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

Visualization of Dimensional Effects in Collective Excitations of Optically Trapped Quasi-Two-Dimensional Bose Gases

We analyze the macroscopic dynamics of a Bose gas axially confined in an optical lattice with a superimposed harmonic trap, taking into account weak tunneling effect. Our results show that upon transition to the quasi-two-dimensional (2D) regime of the trapped gas, the 3D equation of state and equilibrium density profile acquire corrections from 2D many-body effects. The corresponding frequency shift in the transverse breathing mode is accessible within current facilities, suggesting a direct observation of dimensional effects. Comparisons with other relevant effects are also presented.Comment: 4 pages, 1 figur

Density dependent spin susceptibility and effective mass in interacting quasi-two dimensional electron systems

Motivated by recent experimental reports, we carry out a Fermi liquid many-body calculation of the interaction induced renormalization of the spin susceptibility and effective mass in realistic two dimensional (2D) electron systems as a function of carrier density using the leading-order `ladder-bubble' expansion in the dynamically screened Coulomb interaction. Using realistic material parameters for various semiconductor-based 2D systems, we find reasonable quantitative agreement with recent experimental susceptibility and effective mass measurements. We point out a number of open questions regarding quantitative aspects of the comparison between theory and experiment in low-density 2D electron systems

Effects of fiber/matrix interactions on the properties of graphite/epoxy composites

A state-of-the-art literature review of the interactions between fibers and resin within graphite epoxy composite materials was performed. Emphasis centered on: adhesion theory; wetting characteristics of carbon fiber; load transfer mechanisms; methods to evaluate and measure interfacial bond strengths; environmental influence at the interface; and the effect of the interface/interphase on composite performance, with particular attention to impact toughness. In conjunction with the literature review, efforts were made to design experiments to study the wetting behavior of carbon fibers with various finish variants and their effect on adhesion joint strength. The properties of composites with various fiber finishes were measured and compared to the base-line properties of a control. It was shown that by tailoring the interphase properties, a 30% increase in impact toughness was achieved without loss of mechanical properties at both room and elevated temperatures