Seems silly to break my research in separate categories of linear equations and eigenvalues, because much of my work has focused on computing eigenvalues at the same time as solving linear equations and using the eigenvectors to speed up the linear equations convergence.  But, anyway, here’s a breakdown:

  Large Systems of Linear Equations

I am interested in improving the convergence of GMRES and CG by deflating some eigenvalues with an augmented Krylov subspace.

• This started with a GMRES method augmented by approximate eigenvectors,
• then an Implicitly Restarted GMRES method,
• and then a related method called GMRES with Deflated Restarting (GMRES-DR).
• There also is a deflated block GMRES method.
• There is a paper on an intersting property of restarted GMRES, called complementary cycles.
• Also, Lan-DR, a deflated, restarted CG/Lanczos method.
• And a deflated, restarted BiCG/nonsymm Lanczos for computing both right and left eigenvectors and simultaneously solving linear equations.
• A couple papers have focused on explaining deflation of eigenvalues for GMRES for highly non-normal matrices (using pseudoeigenvectors and using embreeism).
• Polynomial Preconditioning – it appears to have gotten a bad rap – it can be very helpful.
• Two-grid Krylov methods, including a deflated, restarted BiCGStab method (yes, BiCGStab is restarted, and it works surprisingly well).
• Polynomial approximation to the inverse of a large matrix – amazing that it is possible.  Applications include multiple right-hand sides and the next item.
• Multipolynomial Monte Carlo with deflation for trace estimation for the inverse of a large QCD matrix. 

 Linear equations with multiple right-hand sides.  One of the major applications is to lattice QCD problems and for this I have collaborated with Walter Wilcox from Baylor physics. 

• using eigenvector information from solving the first right-hand side to assist GMRES or BiCGStab for the other right-hand sides.
• the deflated block GMRES method mentioned above.
• deflated CG for symmetric and Hermitian problems,
• improved seed methods for symmetric and Hermitian matrices.
• Planned: a hybrid GMRES/Richardson method for nonsymmetric matrices with eigenvalue deflation (this has now been in the developmental stage for 25 years; can it make 25 more?).
• Coming Soon:  Simple CG for multiple right-hand sides. 

  Linear Equations with Changing Matrices

• something coming soon (or not soon)

  Large Eigenvalue Problems

• My work has focused on:
  preconditioning techniques for eigenvalue problems, including Davidson’s method and a preconditioned Lanczos method.
• methods for interior eigenvalue problems, including harmonic projection techniques (harmonic Rayleigh-Ritz).
• efficient restarting for the Arnoldi method.
• restarting symmetric and nonsymmetric Lanczos (as mentioned above).
• Two-grid and multiple-grid Arnoldi
• Polynomial preconditioned Arnoldi – this is simpler to implement than previous approaches, because it uses the GMRES polynomial, and it has stability control.
• Coming soon: Arnoldi for Rank-One Updates of a matrix.