Stability analysis of inexact linear solves in model order reduction

Abstract

A dynamical system describes a relation between two or more measurable quantities by a set of differential equations. We focus on first-order non-parametric as well as parametric dynamical systems with varying linearity (linear and bilinear). In general, dynamical systems corresponding to real-world applications are extremely large in size. Simulation and computation with such systems require a large amount of space and time. By using Model Order Reduction (MOR) techniques, these large dynamical systems are reduced into a smaller size, which makes the simulation and computation easier. MOR can be done in many ways, i.e., by using balanced truncation, Hankel approximations or Krylov projection. Projection methods obtain the reduced model by projecting the original full model on a lower-dimensional subspace and are quite popular. Interpolation is usually used to obtain the subspaces involved in the projection. Thus, these methods are referred to as interpolatory projection based MOR algorithms, which we specifically focus on.

In most of these MOR algorithms, people often use direct methods like LU-factorization, etc., to solve the arising linear systems, which have a high time complexity (cubic in terms of the system size). A common solution to this scaling problem is to use iterative methods like Krylov subspace methods, etc., which have a reduced time complexity (between linear and quadratic in terms of the system size), where nnz is the number of nonzeros in the system matrix). Although iterative methods are cheap, they are inexact too. Hence, studying the stability of the underlying MOR algorithms with respect to such approximate (inexact) linear solves becomes important.

One of the first works that performed such a stability analysis focused on popular MOR algorithms for first-order non-parametric linear dynamical systems. Here, the authors briefly mention that their analysis would be easily carried from the first-order to the second-order case. Some researchers expanded this stability analysis to reducing second-order non-parametric linear dynamical systems. Apart from this, a different kind of stability analysis for MOR of second-order non-parametric linear dynamical systems has also been done in literature. In this, the authors first show that the SOAR algorithm (second order Arnoldi) is unstable with respect to the machine precision errors (and not inexact linear solves). Then, they propose a Two-level orthogonal Arnoldi (TOAR) algorithm that cures this instability of SOAR.

Since our focus is on first-order systems, we extend the stability analysis done for the reduction of non-parametric linear dynamical systems to the reduction of the following classes of dynamical systems: non-parametric bilinear and parametric linear. Our analyses can be easily extended to MOR of parametric bilinear dynamical systems, leading to coverage of most of the existing MOR algorithms.

The innovative aspects of this work are as follows: capturing the behavior of bilinear terms in the stability conditions, providing two different sets of constraints for achieving backward stable algorithms, and easily satisfying the extra-orthogonality constraints imposed while achieving stability.