A Symmetry Preserving Alternating Projection Method for Matrix Model Updating

The Matrix Model Updating Problem (MMUP), considered in this paper, concerns updating a symmetric second-order finite element model so that the updated model reproduces a given set of desired eigenvalues and eigenvectors by replacing the corresponding ones from the original model, and preserves the symmetry of the original model. In an optimization setting, this is a constrained nonlinear optimization problem. Taking advantage of the special structure of the constraint sets, it is first shown that the MMUP can be formulated as an optimization problem over the intersection of some special subspaces and linear varieties on the space of matrices. Using this formulation, an alternating projection method is then proposed and analyzed. The projections onto the involved subspaces and linear varieties are characterized. To the best of our knowledge, an alternating projection method for MMUP has not been proposed in the literature earlier. A distinct practical feature of the proposed method is that it is implementable using only a few measured eigenvalues and eigenvectors. No knowledge of the eigenvalues and eigenvectors of the associated quadratic matrix pencil is required. The results of our numerical experiments on both illustrative and benchmark problems show that the algorithm works well. The paper concludes with some future research problems.