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The design of iterative schemes for sparse matrix computations often leads to constrained polynomial approximation problems on sets in the complex plane. For the case of ellipses, we introduce a new class of complex polynomials which are in general very good approximations to the best polynomials and even optimal in most cases.
The Kepler Science Operations Center stores pixel values on approximately six million pixels collected every 30-minutes, as well as data products that are generated as a result of running the Kepler science processing pipeline. The Kepler Database (Kepler DB) management system was created to act as the repository of this information. After one year of ight usage, Kepler DB is managing 3 TiB of data and is expected to grow to over 10 TiB over the course of the mission. Kepler DB is a non-relational, transactional database where data are represented as one dimensional arrays, sparse arrays or binary large objects. We will discuss Kepler DB's APIs, implementation, usage and deployment at the Kepler Science Operations Center.
Monthly composite images from the global coastal zone color scanner (CZCS) data set are used to provide an initial illustration and comparison of seasonal and interannual variability of phytoplankton pigment concentration along the western coasts of South and North America in the Peru Current system (PCS) and California Current system (CCS). The analysis utilizes the entire time series of available data (November 1978 to June 1986) to form a mean annual cycle and an index of interannual variability for a series of both latitudinal and cross-shelf regions within each current system. Within 100 km of the coast, the strongest seasonal cycles in the CCS are in two regions, one between 34 deg and 45 deg N and the second between 24 deg and 29 deg N, each with maximum concentrations (greater than 3.0 mg m(exp-3)) in May-June. Weaker seasonal variability is present north of 45 deg N and in the Southern California Bight region (32 deg N). Within the PCS, in the same 100-km-wide coastal region, highest (greater than 45 deg S) and lowest (less than 20 deg S) latitude regions have a similar seasonal cycle with maximum concentrations (greater than 1.5 mg m(exp -3)) during the austral spring, summer, and fall, matching that evident throughout the CCS. Between these regions, off northern and central Chile, the seasonal maximum occurs during July-August (austral winter), contrary to the influence of upwelling favorable winds. Within the CCS, the dominant feature of interannual variability in the 8-year time series is a strong negative concentration anomaly in 1983, an El Nino year. The relative value of this negative anomaly is strongest off central California and is followed by an even stronger negative anomaly is strongest off central California and is followed by an even stronger negative anomaly in 1984 off Baja, California. In the PCS, strong negative anomalies during the 1982-1983 El Nino period are evident only off the Peruvian coast and are evident there only in the regions 100 km or more from the coast. Although negative anomalies associated with the El Nino were not present at higher latitudes (more than approximately 20 deg S) in the PCS, the extremely sparse sampling weakens our confidence in the results of the interannual analysis in this region. An upper estimate of the systematic winter bias remaining in the global CZCS data after reprocessing with the multiple scattering algorithm is given in the appendix.
An incremental iterative formulation together with the well-known spatially split approximate-factorization algorithm, is presented for solving the large, sparse systems of linear equations that are associated with aerodynamic sensitivity analysis. This formulation is also known as the 'delta' or 'correction' form. For the smaller two dimensional problems, a direct method can be applied to solve these linear equations in either the standard or the incremental form, in which case the two are equivalent. However, iterative methods are needed for larger two-dimensional and three dimensional applications because direct methods require more computer memory than is currently available. Iterative methods for solving these equations in the standard form are generally unsatisfactory due to an ill-conditioned coefficient matrix; this problem is overcome when these equations are cast in the incremental form. The methodology is successfully implemented and tested using an upwind cell-centered finite-volume formulation applied in two dimensions to the thin-layer Navier-Stokes equations for external flow over an airfoil. In three dimensions this methodology is demonstrated with a marching-solution algorithm for the Euler equations to calculate supersonic flow over the High-Speed Civil Transport configuration (HSCT 24E). The sensitivity derivatives obtained with the incremental iterative method from a marching Euler code are used in a design-improvement study of the HSCT configuration that involves thickness. camber, and planform design variables.
