Hierarchical interpolative factorization for elliptic operators: differential equations

This paper introduces the hierarchical interpolative factorization for elliptic partial differential equations (HIF-DE) in two (2D) and three dimensions (3D). This factorization takes the form of an approximate generalized LU/LDL decomposition that facilitates the efficient inversion of the discretized operator. HIF-DE is based on the multifrontal method but uses skeletonization on the separator fronts to sparsify the dense frontal matrices and thus reduce the cost. We conjecture that this strategy yields linear complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity in 3D can be achieved by skeletonizing the compressed fronts themselves, which amounts geometrically to a recursive dimensional reduction scheme. Numerical experiments support our claims and further demonstrate the performance of our algorithm as a fast direct solver and preconditioner. MATLAB codes are freely available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math. arXiv admin note: substantial text overlap with arXiv:1307.266

Hierarchical interpolative factorization for elliptic operators: integral equations

This paper introduces the hierarchical interpolative factorization for integral equations (HIF-IE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretized operator and its inverse. HIF-IE is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features: (1) a matrix factorization framework for sparsifying structured dense matrices and (2) a recursive dimensional reduction strategy to decrease the cost. Thus, higher-dimensional problems are effectively mapped to one dimension, and we conjecture that constructing, applying, and inverting the factorization all have linear or quasilinear complexity. Numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method, direct solver, and preconditioner. HIF-IE is compatible with geometric adaptivity and can handle both boundary and volume problems. MATLAB codes are freely available.Comment: 39 pages, 14 figures, 13 tables; to appear, Comm. Pure Appl. Mat

A fast semi-direct least squares algorithm for hierarchically block separable matrices

We present a fast algorithm for linear least squares problems governed by hierarchically block separable (HBS) matrices. Such matrices are generally dense but data-sparse and can describe many important operators including those derived from asymptotically smooth radial kernels that are not too oscillatory. The algorithm is based on a recursive skeletonization procedure that exposes this sparsity and solves the dense least squares problem as a larger, equality-constrained, sparse one. It relies on a sparse QR factorization coupled with iterative weighted least squares methods. In essence, our scheme consists of a direct component, comprised of matrix compression and factorization, followed by an iterative component to enforce certain equality constraints. At most two iterations are typically required for problems that are not too ill-conditioned. For an $M \times N$ HBS matrix with $M \geq N$ having bounded off-diagonal block rank, the algorithm has optimal $\mathcal{O} (M + N)$ complexity. If the rank increases with the spatial dimension as is common for operators that are singular at the origin, then this becomes $\mathcal{O} (M + N)$ in 1D, $\mathcal{O} (M + N^{3/2})$ in 2D, and $\mathcal{O} (M + N^{2})$ in 3D. We illustrate the performance of the method on both over- and underdetermined systems in a variety of settings, with an emphasis on radial basis function approximation and efficient updating and downdating.Comment: 24 pages, 8 figures, 6 tables; to appear in SIAM J. Matrix Anal. App

The Candida albicans transcription factor Cas5 couples stress responses, drug resistance and cell cycle regulation

We thank Cowen lab members for helpful discussions. We also thank David Rogers (University of Tennessee) for sharing microarray analysis of the CAS5 homozygous mutant, and Li Ang (University of Macau) for assistance in optimizing the ChIP-Seq experiments. J.L.X. is supported by a Canadian Institutes of Health Research Doctoral award and M.D.L. is supported by a Sir Henry Wellcome Postdoctoral Fellowship (Wellcome Trust 096072). B.T.G. holds an Ontario Graduate Scholarship. C.B. and B.J.A. are supported by the Canadian Institutes of Health Research Foundation Grants (FDN-143264 and -143265). D.J.K. is supported by a National Institute of Allergy and Infectious Diseases grant (1R01AI098450) and J.D.L.C.D. is supported by the University of Rochester School of Dentistry and Medicine PREP program (R25 GM064133). A.S. is supported by the Creighton University and the Nebraska Department of Health and Human Services (LB506-2017-55). K.H.W. is supported by the Science and Technology Development Fund of Macau S.A.R. (FDCT; 085/2014/A2). L.E.C. is supported by the Canadian Institutes of Health Research Operating Grants (MOP-86452 and MOP-119520), the Natural Sciences and Engineering Council (NSERC) of Canada Discovery Grants (06261 and 462167), and an NSERC E.W.R. Steacie Memorial Fellowship (477598).Peer reviewedPublisher PD