Adaptive rational Krylov subspaces for large-scale dynamical systems

The rational Krylov space is recognized as a powerful tool within model order reduction techniques for linear dynamical systems. However, its success has been hindered by the lack of a parameter-free procedure, which would effectively generate the sequence of shifts used to build the space. In this paper we propose an adaptive computation of these shifts. The whole procedure only requires us to inject some initial rough estimate of the spectral region of the matrix, while further information is automatically generated during the process. The approach is a full generalization to the nonsymmetric case of the idea first proposed in Druskin et al. (2010) [18] and it is used for two important problems in control: the approximation of the transfer function and the numerical solution of large Lyapunov equations. The procedure can be naturally extended to other related problems, such as the solution of the Sylvester equation, and parametric or higher order systems. Several numerical experiments are proposed to assess the quality of the rational projection space over its most natural competitors.

[1]  Axel Ruhe.  Rational Krylov sequence methods for eigenvalue computation , 1984 .

[2]  Raf Vandebril,et al.  On the Convergence of Rational Ritz Values , 2010, SIAM J. Matrix Anal. Appl..

[3]  Vladimir Druskin,et al.  On Optimal Convergence Rate of the Rational Krylov Subspace Reduction for Electromagnetic Problems in Unbounded Domains , 2009, SIAM J. Numer. Anal..

[4]  Eugene L. Wachspress,et al.  New ADI Model Problem Applications , 1986, FJCC.

[5]  Stefan Güttel,et al.  Rational Krylov Methods for Operator Functions , 2010 .

[6]  A. Antoulas,et al.  H 2 Model Reduction for Large-scale Linear Dynamical Systems * , 2022 .

[7]  S. Hammarling.  Numerical Solution of the Stable, Non-negative Definite Lyapunov Equation , 1982 .

[8]  E. Tyrtyshnikov.  Mosaic-Skeleton approximations , 1996 .

[9]  Eugene L. Wachspress,et al.  Solution of lyapunov equations by alternating direction implicit iteration , 1991 .

[10]  Axel Ruhe.  The rational Krylov algorithm for nonsymmetric eigenvalue problems. III: Complex shifts for real matrices , 1994 .

[11]  Ibrahim M. Elfadel,et al.  A block rational Arnoldi algorithm for multipoint passive model-order reduction of multiport RLC networks , 1997, 1997 Proceedings of IEEE International Conference on Computer Aided Design (ICCAD).

[12]  Danny C. Sorensen,et al.  Discrete Empirical Interpolation for nonlinear model reduction , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[13]  Y. Saad,et al.  Numerical solution of large Lyapunov equations , 1989 .

[14]  Wu-Shiung Feng,et al.  Model-order reductions for MIMO systems using global Krylov subspace methods , 2008, Math. Comput. Simul..

[15]  Peter Benner,et al.  Model Order Reduction for Systems with Non-Rational Transfer Function Arising in Computational Electromagnetics , 2010 .

[16]  Vladimir Druskin,et al.  Optimal finite difference grids and rational approximations of the square root I. Elliptic problems , 2000 .

[17]  R. Freund.  Krylov Subspaces Associated with Higher-Order Linear Dynamical Systems , 2005, math/0501484.

[18]  R. Freund.  Krylov-subspace methods for reduced-order modeling in circuit simulation , 2000 .

[19]  Daniel Kressner,et al.  Block variants of Hammarling's method for solving Lyapunov equations , 2008, TOMS.

[20]  M. B. Van Gijzen,et al.  An elegant IDR(s) variant that efficiently exploits bi-orthogonality properties , 2010 .

[21]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control) , 2005 .

[22]  Peter Benner,et al.  Dimension Reduction of Large-Scale Systems , 2005 .

[23]  E. Saff,et al.  Logarithmic Potentials with External Fields , 1997, Grundlehren der mathematischen Wissenschaften.

[24]  P. Benner,et al.  A MATLAB repository for model reduction based on spectral projection , 2006, 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control.

[25]  J. Boyle,et al.  HSL MI 20 : an efficient AMG preconditioner , 2007 .

[26]  Lothar Reichel,et al.  Error Estimates and Evaluation of Matrix Functions via the Faber Transform , 2009, SIAM J. Numer. Anal..

[27]  Valeria Simoncini,et al.  Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov Equation , 2011, SIAM J. Numer. Anal..

[28]  Eric James Grimme,et al.  Krylov Projection Methods for Model Reduction , 1997 .

[29]  Daniel Skoogh.  A rational Krylov method for model order reduction , 1998 .

[30]  Luís Miguel Silveira,et al.  ARMS - Automatic residue-minimization based sampling for multi-point modeling techniques , 2009, 2009 46th ACM/IEEE Design Automation Conference.

[31]  Thilo Penzl,et al.  A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations , 1998, SIAM J. Sci. Comput..

[32]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[33]  Richard H. Bartels,et al.  Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4] , 1972, Commun. ACM.

[34]  Zhaojun Bai,et al.  A two-directional Arnoldi process and its application to parametric model order reduction , 2009 .

[35]  Valeria Simoncini,et al.  Extended Krylov subspace for parameter dependent systems , 2010 .

[36]  Vladimir Druskin,et al.  Solution of Large Scale Evolutionary Problems Using Rational Krylov Subspaces with Optimized Shifts , 2009, SIAM J. Sci. Comput..

[37]  Axel Ruhe,et al.  Rational Krylov for eigenvalue computation and model order reduction , 2006 .

[38]  L. Hogben.  Handbook of Linear Algebra , 2006 .

[39]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[40]  Chad Lieberman,et al.  On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems , 2010, SIAM J. Sci. Comput..

[41]  Daniel Sundström,et al.  Mathematics in industry , 1985 .

[42]  Z. Bai.  Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems , 2002 .

[43]  C.-J. Richard Shi,et al.  Model-order reduction by dominant subspace projection: error bound, subspace computation, and circuit applications , 2005, IEEE Transactions on Circuits and Systems I: Regular Papers.

[44]  Valeria Simoncini,et al.  A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations , 2007, SIAM J. Sci. Comput..

[45]  Luís Miguel Silveira,et al.  Resampling Plans for Sample Point Selection in Multipoint Model-Order Reduction , 2006, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[46]  Vladimir Druskin,et al.  On convergence of Krylov subspace approximations of time-invariant self-adjoint dynamical systems , 2012 .

[47]  L. Knizhnerman,et al.  Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related Functions , 1998, SIAM J. Matrix Anal. Appl..

[48]  I. Elfadel,et al.  A block rational Arnoldi algorithm for multipoint passive model-order reduction of multiport RLC networks , 1997, ICCAD 1997.

[49]  Paul Van Dooren,et al.  A rational Lanczos algorithm for model reduction , 1996, Numerical Algorithms.

[50]  Karen Willcox,et al.  Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space , 2008, SIAM J. Sci. Comput..

[51]  J. Scott,et al.  HSL_MI20: An efficient AMG preconditioner for finite element problems in 3D , 2010 .

[52]  Serkan Gugercin,et al.  H2 Model Reduction for Large-Scale Linear Dynamical Systems , 2008, SIAM J. Matrix Anal. Appl..

[53]  M. A. Kaashoek,et al.  Signal processing, scattering and operator theory, and numerical methods , 1990 .

[54]  E. Grimme,et al.  Pade approximation of large-scale dynamic systems with Lanczos methods , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.