Stable linear solves with preconditioner updates for model reduction

Abstract

Simulation of large dynamical systems can be unmanageable due to high demands on computational resources. These large systems can be reduced into a smaller di mension by using Model Order Reduction (MOR) techniques. The reduced system has approximately the same characteristics as the original system but it requires sig nificantly less computational effort in simulation. MOR can be done in many ways such as balanced truncation, Hankel approximations, and Krylov projection. Among these, the projection methods are quite popular, and hence, we focus on them. We work with a wide array of MOR algorithms for reducing an extensive range of linear dynamical systems. That is, parametric/non-parametric as well as first-order and second-order. In these MOR algorithms, sequences of very large and sparse linear systems arise during the model reduction process. Solving such linear systems is the main computa tional bottleneck in efficient scaling of these MOR algorithms for reducing extremely large dynamical systems. Preconditioned iterative methods are often used for solving such linear systems. These iterative methods introduce errors because they solve the linear systems up to a certain tolerance. Hence, our first focus is to analyze the stability of MOR algorithms when using inexact linear solves. Further, in these MOR algorithms, the change from one linear system to the next is usually very small, and hence, the applied preconditioner could be reused, which is our second focus. Here, by using our tech niques we demonstrate that a 1.2 million problem can be reduced in 3 days instead of earlier 8 days.