E i g e n v a l u e   C l i n i c

VIGRE Seminar, CAAM 699.010

Monday 11am-12pm (DH 2014)

The Eigenvalue Clinic is a weekly workshop where participants pose research problems related to applications of spectral theory in differential equations, numerical linear algebra, and control theory.

This semester, we are methodically working through the steps one makes when moving from a physical application (e.g., vibration or stability problem) to numerically computed eigenvalues.

For more information, see the syllabus or contact Mark Embree.

Spring 2013

Reid Atcheson, Jonathan Baker, Russell Carden, Mark Embree, Jeffrey Hokanson, Cosmin Ionita, Paul Munger, Charles Puelz

11.    April 10: Relative accuracy in eigenvalue computations
- lapacc.m: Laplacian example (finite differences; bad!)
- beamacc.m: beam example (finite differences; even worse!)
- sm_lapacc.m: Laplacian example (spectral method; better), needs chebab.m from Trefethen
- sm_beamacc.m: beam example (spectral method; better?), needs chebab.m from Trefethen
- Demmel and Veselić, Jacobi's method is more accurate than QR, 1992
- Ye, Relative perturbation bounds for eigenvalues ... , 2009
10.    Apr 1: Similarity transformations for improving eigenvalue computations
- morgan_ex.m: the perils of balancing, an example from Ron Morgan
- Parlett and Reinsch, Balancing a matrix for calculation of eigenvalues and eigenvectors, 1969.
- Higham, Mackey, Tisseur, Garvey, Scaling, sensitivity and stability in the numerical solution of QEPs, 2008
9.    Mar 25: Arnoldi convergence for operators
- arnoldi_laplace.nb: inverse Arnoldi method for the Laplacian (Mathematica)
- arnoldi_advdiff_gf.nb: inverse Arnoldi method for an advection diffusion operator (Mathematica)
- cf_laplace.nb: inverse Arnoldi method for the Laplacian (Chebfun)
- cf_advdiff.nb: inverse Arnoldi method for an advection diffusion operator (Chebfun)
- The Chebfun software
Note: the advection-diffusion examples do not perform well!
8.    Mar 18: Pseudospectra and convergence rates for linear solvers
- Spectra and Pseudospectra: Trefethen and Embree, 2005.
7.    Mar 11: Eigenvalue computation for non-self-adjoint operators
- make_davies.m: Chebyshev pseudospectral discretization of Davies' anharnmoic oscillator (and chebab.m) from Trefethen
6.    Mar 4: An example of classical convergence theory
- Atkinson, Convergence rates for approximate eigenvalues of compact integral operators, 1975
Some supporting work:
- Atkinson, The numerical solution of the eigenvalue problem for compact integral operators, 1967
- Anselone, Collectively compact operator approximations, 1967

See also the work of John Osborn, e.g.:
- Osborn, Spectral approximation for compact operators, 1975

Some MATLAB examples for an integral equation from laser theory:
- laser_trap.m: Nystrom discretization via the trapezoid rule
- laser_cc.m: Nystrom discretization via the Clenshaw-Curtis rule (clencurtab.m from Trefethen)
5.    Feb 18: Spectral pollution, continued
- Davies, Wild spectral behavior of anharmonic oscillators, 1999
4.    Feb 11: Spectral pollution
- Davies and Plum, Spectral pollution, 2004
- Deift and Hempel, On the existence of eigenvalues of the Schrödinger operator..., 1986
- slides from Lyonell Boulton (Heriot-Watt University, Edinburgh)
- pollute.m: demonstration with a multiplication operator
3.    Feb 4: Basics of spectral theory
- Basic results can be found in most functional analysis texts,
   elementary presentations: N. Young, E. Kreyszig
   see also: Reed and Simon, Davies, Edmunds and Evans
- ex1trap.m: discretization of a compact integral operator
2.    Jan 28: Nonlinear eigenvalue problems
- Michiels and Niculescu on eigenvalue problems in delay differential operators
- nlcont.m, nlpoly.m: approximate nonlinear EVP with polynomial EVP
- Betcke, Higham, Mehrmann, Schröder and Tisseur
   NLEVP: A Collection of Nonlinear Eigenvalue Problems
   Download the collection here.
- Higham, Mackey, and Tisseur, The conditioning of linearizations of matrix polynomials, 2007
- Higham, Mackey, Tisseur, Garvey, Scaling, sensitivity and stability in the numerical solution of QEPs, 2008
1.    Jan 21: From vibrations to eigenvalue problems
- Antman, Nonlinear Problems in Elasticity, 2005
- Three discretizations of damped wave operators:
   makeAconst.m: viscous damping
   makeAkvconst.m: Kelvin-Voigt damping
   makeAmag.m: magnetic damping
   chebab.m, clencurtab.m: auxiliary routines adapted from Trefethen