A multilevel augmentation method is considered to solve parameter identification problems in elliptic systems. With the help of the natural linearization technique, the identification problems can be transformed into a linear ill-posed operation equation, where noise exists not only in RHS data but also in operators. Based on multiscale decomposition in solution space, the multilevel augmentation method leads to a fast algorithm for solving discretized ill-posed problems. Combining with Tikhonov regularization, in the implementation of the multilevel augmentation method, one only needs to invert the same matrix with a relatively small size and perform a matrix-vector multiplication at the linear computational complexity. As a result, the computation cost is dramatically reduced. The a posteriori regularization parameter choice rule and the convergence rate for the regularized solution are also studied in this work. Numerical tests illustrate the proposed algorithm and the theoretical estimates.
© 2013 by Walter de Gruyter Berlin Boston