Understanding the permeability of the subsurface is a crucial step to simulate fluid flow in the subsurface. A parameter estimation problem for the flow equations can be solved to find the permeability. The robust identification of material parameters remains a significant challenge. In classical approaches, the non-linear least-squares problem is formulated as a non-linear optimization problem in which the partial differential equation governing the permeability field acts as a constraint. These approaches lead to large scale problems and are, therefore, computationally challenging. This thesis proposes a new approach based on mesh decoupling of state and design variables. This approach allows treating the design variables on various scales of resolution without comprising the accuracy of the state and adjoint solver. The method is implemented on a
one-dimensional and two-dimensional Poisson problem taken from the literature. The first numerical results show that the multilevel approach is able to accelerate the initial stages of the search procedure significantly.