Preconditioned Linear Solves for Parametric Model Order Reduction

TL;DR

This paper presents an efficient approach for solving large parametric linear systems in model order reduction using block iterative methods and an innovative, cost-effective SPAI preconditioner update technique, significantly reducing computation time.

Contribution

It introduces a problem-dependent choice of iterative solvers, advocates for block methods with SPAI preconditioning, and proposes a novel, inexpensive SPAI update method for parametric systems.

Findings

80% reduction in computation time with block iterative methods

70% reduction in time using SPAI preconditioner updates

Effective handling of parametric system variations

Abstract

The main computational cost of algorithms for computing reduced-order models of parametric dynamical systems is in solving sequences of very large and sparse linear systems. We focus on efficiently solving these linear systems, arising while reducing second-order linear dynamical systems, by iterative methods with appropriate preconditioners. We propose that the choice of underlying iterative solver is problem dependent. We propose the use of block variant of the underlying iterative method because often all right-hand-side are available together. Since, Sparse Approximate Inverse (SPAI) preconditioner is a general preconditioner that can be naturally parallelized, we propose its use. Our most novel contribution is a technique to cheaply update the SPAI preconditioner, while solving the parametrically changing linear systems. We support our proposed theory by numerical experiments where we first show benefit of 80% in time by using a block iterative method, and a benefit of 70% in time by using SPAI updates.