Available methods for linear ODEs are listed in Table (according to , an overview from an engineering perspective is in ).
SVD-based (Truncated Balanced Approximation, Singular Perturbation Approximation, Hankel-Norm Approximation).
Have a global error estimate, can be used in a fully automatic manner.
Computational complexity of conventional implementations is O(N^3), can be used for systems with order less than a few thousand unknowns only.
Low-rank Gramian approximants and matrix sign function method.
Have a global error estimate and the computational complexity is acceptable.
Currently under development.
Pade approximants (moment matching) via Krylov subspaces by means of either the Arnoldi or Lanczos process.
Very advantageous computationally, can be applied to very high-dimensional linear systems.
Does not have a global error estimate. It is necessary to select the order of the reduced system manually or with some engineering tricks.
It happens that for high-dimensional finite element models, implicit moment matching is working extremely well. Examples for different domains can be found at the Applications page. Model Reduction inside ANSYS (see Software) can perform implicit moment matching directly for ANSYS models. It can be used for other software as well provided that system matrices are available.
The lack of error estimates for implicit moment matching can be overcomed with the use of error indicators, as described in
T. Bechtold, E. B. Rudnyi and J. G. Korvink,
Error indicators for fully automatic extraction of heat-transfer macromodels for MEMS.
Journal of Micromechanics and Microengineering 2005, v. 15, N 3, pp. 430-440.
Preprint, Final paper at IOP.
Publications in reverse chronological order.
E. B. Rudnyi and J. G. Korvink.
Model Order Reduction for Large Scale Engineering Models Developed in ANSYS.
Lecture Notes in Computer Science, v. 3732, pp. 349-356, 2006.
Title: Applied Parallel Computing. State of the Art in Scientific Computing: 7th International Conference, PARA 2004, Lyngby, Denmark, June 20-23, 2004. Revised Selected Papers.
Final paper at Springer.
We present the software mor4ansys that allows engineers to employ modern model reduction techniques to finite element models developed in ANSYS. We focus on how one extracts the required information from ANSYS and performs model reduction in a C++ implementation that is not dependent on a particular sparse solver. We discuss the computational cost with examples related to structural mechanics and thermal finite element models.
E. B. Rudnyi and J. G. Korvink.
Model Order Reduction of MEMS for Efficient Computer Aided Design and System Simulation.
MTNS2004, Sixteenth International Symposium on Mathematical Theory of Networks and Systems, Katholieke Universiteit Leuven, Belgium, July 5-9, 2004
In the present paper we focus on the application aspect of Krylov-subspace-based model order reduction. We present the software mor4ansys that allows us to directly apply model reduction to ANSYS finite element models. We show that, for many MEMS thermal and structural mechanics problems, model reduction is very efficient means to generate compact models for system-level simulation. Finally, we discuss on how one can use model reduction during the engineering design process.
E. B. Rudnyi, J. G. Korvink.
Model Order Reduction for Large Scale Finite Element Engineering Models.
ECCOMAS CFD 2006, Europian Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, September 5-8, 2006.
Software MOR for ANSYS has been developed at IMTEK in 2003. It allows us to perform model reduction directly to finite element models developed in ANSYS. The goal of the present paper is to describe progress achieved for the last two years and review our publications with application of MOR for ANSYS to various engineering problems for different domains: heat transfer, structural mechanics, thermomechanical models, and acoustics including fluid-structure interaction. We also discuss computational scalability of model reduction and the advanced development such as parametric and weakly nonlinear model reduction.
P. Benner, Lihong Feng, E. B. Rudnyi.
Using the Superposition Property for Model Reduction of Linear Systems with a Large Number of Inputs.
MTNS2008, Proceedings of the 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS2008), Virginia Tech, Blacksburg, Virginia, USA, July 28-August 1, 2008, 12 pages, 2008.
 Athanasios C. Antoulas,
Approximation of Large-Scale Dynamical Systems.
Society for Industrial and Applied Mathematic, 2005, ISBN: 0898715296.
Book at Amazon or at SIAM.
 E. B. Rudnyi, J. G. Korvink,
Review: Automatic Model Reduction for Transient Simulation of MEMS-based Devices.
Sensors Update v. 11, p. 3-33, 2002.
Final paper at Wiley.
Krylov Projection Methods for Model Reduction
Pieter J. Heres,
Robust and efficient Krylov subspace methods for model order reduction
Model Reduction of Linear Systems, an Interpolation Point of View
Numerical Methods for Large Scale Matrix Equations with Applications in LTI System Model Reduction
Athanasios C. Antoulas
Biswa Nath Datta
Roland W. Freund
Enrique S. Quintana-Ortí
Danny C Sorensen
Evgenii B. Rudnyi