An algorithm that performs sparse linear discriminant analysis (Sparse-LDA) finds near-optimal solutions in far less time than the prior art when specialized to binary classification (of 2 classes). Sparse-LDA is a type of feature- or variable- selection problem with numerous applications in statistics, machine learning, computer vision, computational finance, operations research, and bio-informatics. Because of its combinatorial nature, feature- or variable-selection problems are NP-hard or computationally intractable in cases involving more than 30 variables or features. Therefore, one typically seeks approximate solutions by means of greedy search algorithms. The prior Sparse-LDA algorithm was a greedy algorithm that considered the best variable or feature to add/ delete to/ from its subsets in order to maximally discriminate between multiple classes of data. The present algorithm is designed for the special but prevalent case of 2-class or binary classification (e.g. 1 vs. 0, functioning vs. malfunctioning, or change versus no change). The present algorithm provides near-optimal solutions on large real-world datasets having hundreds or even thousands of variables or features (e.g. selecting the fewest wavelength bands in a hyperspectral sensor to do terrain classification) and does so in typical computation times of minutes as compared to days or weeks as taken by the prior art. Sparse LDA requires solving generalized eigenvalue problems for a large number of variable subsets (represented by the submatrices of the input within-class and between-class covariance matrices). In the general (fullrank) case, the amount of computation scales at least cubically with the number of variables and thus the size of the problems that can be solved is limited accordingly. However, in binary classification, the principal eigenvalues can be found using a special analytic formula, without resorting to costly iterative techniques. The present algorithm exploits this analytic form along with the inherent sequential nature of greedy search itself. Together this enables the use of highly-efficient partitioned-matrix-inverse techniques that result in large speedups of computation in both the forward-selection and backward-elimination stages of greedy algorithms in general.
In this study involving advanced fluid flow codes, an incremental iterative formulation (also known as the delta or correction form) together with the well-known spatially-split approximate factorization algorithm, is presented for solving the very large sparse systems of linear equations which are associated with aerodynamic sensitivity analysis. For smaller 2D problems, a direct method can be applied to solve these linear equations in either the standard or the incremental form, in which case the two are equivalent. Iterative methods are needed for larger 2D and future 3D applications, however, because direct methods require much more computer memory than is currently available. Iterative methods for solving these equations in the standard form are generally unsatisfactory due to an ill-conditioning of the coefficient matrix; this problem can be overcome when these equations are cast in the incremental form. These and other benefits are discussed. The methodology is successfully implemented and tested in 2D using an upwind, cell-centered, finite volume formulation applied to the thin-layer Navier-Stokes equations. Results are presented for two sample airfoil problems: (1) subsonic low Reynolds number laminar flow; and (2) transonic high Reynolds number turbulent flow.
Successive approximations formed in Fourier space. Algorithm extracts periodic signals from sparse, irregularly sampled sets of measurement data. Pertains to data processed via fast Fourier transforms (FFTs). Data represents signal components with initially unknown frequencies spanning large spectral range and includes frequencies not integer multiples of minimum FFT frequency.
A simplified procedure is presented for analysis of ducts with discretely nonuniform properties. The analysis uses basis functions as the generalized coordinates. The duct eigenfunctions are approximated by finite series of these functions. The emphasis is on solution of the resulting large sparse set of linear equations. Characteristics of sparse matrix algorithms are outlined and some criteria for application are established. Analogies with structural methods are used to illustrate variations which can increase efficiency in generating values for design optimization routines. The effects of basis function selection, number of eigenfunctions and identification and ordering of equations on the sparsity and solution stability are included.
We present our analysis of archival Hubble Space Telescope Wide Field Planetary Camera 2 (WFPC2) observations in F555W (approximately V) and F814W (approximately I) of the central region of the Ursa Minor dwarf spheroidal galaxy. The V versus V - I color-magnitude diagram features a sparsely populated blue horizontal branch, a steep thin red giant branch, and a narrow subgiant branch. The main sequence reaches approximately 2 magnitudes below the main-sequence turnoff (V(sup UMi, sub TO) approximately equals 23.27 +/- 0.11 mag) of the median stellar population. We compare the fiducial sequence of the Galactic globular cluster M92 (NGC 6341). The excellent match between Ursa Minor and M92 confirms that the median stellar population of the UMi dSph galaxy is metal poor ([Fe/H](sub UMi) approximately equals [Fe/H](sub M92) approximately equals -2.2 dex) and ancient (age(sub UMi)approximately equalsage(sub M92) approximately equals 14 Gyr). The B - V reddening and the absorption in V are estimated to be E(B - V) = 0.03 +/- 0.01 mag and A(sup UMi, sub V) = 0.09 +/- 0.03 mag. A new estimate of the distance modulus of Ursa Minor, (m - M)(sup UMi, sub 0) = 19.18 +/- 0.12 mag, has been derived based on fiducial-sequence fitting M92 [DELTA.V(sub UMi - M92) = 4.60 +/- 0.03 mag and DELTA(V - I)(sub UMi - M92) = 0.010 +/- 0.005 mag] and the adoption of the apparent V distance modulus for M92 of (m - M)(sup M92, sub V) = 14.67 +/- 0.08 mag (Pont et al. 1998, A&A, 329, 87). The Ursa Minor dwarf spheroidal galaxy is then at a distance of 69 +/- 4 kpc from the Sun. These HST observations indicate that Ursa Minor has had a very simple star formation history consisting mainly of a single major burst of star formation about 14 Gyr ago which lasted approximately < 2 Gyr. While we may have missed minor younger stellar populations due to the small field-of-view of the WFPC2 instrument, these observations clearly show that most of the stars in the central region Ursa Minor dwarf spheroidal galaxy are ancient. If the ancient Galactic globular clusters, like M92, formed concurrently with the early formation of the Milky Way galaxy itself, then the Ursa Minor dwarf spheroidal is probably as old as the Milky Way.
