ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

Contributed talks on Dynamical Systems and Control

Tue 15:00–15:30, Auditorium Max Weber, Chair: Pieter Lietaert
Symmetrically structured linearizations for model reduction
Pieter Lietaert (KU Leuven)
Joint work with Karl Meerbergen (KU Leuven); Françoise Tisseur (The University of Manchester)

There are more and more applications of single-input-single-output systems whose matrix is a nonlinear function of the Laplace variable. Examples are thermo-acoustics, acoustics in porous materials, and PML boundary layers. In these applications, one is interested in obtaining a linear reduced model for coupling with time dependent models.

Model order reduction of linear single-input-single-output systems is well understood. The reduction techniques are often based on rational Krylov spaces, e.g., moment matching, IRKA and balanced truncation. Model order reduction of systems with nonlinear frequency dependencies is still an unsolved problem. For one-sided models, methods developed for nonlinear eigenvalue problems can be used. These methods rely on 'linearization' of a matrix polynomial that is expressed in some polynomial basis, e.g., Newton or Lagrange. These techniques can additionally be used for two-sided model reduction, but the output related subspace structure is difficult to efficiently exploit. Therefore, we propose symmetric structured Fiedler-like linearizations. The resulting model is reduced using a rational Krylov method, requiring only system solutions with the matrix of the original nonlinear system. This work is an extension of the CORK method for one-sided systems [1]. When monomials or (rational) Newton polynomials are used, the advantage of this structured linearization is the ability to add interpolation points adaptively during the Krylov steps. The obtained reduced model is linear and interpolates the transfer function in points of the Newton polynomials.

  1. R. Van Beeumen, K. Meerbergen, and W. Michiels, Compact rational Krylov methods for nonlinear eigenvalue problems, SIAM Journal on Matrix Analysis and Applications, 36(2) (2015), pp. 820–838.