CAAM 651: Topics in Numerical Linear Algebra

Numerical Methods for Dimension Reduction of Dynamical Systems

Spring 2011 · Rice University

Admin:	Tuesdays/Thursdays 4-5:20pm, Duncan Hall 1046; Syllabus; Outline
Lecture 28:	Model reduction for descriptor systems Project for Part 4 on Nonlinear Model Reduction: Homework tasks for nonlinear model reduction modred_tasks4.tex: LaTeX source. Systems and MATLAB routines for use with this project are linked below.
Lecture 27:	Matrix pencils Project for Part 3 on Balanced Truncation: Homework tasks for balanced truncation model reduction modred_tasks3.zip: zip file containing LaTeX source. Systems and MATLAB routines for use with this project are linked below.
Lecture 26:	SOAR (Second Order Arnoldi Reduction) for second-order systems
Lecture 25:	DEIM: computational examples
Lecture 24:	DEIM: Analysis of oblique projection
Lecture 23:	A GMRES-like method with DEIM-based quasi-minization
Lecture 22:	The Discrete Empirical Interpolation Method (DEIM) S. Chaturantabut and D. C. Sorensen Nonlinear model reduction via discrete empirical interpolation SIAM J. Sci. Comp. 32 (2010) 2737-2764.
Lecture 21:	Approximate balancing; Introduction to nonlinear model reduction Project for Part 2 on Moment-Matching Model Reduction: Homework tasks for moment matching model reduction modred_tasks2.tex: LaTeX source. Systems and MATLAB routines for use with this project are linked below.
Lecture 20:	Smith-ADI minimax problem; Modified Low-Rank Smith algorithm John Sabino Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method Tech. Report 06-08, CAAM Department, Rice University, 2006.
Lecture 19:	Low-rank Smith, decay of singular values
Lecture 18:	Hammarling's algorithm and Smith's method for Lyapunov equations smith.m: Implementation of Smith's Method [Ryan Nong] lyapU.m, lyapUR.m: dense Lyapunov solver [Sorensen/Zhou] R. A. Smith Matrix equation XA+BX=C SIAM J. Appl. Math. 16 (1968) 198-201.
Lecture 17:	Sylvester operator, Bartels-Stewart method
Lecture 16:	Balanced truncation error bounds
Lecture 15:	Balanced truncation model reduction Bruce C. Moore Principal component analysis in linear systems: control stability, observability, and model reduction IEEE Trans. Auto. Control AC-26 (1981) 17-32.
Lecture 14:	POD and balanced truncation
Lecture 13:	Iterative Rational Krylov Algorithm (IRKA)
Lecture 12:	Moment matching model reduction MORcodes.zip: zip file containing moment matching class demos P. Feldmann and R. W. Freund Efficient linear circuit analysis by Padé aproximation via the Lanczos proces IEEE Trans. CAD of Integrated Circuits and Systems 14 (1994) 639-649. Project for Part 1 on Systems Theory: modred_tasks1.pdf: homework tasks for systems theory section modred_tasks1.zip: file containing LaTeX source and macros for reports sylvester.m: for the optional task, you can use this routine to solve the Lyapunov equation Oberwolfach Model Reduction Benchmark Collection Benchmark problems for model reduction from NICONET (thanks, Cosmin). Sample systems of modest dimension: pick one, or use one from your own research. These examples were collected by Younès Chahlaoui and Paul Van Dooren. CD.mat: model of a CD player swing arm (adapted from NICONET database) iss.mat: model of the Russian service module from the International Space Station (NICONET) Orr-Som.mat: Orr-Sommerfeld system for Couette flow; use random B and C vectors (NICONET) eady.mat: storm track model (NICONET) build.mat: small vibration model for Los Angelas University Hospital (NICONET) beam.mat: clamped beam model (NICONET) tline_mod.mat: transmission line circuit model (adapted from NICONET): beware of the scaling!
Lecture 11:	The error system; moments of the transfer function
Lecture 10:	System norms A. C. Antoulas Approximation of Large-Scale Dynamical Systems SIAM, Philadelphia, 2005.
Lecture 9:	System gramians, Laplace transforms, and the transfer function
Lecture 8:	Controllability and observatility, continued
Lecture 7:	Controllability and observatility Most every book on control theory contains a discussion of controllability and observability, though the definition vary slightly. We have drawn our discussion from: K. Zhou with J. C. Doyle and K. Glover Robust and Optimal Control Prentice-Hall, 1996.
Lecture 6:	The matrix exponential and the total energy of a dynamical system C. Moler and C. Van Loan Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later SIAM Review 45 (2003) 3-49. S. J. Cox Designing for optimal energy absorption, 1: Lumped parameter systems J. Vib. Acoust. 120 (1998) 339-345. sylvester.m: dense Lyapunov equations can be solved using more general code [Sorensen/Zhou]
Lecture 5:	GMRES and other Krylov subspace methods Yousef Saad Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, 2003 second edition from SIAM first edition (free download)
Lecture 4:	GMRES Y. Saad and M. Schultz GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM J. Sci. Stat. Comp. 7 (1986) 856-869. C. Paige, B. Parlett, H. van der Vors Approximate solutions and eigenvalue bounds from Krylov subspaces Num. Lin. Alg. Appl. 2 (1995) 115-133.
Lecture 3:	The Arnoldi Decomposition ArnoldiTest.m: MATLAB code comparing Arnoldi factorization using flavors of Gram-Schmidt orthogonalization: classical, modified, and classical w/reorthogonalization. This routine uses the following three codes. Arnoldi.m: Classical Gram-Schmidt ArnoldiMGS.m: Modified Gram-Schmidt ArnoldiC.m: Classical Gram-Schmidt with DGKS reorthogonalization Y. Saad. Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices Linear Algebra Appl. 34 (1980) 269-295. J. W. Daniel, W. B. Gragg, L. Kaufman, and G. W. Stewart Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization Math. Comp. 30 (1976) 772-795.
Lecture 2:	Sparse Gaussian elimination and classical splitting methods Yousef Saad Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, 2003 second edition from SIAM first edition (free download) MATLAB demos from CAAM 551 website
Lecture 1:	Overview of model reduction; introduction to sparse matrices D. C. Sorensen New Directions in the Application of Model Order Reduction Slides from a talk given at the 2008 MTNS Conference. A. George and J. W. H. Liu The evolution of the minimum degree ordering algorithm SIAM Review 31 (1989) 1-19.