Stability Analysis of Inexact Solves in Model Reduction of Non-parametric Second-order Dynamical systems

Abstract

Here, we focus on Model Order Reduction (MOR) of non-parametric second-order dynamical systems. In these MOR algorithms, sequences of large and sparse linear systems arise during the model reduction process. Solving such linear systems is the main computational bottleneck in efficient scaling of these MOR algorithms for reducing extremely large dynamical systems. Preconditioned iterative methods are often used for solving such linear systems. These iterative methods introduce errors because they solve the linear systems up to a certain tolerance. Hence, our focus is to analyze the stability of these MOR algorithms when using inexact linear solves. Adaptive Iterative Rational Global Arnoldi (AIRGA) is a popular MOR algorithm belonging to this category. We prove that, under four mild conditions, the AIRGA algorithm is backward stable with respect to the errors introduced by these inexact linear solves. Our results easily extend to other MOR algorithms belonging to this category. Our first condition enforces the use of a Ritz-Galerkin based linear solver, where the residual of a linear system is made orthogonal to the corresponding Krylov subspace. Our second condition requires satisfying few extra orthogonalities. We show how to modify the underlying linear solver to achieve these extra orthogonalities. We further demonstrate that using a recycling variant of the underlying linear solver helps us achieve these orthogonalities cheaply and with no code changes. Our third condition involves existence and invertibility of a matrix mostly dependent upon the input dynamical system, with the norm of this matrix bounded by one. Our fourth and final condition involves being able to compute a perturbation from the derived expression and bounding its norm by one as well. The last two conditions are easily satisfied by all our models.