In previous work, we experimented a variant of the two-level ideal algorithm for parametric shape optimization that was proposed by Desideri, and two of three approaches were shown superior. In this report, we try to look them in a more classic way -- in terms of singular value decomposition (SVD) and regularizations, for model problems, i.e., we combine these regularization tools with these two-level ideal algorithms. Numerical results show that regularized two level algorithms are more robust.