The area of preconditioned iterative solvers for eigenvalue computations is rapidly developing. Software implementations of several preconditioned eigensolvers, in particular, the locally optimal block preconditioned conjugate gradient (LOBPCG) method developed by the principal investigator (PI) earlier, are being written. The recent progress opens new opportunities to develop efficient preconditioned iterative solvers for interior eigenvalues and singular value computations. The use of preconditioned eigensolvers in applications raises new area-specific important issues, both practical and theoretical, which need to be resolved. The proposed research addresses these issues based on the success of the previous work of the PI. The PI expects to advance the theory of some known methods, to discover new locally optimal algorithms, and to be able to provide specific recommendations concerning the choice of methods. The following specific and interrelated research projects are proposed: reducing the LOBPCG costs for large block sizes; developing efficient preconditioned solvers for interior eigenvalues and singular value computations; finite element method error analysis for eigenproblems resulting from partial differential equations with highly discontinuous coefficients. The projects form a balanced mix of theoretical research and code development. Numerical simulations are to be performed on modern parallel computing systems, such as the IBM BlueGene/L supercomputer.
The ideas behind the proposed research are original, and build on prior work. The project addresses mathematically difficult and practically important problems. The broader impact resulting from the proposed activity is twofold: Ph.D. education and advances in software. Funds are requested in the proposal to support Ph.D. students. Improvement of the software currently used and development of new codes for scientists and engineers creates potential advances, e.g., in analyzing extremely large data sets, which is important for national security.
