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

Kapil Ahuja, Navneet Pratap Singh

TL;DR

This work addresses the stability of Model Order Reduction for non-parametric second-order dynamical systems when large sparse linear systems are solved with inexact, preconditioned iterative methods. By focusing on the AIRGA algorithm, it develops a backward stability framework requiring four conditions, including Ritz-Galerkin residual orthogonality and extra orthogonality properties, achievable via Recycling CG with modest code changes. The authors derive an explicit bound on the reduced-system error in terms of a conditioning measure κ(H(s)) and the perturbation Z, and show how to compute Z from solver residuals using a low-rank SVD-based approach. Numerical experiments on a 10k-degree beam model and a 17k-degree gyroscope model demonstrate that tighter linear-solver tolerances improve reduced-model accuracy, with conditioning playing a crucial role in the achievable precision. Overall, the paper provides a principled way to perform scalable and reliable MOR for large second-order systems using inexact solves, with practical guidance for implementation and preconditioning.

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.

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

TL;DR

This work addresses the stability of Model Order Reduction for non-parametric second-order dynamical systems when large sparse linear systems are solved with inexact, preconditioned iterative methods. By focusing on the AIRGA algorithm, it develops a backward stability framework requiring four conditions, including Ritz-Galerkin residual orthogonality and extra orthogonality properties, achievable via Recycling CG with modest code changes. The authors derive an explicit bound on the reduced-system error in terms of a conditioning measure κ(H(s)) and the perturbation Z, and show how to compute Z from solver residuals using a low-rank SVD-based approach. Numerical experiments on a 10k-degree beam model and a 17k-degree gyroscope model demonstrate that tighter linear-solver tolerances improve reduced-model accuracy, with conditioning playing a crucial role in the achievable precision. Overall, the paper provides a principled way to perform scalable and reliable MOR for large second-order systems using inexact solves, with practical guidance for implementation and preconditioning.

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.

Paper Structure

This paper contains 19 sections, 73 equations, 1 figure, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Iterative Methods and Preconditioners
  3. Stability Analysis of MOR Algorithms
  4. Stability Analysis of AIRGA
  5. Backward Stability Analysis
  6. First Condition of Stability
  7. Second Condition for Stability
  8. Satisfying Backward Stability Conditions
  9. Achieving Extra Orthogonalities
  10. Adapting the Lanczos Process
  11. Adapting the Ritz-Galerkin Projection
  12. Implementation
  13. Accuracy of the Reduced Systems
  14. Conditioning Expression
  15. Computation of Perturbation
  16. ...and 4 more sections

Figures (1)

  • Figure 1: Accuracy of the reduced system plotted at each AIRGA iteration for two different stopping tolerances in RCG; one dimensional beam model.