The difficulties of constructing efficient algorithms for three-dimensional flow are discussed. Reasonable candidates are analyzed and tested, and most are found to have obvious shortcomings. Yet, there is promise that an efficient class of algorithms exist between the severely time-step sized-limited explicit or approximately factored algorithms and the computationally intensive direct inversion of large sparse matrices by Gaussian elimination.
Up to today, preconditioning methods on massively parallel systems have faced a major difficulty. The most successful preconditioning methods in terms of accelerating the convergence of the iterative solver such as incomplete LU factorizations are notoriously difficult to implement on parallel machines for two reasons: (1) the actual computation of the preconditioner is not very floating-point intensive, but requires a large amount of unstructured communication, and (2) the application of the preconditioning matrix in the iteration phase (i.e. triangular solves) are difficult to parallelize because of the recursive nature of the computation. Here we present a new approach to preconditioning for very large, sparse, unsymmetric, linear systems, which avoids both difficulties. We explicitly compute an approximate inverse to our original matrix. This new preconditioning matrix can be applied most efficiently for iterative methods on massively parallel machines, since the preconditioning phase involves only a matrix-vector multiplication, with possibly a dense matrix. Furthermore the actual computation of the preconditioning matrix has natural parallelism. For a problem of size n, the preconditioning matrix can be computed by solving n independent small least squares problems. The algorithm and its implementation on the Connection Machine CM-5 are discussed in detail and supported by extensive timings obtained from real problem data.
This project is to continue the development of the global simulation capabilities of tropospheric and stratospheric chemistry and aerosols in a unified global model. This is a part of our overall investigation of aerosol-chemistry-climate interaction. In the past year, we have enabled the tropospheric chemistry simulations based on the GEOS-CHEM model, and added stratospheric chemical reactions into the GEOS-CHEM such that a globally unified troposphere-stratosphere chemistry and transport can be simulated consistently without any simplifications. The tropospheric chemical mechanism in the GEOS-CHEM includes 80 species and 150 reactions. 24 tracers are transported, including O3, NOx, total nitrogen (NOy), H2O2, CO, and several types of hydrocarbon. The chemical solver used in the GEOS-CHEM model is a highly accurate sparse-matrix vectorized Gear solver (SMVGEAR). The stratospheric chemical mechanism includes an additional approximately 100 reactions and photolysis processes. Because of the large number of total chemical reactions and photolysis processes and very different photochemical regimes involved in the unified simulation, the model demands significant computer resources that are currently not practical. Therefore, several improvements will be taken, such as massive parallelization, code optimization, or selecting a faster solver. We have also continued aerosol simulation (including sulfate, dust, black carbon, organic carbon, and sea-salt) in the global model to cover most of year 2002. These results have been made available to many groups worldwide and accessible from the website http://code916.gsfc.nasa.gov/People/Chin/aot.html.
The Sparse Matrix approach for obtaining lead crystallization conditions has proven to be very fruitful for the crystallization of proteins and nucleic acids. Here we report a Sparse Matrix developed specifically for the crystallization of protein-DNA complexes. This method is rapid and economical, typically requiring 2.5 mg of complex to test 48 conditions. The method was originally developed to crystallize basic fibroblast growth factor (bFGF) complexed with DNA sequences identified through in vitro selection, or SELEX, methods. Two DNA aptamers that bind with approximately nanomolar affinity and inhibit the angiogenic properties of bFGF were selected for co-crystallization. The Sparse Matrix produced lead crystallization conditions for both bFGF-DNA complexes.
Constrained Chebyshev approximation problems of the type with minimum (p is an element of Pi(sub n):p(c)=1) and maximum (z is an element of E) with /p(z)/ are considered. Here Pi(sub n) denotes the set of all complex polynomials of degree at most n, E is any ellipse in the complex plane, and c is an element of C/E. Such approximation problems arise in the context of optimizing semi-iterative methods for the solution of large, sparse systems of linear equations Ax=b with complex non-Hermitian coefficient matrices A. The problem of obtaining optimal polynomial preconditioners for conjugate gradient type methods for Ax=b also leads to problems of this type. A new family of polynomials -- q(sub n)(z;c), n is an element of N, and c is an element of C/E -- are introduced as the polynomials which are optimal for a modified version of the Chebyshev approximation problem with Pi(sub n) replaced by a certain subfamily. Some simple properties of q(sub n) are also listed. A necessary and sufficient condition for q(sub n) to be the extremal polynomial for the approximation problem is then derived. Finally, it is shown that q(sub n) is indeed optimal for the problem for all fixed n whenever the distance between c and E is sufficiently large. Results of some numerical tests are presented.
