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

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

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. VIII. On the Relationship Between Disk Stability and Star Formation

We study the relationship between the stability level of late-type galaxy disks and their star-formation activity using integral-field gaseous and stellar kinematic data. Specifically, we compare the two-component (gas+stars) stability parameter from Romeo & Wiegert (Q_RW), incorporating stellar kinematic data for the first time, and the star-formation rate estimated from 21cm continuum emission. We determine the stability level of each disk probabilistically using a Bayesian analysis of our data and a simple dynamical model. Our method incorporates the shape of the stellar velocity ellipsoid (SVE) and yields robust SVE measurements for over 90% of our sample. Averaging over this subsample, we find a meridional shape of sigma_z/sigma_R = 0.51^{+0.36}_{-0.25} for the SVE and, at 1.5 disk scale lengths, a stability parameter of Q_RW = 2.0 +/- 0.9. We also find that the disk-averaged star-formation-rate surface density (Sigma-dot_e,*) is correlated with the disk-averaged gas and stellar mass surface densities (Sigma_e,g and Sigma_e,*) and anti-correlated with Q_RW. We show that an anti-correlation between Sigma-dot_e,* and Q_RW can be predicted using empirical scaling relations, such that this outcome is consistent with well-established statistical properties of star-forming galaxies. Interestingly, Sigma-dot_e,* is not correlated with the gas-only or star-only Toomre parameters, demonstrating the merit of calculating a multi-component stability parameter when comparing to star-formation activity. Finally, our results are consistent with the Ostriker et al. model of self-regulated star-formation, which predicts Sigma-dot_e,*/Sigma_e,g/sqrt(Sigma_e,*). Based on this and other theoretical expectations, we discuss the possibility of a physical link between disk stability level and star-formation rate in light of our empirical results.Comment: Accepted for publication in ApJ. 15 pages, 6 figures, 2 tables. An electronic version of Table 1 is available by request, or at http://www.astro.rug.nl/~westfall/research/dmVIII_table1.tx