A multilevel model correction method for parameter identification

Output least-squares formulation with Tikhonov regularization is one of the most frequently used and reliable methods for parameter identification in PDEs. As the discretized optimization system of the formulation is nonlinear, ill-conditioned, and constrained with some PDEs, its numerical solution is very time-consuming and often unstable. In this paper, we present a multilevel model correction method, which aims at solving the general nonlinear output least-squares optimization system in an efficient and robust manner. The multilevel method is conducted iteratively so that larger part of its computations is done on a sequence of nested coarse meshes defined on the concerned physical domain. The key ingredient of the method is to update the output least-squares objective functional on each coarse mesh by using the most updated information of the identifying parameter from a finer mesh. Numerical experiments are provided to demonstrate the robustness and efficiency of the method.