448 research outputs found
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 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 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 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, 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
An Empirical Relation Between The Large-Scale Magnetic Field And The Dynamical Mass In Galaxies
The origin and evolution of cosmic magnetic fields as well as the influence
of the magnetic fields on the evolution of galaxies are unknown. Though not
without challenges, the dynamo theory can explain the large-scale coherent
magnetic fields which govern galaxies, but observational evidence for the
theory is so far very scarce. Putting together the available data of
non-interacting, non-cluster galaxies with known large-scale magnetic fields,
we find a tight correlation between the integrated polarized flux density,
S(PI), and the rotation speed, v(rot), of galaxies. This leads to an almost
linear correlation between the large-scale magnetic field B and v(rot),
assuming that the number of cosmic ray electrons is proportional to the star
formation rate, and a super-linear correlation assuming equipartition between
magnetic fields and cosmic rays. This correlation cannot be attributed to an
active linear alpha-Omega dynamo, as no correlation holds with global shear or
angular speed. It indicates instead a coupling between the large-scale magnetic
field and the dynamical mass of the galaxies, B ~ M^(0.25-0.4). Hence, faster
rotating and/or more massive galaxies have stronger large-scale magnetic
fields. The observed B-v(rot) correlation shows that the anisotropic turbulent
magnetic field dominates B in fast rotating galaxies as the turbulent magnetic
field, coupled with gas, is enhanced and ordered due to the strong gas
compression and/or local shear in these systems. This study supports an
stationary condition for the large-scale magnetic field as long as the
dynamical mass of galaxies is constant.Comment: 23 pages, 4 figures, accepted for publication in the Astrophysical
Journal Letter
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