Computational Methods for Model Reduction of Large-Scale Sparse Structured Descriptor Systems

Abstract

Currently descriptor systems, i.e., the systems whose dynamics obey differentialalgebraic equations (DAEs), play important roles in various disciplines of science and technology. In general, such systems are generated by finite element or finite difference methods. If the grid resolution becomes very fine, because many details must be resolved, the systems become very large. Moreover they are sparse, i.e., most of the elements in the matrices of the system are zero, which are not stored. A high dimensional system will always be complex, requiring a great deal of memory, thereby hindering computational performance significantly in simulation. Sometimes the systems are too large to store due to memory restrictions. Therefore, we seek to reduce the complexity of the model by applying model order reduction (MOR), i.e., we seek an approximation of the original model that well-approximates the behavior of the original model, yet is much faster to evaluate. We investigate efficient model reduction of sparse large-scale descriptor systems. We focus on the balancing based method balanced truncation (BT). A balanced truncation based method for such systems is introduced by Stykel (see, e.g., her PhD thesis, published in 2002). The author discusses a general framework of the BT method for a descriptor system. In general, the method is based on explicit computation of the spectral projectors onto the left and right deflating subspaces of the matrix pencil corresponding to the finite and infinite eigenvalues. Although these projectors are available for particular systems, computation is expensive. In this thesis, we focus on how to avoid computing such kind of projectors explicitly. Besides balanced truncation, the idea of avoidance of the projectors is extended to interpolation of transfer function, via iterative rational Krylov algorithms (IRKA) and projection onto dominant eigenspace, of the Gramian (PDEG) based model reduction methods. First, we discuss the model reduction problem for index 2 first order unstable descriptor systems arising from spatially discretized linearized Navier-Stokes equations. We apply our algorithms to the linearization of the von Karman vortex shedding at a moderate Reynolds number. We demonstrate that the resulting reduced model can be used to accurately simulate the unstable linearized model and to design a stabilizing controller. Future work will include the realization of the resulting control law for the full nonlinear model. Second, we investigate model reduction of a finite element model of a spindle head configuration in a machine tool. The special feature of this spindle head is that it is partially driven by a set of piezo actuators. Due