Adaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction

• Published in: Advances in computational mathematics Vol. 50; no. 4
• Main Authors: Aumann, Quirin, Werner, Steffen W. R.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2024; Springer Nature B.V
• Subjects: Algorithms; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Eigenvalues; Frequency ranges; Interpolation; Iterative methods; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Model reduction; Nonlinear systems; Reduced order models; Visualization; 93A15; Structure preservation; 41A30; Projection methods; Structured interpolation; 30E05; 65D05; Model order reduction; 93C80; Dynamical systems
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-024-10166-z

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures.