Multivariate Interpolation Methods for Parametric Model Reduction (MIM4PMOR)

• Applicants: Professor Dr. Athanasios C. Antoulas; Professor Dr. Peter Benner
• Subject Area: Mathematics
• Term: from 2011 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 201063239

• Final Report Year: 2020

Final Report Abstract

At MPI Magdeburg, the connection between bilinear and linear parameter-varying systems was derived and its use for model order reduction was further investigated. The connection was employed to derive optimal interpolation conditions. In this work, a new interpolation-based H2 -framework for bilinear control systems and a new way of computing the H2 -norm of such bilinear systems has been introduced. As a generalization of the iterative rational Krylov algorithm (IRKA), two iterative algorithms have been introduced which compute reduced-order systems that satisfy local H2-optimality conditions. This built the basis for new IRKA-like algorithms which also compute H2-optimal ROMs with guaranteed stability preservation. Moreover, a connection with generalized Sylvester equations for deriving a reduced-order model satisfying the H2-optimal interpolation conditions was investigated. Here also efficient iterative algorithms for the solution of these generalized Sylvester equations were proposed, using similar developments for bilinear Lyapunov equation. The team at JU Bremen developed a framework for multi-variate rational interpolation. The main tool is a generalization of the single-variable Loewner matrix. A key result states that as before, all sufficiently large Loewner matrices having a particular structure, reveal information about the minimal complexity of the underlying rational function. Again, the null space contains the information needed to reconstruct the underlying two-variable rational functions or reduced versions thereof. Furthermore, using the observation that linear parametric systems can be re-interpreted as bilinear systems, the problem of model reduction of bilinear systems from data was addressed. The data consists of values of high order transfer functions. The underlying philosophy is: collect data and extract the desired information. This was accomplished by extending the Loewner framework to the reduction of bilinear systems. Its main features are: (i) Given input/output data, we can construct with no computation, a singular high order model in generalized (descriptor) state space form.

Publications

• Interpolation-based H2 -model reduction of bilinear control systems, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 859–885
  Benner, Peter & Breiten, Tobias
  
• On two-variable rational interpolation, Lin. Alg. Appl., 436 (2012), pp. 2889–2915
  Antoulas, A.C.; Ionita, A.C. & Lefteriu, S.
  
• Low rank methods for a class of generalized Lyapunov equations and related issues, Numerische Mathematik, 124 (2013), pp. 441–470
  Benner, Peter & Breiten, Tobias
  
• Case Study: Parametrized Reduction Using Reduced-Basis and the Loewner Framework, in Reduced Order Methods for Modeling and Computational Reduction, A. Quarteroni and G. Rozza, eds., Modeling, Simulation and Applications, Springer–Verlag, pp. 51–66, 2013
  Ionita, Antonio C. & Antoulas, Athanasios C.
  
• Data-driven parametrized model reduction in the Loewner framework, SIAM J. Sci. Comp., 36:3 (2014), pp. A984–A1007
  Ionita, A. C. & Antoulas, A. C.
  
• Parametric model order reduction of thermal models using the bilinear interpolatory rational Krylov algorithm, Math. Comput. Model. Dyn. Syst., 21 (2015), pp. 103–129
  Bruns, Angelika & Benner, Peter
  
• Model reduction of bilinear systems in the Loewner framework, SIAM A.C. Antoulas, A.C. Ionita, SIAM J. Sci. Comp., 38:5 (2018), pp. B889–B916
  Antoulas, A. C.; Gosea, I. V. & Ionita, A. C.
  
• Chapter 8: A tutorial introduction to the Loewner Framework for Model Reduction, in Model Reduction and Approximation for Complex Systems, in Model Reduction and Approximation: Theory and Algorithms, P. Benner, A. Cohen, M. Ohlberger, and K. Willcox, eds., SIAM, 2017, pp. 335–376
  Antoulas, Athanasios C.; Lefteriu, Sanda & Ionita, A. Cosmin
  
• Comparison of methods for parametric model order reduction of time-dependent problems, in Model Reduction and Approximation: Theory and Algorithms, P. Benner, A. Cohen, M. Ohlberger, and K. Willcox, eds., SIAM, 2017, pp. 377–407
  Baur, Ulrike; Benner, Peter; Haasdonk, Bernard; Himpe, Christian; Martini, Immanuel & Ohlberger, Mario
  
• Implicit Volterra series interpolation for model reduction of bilinear systems, J. Comput. Appl. Math., 316 (2017), pp. 15–28
  Ahmad, Mian Ilyas; Baur, Ulrike & Benner, Peter