Reusing Preconditioners in Projection Based Model Order Reduction Algorithms

Abstract

Dynamical systems are pervasive in almost all engineering and scientific applications. Simulating such systems is computationally very intensive. Hence, Model Order Reduction (MOR) is used to reduce them to a lower dimension. Most of the MOR algorithms require solving large sparse sequences of linear systems. Since using direct methods for solving such systems does not scale well in time with respect to the increase in the input dimension, efficient preconditioned iterative methods are commonly used. In one of our previous works, we have shown substantial improvements by reusing preconditioners for the parametric MOR (Singh et al. 2019). Here, we had proposed techniques for both, the non-parametric and the parametric cases, but had applied them only to the latter. We have three main contributions here. First, we demonstrate that preconditioners can be reused more effectively in the non-parametric case as compared to the parametric one. Second, we show that reusing preconditioners is an art via detailed algorithmic implementations in multiple MOR algorithms. Third and final, we demonstrate that reusing preconditioners for reducing a real-life industrial problem (of size 1.2 million), leads to relative savings of up to 64 % in the total computation time (in absolute terms a saving of 5 days).