[Paper Review] Stability-Preserving, Adaptive Model Order Reduction of DAEs by Krylov-Subspace Methods

Alessandro Castagnotto, Heiko K. F. Panzer+2 more|arXiv (Cornell University)|Aug 28, 2015

Model Reduction and Neural Networks14 references3 citations

TL;DR

This paper presents two Krylov-subspace-based model order reduction (MOR) techniques for index-1 differential-algebraic equations (DAEs) that preserve stability and structural properties. The methods enable adaptive, stability-preserving MOR without explicit transformation to ODEs, with one method ensuring strict dissipativity and the other enabling H₂-pseudo-optimal reduction with adaptive parameter selection.

ABSTRACT

Systems of differential-algebraic equations (DAEs) represent a widespread formalism in the modeling of constrained mechanical systems and electrical networks. Due to the automatic, object-oriented generation of the equations of motion and the resulting redundancies in the descriptor variables, DAE systems often reach a very high order. This motivates the use of model order reduction (MOR) techniques that capture the relevant input-output dynamics in a reduced model of much smaller order, while satisfying the constraints and preserving fundamental properties. Due to their particular structure, new MOR techniques designed to work directly on the DAE are required that reduce the dynamical part while preserving the algebraic. In this contribution, we exploit the specific structure of index-1 systems in semi-explicit form and present two different methods for stability-preserving MOR of DAEs. The first technique preserves strictly dissipativity of the underlying dynamics, the second takes advantage of H2-pseudo-optimal reduction and further allows for an adaptive selection of reduction parameters such as reduced order and Krylov shifts.

Motivation & Objective

• Address the lack of stability-preserving model order reduction (MOR) techniques for differential-algebraic equations (DAEs), particularly in high-order systems.
• Develop MOR methods that preserve the algebraic constraints and intrinsic stability properties of index-1 DAEs without requiring explicit solution of the underlying ODE.
• Enable adaptive selection of reduced order and Krylov shifts in MOR to improve accuracy and efficiency.
• Ensure strict dissipativity and H₂-optimality in the reduced model while maintaining the structural properties of the original DAE.
• Provide a framework for MOR of large-scale DAEs arising in mechanical and electrical systems without sacrificing numerical stability or constraint satisfaction.

Proposed method

• Utilize Petrov-Galerkin projection with Krylov-subspace methods to construct reduced-order models (ROMs) directly from the DAE system in semi-explicit index-1 form.
• Apply orthogonal projection (V = W) under conditions ensuring that the reduced system preserves the descriptor structure and algebraic constraints.
• Design a first method that guarantees strict dissipativity of the reduced system by preserving the negative definiteness of A + A^T.
• Implement a second method based on H₂-pseudo-optimal moment matching, which allows adaptive selection of reduced order and Krylov shifts via residual minimization.
• Use Schur complement techniques to analyze and preserve the stability properties of the reduced system by ensuring A₁ + A₁^T ≺ 0 in the reduced dynamical subsystem.
• Construct the reduced system via projection matrices V and W derived from Krylov subspaces spanned by (sE - A)^{-1}B, enabling moment matching at selected shift parameters.

Experimental results

Research questions

• RQ1 Can Krylov-subspace-based MOR for DAEs preserve the stability of the original system without transforming to an ODE?
• RQ2 How can strict dissipativity be preserved in the reduced-order model of a DAE without explicit knowledge of the underlying ODE?
• RQ3 Can H₂-pseudo-optimal reduction be adapted for DAEs to allow adaptive selection of reduced order and Krylov shifts?
• RQ4 What conditions ensure that the Petrov-Galerkin projection preserves the algebraic constraints and index-1 structure in the reduced model?
• RQ5 How do the proposed methods compare in accuracy and stability to standard Krylov-based MOR applied to ODEs or DAEs without stability guarantees?

Key findings

• The first method ensures that if the original DAE is strictly dissipative (A + A^T ≺ 0), then the reduced system also satisfies A_r + A_r^T ≺ 0, preserving stability.
• The second method achieves H₂-pseudo-optimality by matching moments at selected shift parameters, enabling adaptive tuning of reduced order and Krylov shifts.
• Numerical experiments demonstrate that both methods preserve the algebraic constraints and maintain stability even when standard moment-matching techniques fail.
• The use of Schur complement analysis confirms that the stability of the reduced dynamical subsystem depends on the invertibility and definiteness of the Schur complement A₁.
• The proposed techniques avoid the need to compute the underlying ODE, enabling direct MOR of DAEs while preserving structural and stability properties.
• The transmission line SE-DAE case study confirms that the methods maintain accuracy and stability across varying system sizes and reduction levels.