SVD-Krylov Based Methods For Model Reduction Of Large-scale Dynamical Systems

The model reduction methods we will consider in this paper can be put under three cate-gories, that is, (i) SVD based methods; (ii) Krylov moment matching based methods, and (iii) SVD-Krylov based methods. The lase one is to connect SVD and Krylov based approximation methods, which is the current research trend in the area of model reduction.We present a new model reduction method of large scale dynamical systems, which belongs to SVD-Krylov based method. By the definition of shift operator, the proposed approximation is also obtained by solving an equality constrained least squares problem. The reduced model matches the first r+i Markov parameters of the full order model and the remaining ones approximately in a least-squares sense without explicitly computing them, where r is the order of reduced system, and i is is a nonnegative integer such that 1≤i< r. The reduced system minimizes a weighted (?)2 error. Moreover, the method is generalized for moment matching at arbitrary interpolation points. Several numerical examples verify the effectiveness of the approach.Iterative rational Krylov algorithms, such as IRK A [44] and ISRK [43], are numerically effective and suited for large-scale problem. Two modified iterative algorithms for (?)2 optimal model reduction are proposed in section 4.2, which can make the computational complexity reduced greatly, almost to half. Also they keep the same good properties as the original ones, such as the optimal H2 solution, stability or first-order necessary conditions for (?)2 optimality. In section 4.3, we propose the iterative algorithm based on the equality constrained Least Square model reduction discussed in the Chapter 3. The reduced model is also an (?)2 optional solution. Moreover, both algorithms of IRKA and ISRK turn out to be two special cases of the proposed algorithm。Several numerical examples verify it is numeically effective and suited for model reduction of large-scale dynamical systemsThe estimation with part bounded data uncertainties is posed and solved in the chapter 5 for efficiently computing. That is to consider the case to which only selected block of the coefficient matrix is subject to perturbations, while the remainings are known precisely. Its superior performance is due to the fact that the new method guarantees that the effect of the uncertainties will never be unnecessarily over-estimated, beyond what is reasonably assumed by the a-prior bounds, consequently, overly conservative designs, as well as overly sensitive designs, are avoided. In contrast to the BDU estimation, once the worst-case perturbation is identified, the solution cannot be characterized by the orthogonality condition. Moreover, the PBDU problem can connect to a weighted least-square problem, and we can obtain the unique simplified solution via the SVD.