Scientific Computing Seminar

Date:

Tuesday, January 18, 2005

Time:

2:00pm-3:00pm

Location:

50D-3416

Seminar Speaker:

Constantine Bekas
University of Minnesota
http://www-users.cs.umn.edu/~bekas

Title:

AMLS, Spectral Schur Complements and Iterative Computation of Eigenvalues

Abstract:

In the last few decades, Krylov projection methods such as the Lanczos algorithm and its variants, have dominated the scene of algorithms for eigenvalue problems. Recently, an alternative approach has emerged in structural engineering as a competitor to the standard shift-and-invert Lanczos approach. The algorithm, called Automated Multilevel Substructuring method (AMLS) [Bennighof-Lehoucq, 03] is rooted in a domain decomposition framework. It has been reported as being capable of computing thousands of the smallest normal modes of dynamic structures on commodity workstations and of being orders of magnitude faster than the standard approach [Kropp-Heiserer, 02].
In this talk we adopt a purely algebraic viewpoint and demonstrate that AMLS can be viewed as a method which exploits a first order approximation to a nonlinear eigenvalue problem in order to extract a good subspace for a Rayleigh-Ritz projection process. This technique leads to approximations from a single Schur complement derived from a domain decomposition of the physical problem. Exploiting this observation, we have devised several possible enhancements in two main directions. The first introduces Krylov subspaces to the technique, and the second considers a more accurate (second order instead of first order) scheme, which is based on a quadratic eigenvalue problem. Finally, combinations of the above two strategies have been considered with a goal of enhancing robustness.

Currently, AMLS is a one-shot algorithm in the sense that certain approximate eigenvectors are built from the last level up to the highest level and no further refinements are made. The current framework does not iteratively refine these approximations. We will discuss this issue and explore the feasibility of an iterative scheme based on AMLS. We will show how one can devise an AMLS based method that adopts the shifting strategies of shift-and-invert Lanczos resulting in a highly robust method.

Sponsor of Seminar:

Esmond Ng

Scientific Computing

Contact Esmond G. Ng EGNg@lbl.gov