Evaluating the Impact of SDC on the GMRES Iterative Solver

Increasing parallelism and transistor density, along with increasingly tighter energy and peak power constraints, may force exposure of occasionally incorrect computation or storage to application codes. Silent data corruption (SDC) will likely be infrequent, yet one SDC suffices to make numerical algorithms like iterative linear solvers cease progress towards the correct answer. Thus, we focus on resilience of the iterative linear solver GMRES to a single transient SDC. We derive inexpensive checks to detect the effects of an SDC in GMRES that work for a more general SDC model than presuming a bit flip. Our experiments show that when GMRES is used as the inner solver of an inner-outer iteration, it can "run through" SDC of almost any magnitude in the computationally intensive orthogonalization phase. That is, it gets the right answer using faulty data without any required roll back. Those SDCs which it cannot run through, get caught by our detection scheme

Democratic Representations

Minimization of the $\ell_{\infty}$ (or maximum) norm subject to a constraint that imposes consistency to an underdetermined system of linear equations finds use in a large number of practical applications, including vector quantization, approximate nearest neighbor search, peak-to-average power ratio (or "crest factor") reduction in communication systems, and peak force minimization in robotics and control. This paper analyzes the fundamental properties of signal representations obtained by solving such a convex optimization problem. We develop bounds on the maximum magnitude of such representations using the uncertainty principle (UP) introduced by Lyubarskii and Vershynin, and study the efficacy of $\ell_{\infty}$-norm-based dynamic range reduction. Our analysis shows that matrices satisfying the UP, such as randomly subsampled Fourier or i.i.d. Gaussian matrices, enable the computation of what we call democratic representations, whose entries all have small and similar magnitude, as well as low dynamic range. To compute democratic representations at low computational complexity, we present two new, efficient convex optimization algorithms. We finally demonstrate the efficacy of democratic representations for dynamic range reduction in a DVB-T2-based broadcast system.Comment: Submitted to a Journa

Toward a Robust Sparse Data Representation for Wireless Sensor Networks

Compressive sensing has been successfully used for optimized operations in wireless sensor networks. However, raw data collected by sensors may be neither originally sparse nor easily transformed into a sparse data representation. This paper addresses the problem of transforming source data collected by sensor nodes into a sparse representation with a few nonzero elements. Our contributions that address three major issues include: 1) an effective method that extracts population sparsity of the data, 2) a sparsity ratio guarantee scheme, and 3) a customized learning algorithm of the sparsifying dictionary. We introduce an unsupervised neural network to extract an intrinsic sparse coding of the data. The sparse codes are generated at the activation of the hidden layer using a sparsity nomination constraint and a shrinking mechanism. Our analysis using real data samples shows that the proposed method outperforms conventional sparsity-inducing methods.Comment: 8 page

A Fast Chi-squared Technique For Period Search of Irregularly Sampled Data

A new, computationally- and statistically-efficient algorithm, the Fast $\chi^2$ algorithm, can find a periodic signal with harmonic content in irregularly-sampled data with non-uniform errors. The algorithm calculates the minimized $\chi^2$ as a function of frequency at the desired number of harmonics, using Fast Fourier Transforms to provide $O (N \log N)$ performance. The code for a reference implementation is provided.Comment: Source code for the reference implementation is available at http://public.lanl.gov/palmer/fastchi.html . Accepted by ApJ. 24 pages, 4 figure