A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method

A numerical method for solving elliptic PDEs with variable coefficients on two-dimensional domains is presented. The method is based on high-order composite spectral approximations and is designed for problems with smooth solutions. The resulting system of linear equations is solved using a direct (as opposed to iterative) solver that has optimal O(N) complexity for all stages of the computation when applied to problems with non-oscillatory solutions such as the Laplace and the Stokes equations. Numerical examples demonstrate that the scheme is capable of computing solutions with relative accuracy of $10^{-10}$ or better, even for challenging problems such as highly oscillatory Helmholtz problems and convection-dominated convection diffusion equations. In terms of speed, it is demonstrated that a problem with a non-oscillatory solution that was discretized using $10^{8}$ nodes was solved in 115 minutes on a personal work-station with two quad-core 3.3GHz CPUs. Since the solver is direct, and the "solution operator" fits in RAM, any solves beyond the first are very fast. In the example with $10^{8}$ unknowns, solves require only 30 seconds.Comment: arXiv admin note: text overlap with arXiv:1302.599

Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions

Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed---either explicitly or implicitly---to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis

Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions

Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an m × n matrix. (i) For a dense input matrix, randomized algorithms require O(mn log(k)) floating-point operations (flops) in contrast to O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data

Publishing in paleontology

La estructura de la publicación paleontológica depende básicamente del hecho de que la Paleontologia a) representa un tema muy amplio peroemplea relativamente pocos especialistas, b) necesita tanto representación ideográfica masiva como un aumento proporcionalde discusiónnomotética y c) estádividida entre las ciencias de la tierra y de la vida. La publicación se lleva aún a cabo a traves de series anticuadas que incluyen temas variados, y los paleontólogos empiezan lentamente a comprender la  necesidad de una presentación estructurada y de la canalización de los resultados de la investigación. Los volúmenes de Symposios contribuyen considerablemente al deterioro de la publicación de la  Paleontologia debido a su insuficiente circulación, al inadecuado control de calidad y a la insuficiente  accesibilidad a los articulas a través de servicios secundarios. La divulgación insuficiente es, no obstante,  admirablemente compensada a través de la circulación de separatas canalizada por catálogos y noticiarios. La publicación sinoptica ofrece una solución inminente al problema económico de  Ia Paleontologia ideográfica, pero no gana terreno. No obstante, el enterramiento de la Paleontologia ideográfica en la ((literatura gris» aún no ha finalizado. La disminucion de las exigencias de la educación escolar acarrea repercusiones en el estilo literario, la tenninologia y la nomenclatura. El  internacionalismo gana terreno y ha de ser promovido. La Paleontologia ideográfica avanzará mas lentamente que otras ramas de las Ciencias Naturales en la adaptación de la impresión en papel a lasmicrofichas y a la comunicación electrónica. Esto esdebido a la necesidad inherente de ilustraciones adecuadas y de comparación simultánea, e igualmente a la falta de procedimientos para el tratamiento del material sucesivamente modernizado y de las exigencias de códigos de nomenclatura biológica

The DiskMass Survey. X. Radio synthesis imaging of spiral galaxies

We present results from 21 cm radio synthesis imaging of 28 spiral galaxies from the DiskMass Survey obtained with the VLA, WSRT, and GMRT facilities. We detail the observations and data reduction procedures and present a brief analysis of the radio data. We construct 21 cm continuum images, global HI emission-line profiles, column-density maps, velocity fields, and position-velocity diagrams. From these we determine star formation rates (SFRs), HI line widths, total HI masses, rotation curves, and azimuthally-averaged radial HI column-density profiles. All galaxies have an HI disk that extends beyond the readily observable stellar disk, with an average ratio and scatter of R_{HI}/R_{25}=1.35+/-0.22, and a majority of the galaxies appear to have a warped HI disk. A tight correlation exists between total HI mass and HI diameter, with the largest disks having a slightly lower average column density. Galaxies with relatively large HI disks tend to exhibit an enhanced stellar velocity dispersion at larger radii, suggesting the influence of the gas disk on the stellar dynamics in the outer regions of disk galaxies. We find a striking similarity among the radial HI surface density profiles, where the average, normalized radial profile of the late-type spirals is described surprisingly well with a Gaussian profile. These results can be used to estimate HI surface density profiles in galaxies that only have a total HI flux measurement. We compare our 21 cm radio continuum luminosities with 60 micron luminosities from IRAS observations for a subsample of 15 galaxies and find that these follow a tight radio-infrared relation, with a hint of a deviation from this relation at low luminosities. We also find a strong correlation between the average SFR surface density and the K-band surface brightness of the stellar disk.Comment: 22 pages + Appendix, 16 figures + Atlas, 5 tables. Accepted for publication in Astronomy & Astrophysic