Preconditioned Linear Solves for Parametric Model Order Reduction
For engineers simulating parametric dynamical systems, this provides efficient linear solvers that significantly speed up reduced-order model construction.
This work reduces the computational cost of solving sequences of large sparse linear systems in parametric model order reduction by using block iterative methods and a novel cheap update technique for Sparse Approximate Inverse (SPAI) preconditioners, achieving 80% time savings with block methods and 70% with SPAI updates.
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.