Preconditioned linear solves for parametric model order reduction

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 of equations, which are predominantly dependent on slowly varying parameter values. We focus on efficiently solving these linear systems, specifically those arising in a set of algorithms for reducing linear dynamical systems with the parameters linearly embedded in the system matrices. We propose the use of the block variant of the problem-dependent underlying iterative method because often, all right hand sides 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 parametrically changing linear systems. We support our proposed theory by numerical experiments where-in two different models are reduced by a commonly used parametric model order reduction algorithm called RPMOR. Experimentally, we demonstrate that using a block variant of the underlying iterative solver saves nearly 95% of the computation time over the non-block version. Further, and more importantly, block GCRO with SPAI update saves around 60% of the time over block GCRO with SPAI.