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).

Preconditioned Least-Squares Petrov-Galerkin Reduced Order Models

“…The second category of methods explore the idea of introducing a preconditioner within the ROM workflow to speed up the online evaluation of the ROM. This work is motivated by the observation that the repeated solution of linear systems, characterized by dense matrices, can be the main computational bottleneck in efficient scaling of ROMs [69]. Commonly, the linear systems arising within the ROM workflow are solved using direct methods.…”

“…These preconditioners are derived from preconditioners used in solving the FOM, and are shown to be more efficient than direct solvers for problems depending on a moderately large number of parameters. Singh et al develop strategies for reusing preconditioners in projection-based model reduction algorithms for parametric [70] and non-parametric [69] ROMs to improve scalability of ROM solution over direct methods. These authors introduce a novel Sparse Approximate Inverse (SPAI) preconditioner for ROMs constructed via moment matching using the Bilinear Iterative Rational Krylov Algorithm (BIRKA) algorithm.…”

Informative Order-Reduction of Underdamped Third-Order Systems

“…The damping ratio of the second-order system part The natural frequency of the second-order part The damping factor of the second-order system The damping frequency of the second-order system 1 The first pole of the third-order system…”

“…However, working at operation regions where the initial assumptions do not coincide with the actual conditions will lead to inaccurate representation. Reduced-order models received more attention in the last decade to work with higher-order models [1]. Unlike second-order differential equations, the third-order differential equations don't have a unique general solution; instead, they have a procedure that must be followed to reach the answers.…”

“…Even for simulation analysis, third-order models are usually used [12]. Model order reduction concept is widely used in many engineering applications and research [1,13,14]. The authors in [15] reduced the order of active magnetic bearing and developed new control strategy.…”