A Schur complement approach to preconditioning sparse linear least-squares problems with some dense rows

The effectiveness of sparse matrix techniques for directly solving large-scale linear least-squares problems is severely limited if the system matrix A has one or more nearly dense rows. In this paper, we partition the rows of A into sparse rows and dense rows (A s and A d ) and apply the Schur complement approach. A potential difficulty is that the reduced normal matrix AsTA s is often rank-deficient, even if A is of full rank. To overcome this, we propose explicitly removing null columns of A s and then employing a regularization parameter and using the resulting Cholesky factors as a preconditioner for an iterative solver applied to the symmetric indefinite reduced augmented system. We consider complete factorizations as well as incomplete Cholesky factorizations of the shifted reduced normal matrix. Numerical experiments are performed on a range of large least-squares problems arising from practical applications. These demonstrate the effectiveness of the proposed approach when combined with either a sparse parallel direct solver or a robust incomplete Cholesky factorization algorithm

Uncovering a Macrophage Transcriptional Program by Integrating Evidence from Motif Scanning and Expression Dynamics

Macrophages are versatile immune cells that can detect a variety of pathogen-associated molecular patterns through their Toll-like receptors (TLRs). In response to microbial challenge, the TLR-stimulated macrophage undergoes an activation program controlled by a dynamically inducible transcriptional regulatory network. Mapping a complex mammalian transcriptional network poses significant challenges and requires the integration of multiple experimental data types. In this work, we inferred a transcriptional network underlying TLR-stimulated murine macrophage activation. Microarray-based expression profiling and transcription factor binding site motif scanning were used to infer a network of associations between transcription factor genes and clusters of co-expressed target genes. The time-lagged correlation was used to analyze temporal expression data in order to identify potential causal influences in the network. A novel statistical test was developed to assess the significance of the time-lagged correlation. Several associations in the resulting inferred network were validated using targeted ChIP-on-chip experiments. The network incorporates known regulators and gives insight into the transcriptional control of macrophage activation. Our analysis identified a novel regulator (TGIF1) that may have a role in macrophage activation

Optimal solvers for PDE-constrained optimization

Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, particularly in problems of design. The solution of such PDE-constrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2- and 3-dimensional Poisson problem is the PDE. The large-dimensional linear systems which result from discretization and which need to be solved are of saddle-point type. We introduce two optimal preconditioners for these systems, which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The op- timality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other PDEs. © 2010 Society for Industrial and Applied Mathematics