Research

1. Materials Computation Center, University of Illinois Duane Johnson and Richard Martin, NSF DMR-9976550Fast Solvers for Sequences of Linear Systems in Electronic Structuresfrom co-PIs: Eric de Sturler and Duane Johnson loading Research Solving large, sparse linear systems is often the most computationally intensive part of many applications. In electronic-structure calculations we solve matrix equations for many right-hand sides. As such, we are developing methods that improve convergence and reduce CPU time by reusing information from previous solves. Preliminary results look promising, scaling linear in system size. Our new methods are also successful in mechanics problems, such as crack propagation in materials. At right is the results of breaking of a plate (symmetric in the x-axis and breaking along the symmetry axis) from a model provided by P. Geubelle (UIUC-AAE). In this application, the CPU time is halved. A paper is in preparation with graduate students M. Parks and G. Mackey. See also Yu, de Sturler, and Johnson, A Block Iterative Solver for Complex Non-Hermitian Systems Applied to Large-Scale Electronic-Structure Calculations, Tech. Report No. UIUCDCS-R-2002-2299, 2002. crack growth Number of iterations per linear system for 20 loading steps. This problem is symmetric positive definite and ill-conditioned, using an incomplete Cholesky preconditioner. The methods GCROT and GCRO-DR were adapted for reuse in a sequence of linear systems (where both left and right sides change).