
Searching for it: Exploring data with numerical linear algebra and approximation theory PI: Y. Saad This work aims at developing new and effective algorithms for performing dimensionality reduction tasks by methods which blend techniques from numerical linear algebra and approximation theory. Current implementations of LSI, and other dimensionality reduction methods, rely on matrix decompositions such as the Singular Value Decomposition (SVD). SVDbased methods compute explicitly the basis of the dominant singular vectors and proceed with a projection of the data on this basis. This has the desirable effect of filtering out noise and redundancy inherent to the data, while retaining its main structural features (e.g., `semantic contents' in LSI). However, SVDbased methods tend to be expensive, both in terms of computational cost and storage, and become impractical for very large data sets. The premise of this proposal is that there is no need to compute the (partial) SVD in order to perform dimensionality reduction The projection of a given vector onto the space associated with the largest singular values can be accurately reproduced by a polynomial filtering technique. This technique, which entails repeated multiplication of a vector by the original data matrix and its transpose, offers several advantages including low computational and storage requirements. Other techniques explored include tensorbased methods, and methods which exploit affinity graphs. 

Algebraic Recursive Multilevel Solvers: advances in scalable
and robust parallel linear system solution methods
PI: Y. Saad, coPI: M. Sosonkina, Ames Lab., Iowa.  The main thrust of this proposal is to develop a class of parallel multilevel ILUtype preconditioning techniques. The algorithms will be developed and tested within one single programming framework. This framework utilizes the Algebraic Recursive Multilevel Solver (ARMS) methods, which have been recently developed in the sequential case. The advantages of a parallel ARMS framework are numerous. Recursive Multilevel ILU methods allow to develop many of the standard iterative solvers, and even direct solvers, into one single generic code. Their multilevel nature gives them the potential of bridging the gap between the excellent performance obtained from multigrid methods in some situations and the generalpurpose nature of preconditioned Krylov solvers. In addition, they have excellent scalability properties. Thanks to these features, the resulting software will provide a whole class of new and existing algorithms. These include Schwarz procedures, Schur complement type methods, direct solvers, and multilevel techniques. Performance and scalability issues will be examined for each instantiation of an algorithm. We will also conduct considerable testing on realistic problems arising in applications on which the PI and the coPI are involved. 

Scalable methods for electronic excitations and optical responses. PI: Juan Meza  LBL.  The goal of this project is to investigate new algorithms and methodologies for excited states of various nanostructures. The Minnesota team will focus primarily on methods based on timedependent density functional theory (TDDFT). Current work is focussing on two aspects: (1) better eigenvalue algorithms and (2) methods which avoid eigenvalue problems altogether. The goal is to start by testing these approaches in (ground state) Density Functional Theory and then to carry over the techniques developed to excited states (TDDFT). 

Institute for the Theory of Advanced Materials in Information Technology

PI: Prof. J. Chelikowsky, University of Texas at Austin. coPI: Yousef Saad. The main focus of this research is to exploit high performance computers for solving large scale problems that arise in modeling real materials, i.e., materials which cannot be characterized as simple crystals, unreconstructed surfaces, or artificial models such as jellium. On the applications side, the efforts of the team is specifically oriented toward problems related to modeling related to electronic materials, e.g., materials used in building computer devices. 