The information capacity of Kanerva's Sparse Distributed Memory (SDM) and Hopfield-type neural networks is investigated. Under the approximations used here, it is shown that the total information stored in these systems is proportional to the number connections in the network. The proportionality constant is the same for the SDM and Hopfield-type models independent of the particular model, or the order of the model. The approximations are checked numerically. This same analysis can be used to show that the SDM can store sequences of spatiotemporal patterns, and the addition of time-delayed connections allows the retrieval of context dependent temporal patterns. A minor modification of the SDM can be used to store correlated patterns.
The information capacity of Kanerva's Sparse Distributed Memory (SDM) and Hopfield-type neural networks is investigated. Under the approximations used, it is shown that the total information stored in these systems is proportional to the number connections in the network. The proportionality constant is the same for the SDM and Hopfield-type models independent of the particular model, or the order of the model. The approximations are checked numerically. This same analysis can be used to show that the SDM can store sequences of spatiotemporal patterns, and the addition of time-delayed connections allows the retrieval of context dependent temporal patterns. A minor modification of the SDM can be used to store correlated patterns.
Spatial correlation among densely packed particles can substantially change their single-scattering properties, thus making questionable the applicability of the independent scattering approximation in calculations of light scattering by planetary regoliths. The same problem arises in geophysics in light scattering computations for snow, frosts, and bare soil. In this paper, we use a dense-medium light-scattering theory based on the introduction of the static structure factor to calculate asymmetry parameters of the phase function for densely packed particles with real refractive indices 1.31 and 1.66, approximating water ice and soil particles, respectively, and imaginary refractive indices 0, 0.01, and 0.3. For sparsely distributed, independently scattering grains, the calculated asymmetry parameters are always positive and always larger than those for densely packed particles. For densely packed grains, the asymmetry parameters may be negative but only for radius-to-wavelength ratios from about 0.1 to about 0.4. With decreasing particle size, the calculated asymmetry parameters tend to zero independently of the compaction state. In the geometrical optics regime, the asymmetry parameters for densely packed scatterers are positive and very close to those for independently scattering grains. These results may have important implications for remote sensing of the Earth and solid planetary surfaces. In particular, it is demonstrated that negative asymmetry parameters derived with some approximate multiple-scattering theories may be physically irrelevant and can be the result of using an inaccurate bidirectional reflection function combined with the ill-conditionally of the inverse scattering problem.
Simultaneous X-ray, ultraviolet, optical, infrared, and radio monitoring data were used to test and constrain models of continuum emission from the BL Lacertae object PKS 2155-304. Intensively sampled ultraviolet and soft X-ray light curves showed a clear temporal correlation with the X-rays leading the ultraviolet by 2-3 hr. This lag was found to be significantly different from zero after an exhaustive comparison of four different techniques for measuring temporal correlations. Variations in the ultraviolet trough optical wave bands were also strongly correlated, with no measurable lag down to limiting timescales of approximately less than 1-2 hr. This strong correlation extends to the near-infrared, but the less intensive sampling precludes measurement of any lag beyomnd an upper limit of approximately less than 1 day. These lags and limits of the order of hours are much shorter than most rapid observed single-band variations. Because of the very sparse radio sampling, it was not possible to measure quantitatively the correlation and lag with shorter wavelengths, but the data do suggest that the radio may lag the optical/ultraviolet by approximately 1 week, with longer delays and weaker variations to longer radio wavelengths. The epoch-folding Q(exp 2) statistic was used to test for periodicity, and no evidence for strict or quasi-periodicity was found in any of the light curves. Because they lead the lower frequencies, the soft X-rays (approximately less than 1 keV) cannot arise from synchrotron self-Compton scattering. These results also rule out the accretion disk model, which predicts a measurable lag between ultraviolet/optical wavelength bands and a correlation between hardness and brightness, neither of which were seen. They are consistent with the entire radio through X-ray continuum arising from direct synchrotron emission from a relativistic jet. However, the tapered jet model, in which the X-ray emission is produced closer in, has problems explaining the magnitude of the ultraviolet/X-ray lag, because the X-ray-emitting electrons have very short lifetimes (t(sub 1/2) much less than 1 s). The result that the lag is much smaller than the variability timescale suggests instead that the radiation may be produced in a flattened region such as a shock front.