209 research outputs found
Supporting GENP with Random Multipliers
We prove that standard Gaussian random multipliers are expected to stabilize
numerically both Gaussian elimination with no pivoting and block Gaussian
elimination. Our tests show similar results where we applied circulant random
multipliers instead of Gaussian ones.Comment: 14 page
TR-2012004: Solving Linear Systems of Equations with Randomization, Augmentation and Aggregation
TR-2008009: Solving Homogeneous Linear Systems with Weakly Randomized Additive Preprocessing
TR-2012007: Solving Linear Systems of Equations with Randomization, Augmentation and Aggregation II
Seeking a basis for the null space of a rectangular and possibly rank deficient and ill conditioned matrix we apply randomization, augmentation, and aggregation to reduce our task to computations with well conditioned matrices of full rank. Our algorithms avoid pivoting and orthogonalization, preserve matrix structure and sparseness, and in the case of an ill conditioned input perform only a small part of the computations with high accuracy. We extend the algorithms to the solution of nonhomogeneous nonsingular ill conditioned linear systems of equations whose matrices have small numerical nullities. Our estimates and experiments show dramatic progress versus the customary matrix algorithms where the input matrices are rank deficient or ill conditioned. Our study can be of independent technical interest: we extend the known results on conditioning of random matrices to randomized preconditioning, estimate the condition numbers of randomly augmented matrices, and link augmentation to aggregation as well as homogeneous to nonhomogeneous linear systems of equations
The Correspondence between Convergence Peaks from Weak Lensing and Massive Dark Matter Haloes
The convergence peaks, constructed from galaxy shape measurement in weak
lensing, is a powerful probe of cosmology as the peaks can be connected with
the underlined dark matter haloes. However the capability of convergence peak
statistic is affected by the noise in galaxy shape measurement, signal to noise
ratio as well as the contribution from the projected mass distribution from the
large-scale structures along the line of sight (LOS). In this paper we use the
ray-tracing simulation on a curved sky to investigate the correspondence
between the convergence peak and the dark matter haloes at the LOS. We find
that, in case of no noise and for source galaxies at , more than
peaks with (signal to noise ratio) are related to
more than one massive haloes with mass larger than .
Those massive haloes contribute to high peaks ()
with the remaining contributions are from the large-scale structures. On the
other hand, the peaks distribution is skewed by the noise in galaxy shape
measurement, especially for lower SNR peaks. In the noisy field where the shape
noise is modelled as a Gaussian distribution, about high peaks
() are true peaks and the fraction decreases to for
lower peaks (). Furthermore, we find that high peaks
() are dominated by very massive haloes larger than .Comment: 13 pages, 11 figures, 4 tables, accepted for publication in MNRAS.
Our mock galaxy catalog is available upon request by email to the author
([email protected]
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