FINAL REPORT (MILESTONE DATE 9/30/11) FOR SUBCONTRACT NO. B594099 NUMERICAL METHODS FOR LARGE-SCALE DATA FACTORIZATION

Abstract

The following work has been performed by PI Hans De Sterck and graduate student Manda Winlaw for the required tasks 1-5 (as listed in the Statement of Work). Graduate student Manda Winlaw has visited LLNL January 31-March 11, 2011 and May 23-August 19, 2010, working with Van Henson and Mike O'Hara on non-negative matrix factorizations (NMF). She has investigated the dense subgraph clustering algorithm from 'Finding Dense Subgraphs for Sparse Undirected, Directed, and Bipartite Graphs' by Chen and Saad, testing this method on several term-document matrices and adapting it to cluster based on the rank of the subgraphs instead of the density. Manda Winlaw was awarded a first prize in the annual LLNL summer student poster competition for a poster on her NMF research. PI Hans De Sterck has developed a new adaptive algebraic multigrid algorithm for computing a few dominant or minimal singular triplets of sparse rectangular matrices. This work builds on adaptive algebraic multigrid methods that were further developed by the PI and collaborators (including Sanders and Henson) for Markov chains. The method also combines and extends existing multigrid algorithms for the symmetric eigenproblem. The PI has visited LLNL February 22-25, 2011, and has given a CASC seminarmore » 'Algebraic Multigrid for the Singular Value Problem' on this work on February 23, 2011. During his visit, he has discussed this work and related topics with Van Henson, Geoffrey Sanders, Panayot Vassilevski, and others. He has tested the algorithm on PDE matrices and on a term-document matrix, with promising initial results. Manda Winlaw has also started to work, with O'Hara, on estimating probability distributions over undirected graph edges. The goal is to estimate probabilistic models from sets of undirected graph edges for the purpose of prediction, anomaly detection and support to supervised learning. Graduate student Manda Winlaw is writing a paper on the results obtained with O'Hara which will be submitted some time later in 2011 to a data mining conference. PI Hans De Sterck has developed a new optimization algorithm for canonical tensor approximation, formulating an extension of the nonlinear GMRES method to optimization problems. Numerical results for tensors with up to 8 modes show that this new method is efficient for sparse and dense tensors. He has written a paper on this which has been submitted to the SIAM Journal on Scientific Computing. PI Hans De Sterck has further developed his new optimization algorithm for canonical tensor approximation, formulating an extension in terms of steepest-descent preconditioning, which makes the approach generally applicable for nonlinear optimization. He has written a paper on this extension which has been submitted to Numerical Linear Algebra with Applications.« less