the grey level of the different images are nearly equal in every picture-element. This problem arises in the investigation of human brains. The complete inverse problem is ill posed in the sense of Hadamard and nonlinear, so the numerical solution is quite difficult. We solve the inverse problem by a Landweber iteration. In each minimization step an approximate solution for the linearized problem is computed with a multigrid method as an inner iteration. Finally, we present some experimental results for synthetic and real images.

This paper considers problems of distributed parameter estimation from data measurements on solutions of dierential equations. A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e., the model). This functional consists of a data tting term, involving the solution of a nite volume or nite element discretization of the forward dierential equation, and a Tikhonov-type regularization term, involving the discretization of a mix of model derivatives. The grid spacing of the model discretization, as well as the relative weight of the entire regularization term, aect the sort of regularization achieved. We investigate a number of questions arising regarding their relationship, including the degree of nonlinearity of the least squares functional. We also investigate the correct scaling of the regularization matrix, where we rigorously associate the practice of using unscaled regularization matrices with approximations of...