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A computational model of the deterministic and stochastic processes involved in multispectral remote sensing was designed to evaluate the performance of sensor systems and data processing algorithms for spectral feature classification. Accuracy in distinguishing between categories of surfaces or between specific types is developed as a means to compare sensor systems and data processing algorithms. The model allows studies to be made of the effects of variability of the atmosphere and of surface reflectance, as well as the effects of channel selection and sensor noise. Examples of these effects are shown.

The analysis of spectral signals for features that represent physical phenomenon is ubiquitous in the science and engineering communities. There are two main approaches that can be taken to extract relevant features from these high-dimensional data streams. The first set of approaches relies on extracting features using a physics-based paradigm where the underlying physical mechanism that generates the spectra is used to infer the most important features in the data stream. We focus on a complementary methodology that uses a data-driven technique that is informed by the underlying physics but also has the ability to adapt to unmodeled system attributes and dynamics. We discuss the following four algorithms: Spectral Decomposition Algorithm (SDA), Non-Negative Matrix Factorization (NMF), Independent Component Analysis (ICA) and Principal Components Analysis (PCA) and compare their performance on a spectral emulator which we use to generate artificial data with known statistical properties. This spectral emulator mimics the real-world phenomena arising from the plume of the space shuttle main engine and can be used to validate the results that arise from various spectral decomposition algorithms and is very useful for situations where real-world systems have very low probabilities of fault or failure. Our results indicate that methods like SDA and NMF provide a straightforward way of incorporating prior physical knowledge while NMF with a tuning mechanism can give superior performance on some tests. We demonstrate these algorithms to detect potential system-health issues on data from a spectral emulator with tunable health parameters.

The celebrated Cheeger's Inequality establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph. In this paper we introduce a new hypergraph Laplacian operator generalizing the Laplacian matrix of graphs. In particular, the operator is induced by a diffusion process on the hypergraph, such that within each hyperedge, measure flows from vertices having maximum weighted measure to those having minimum. Since the operator is non-linear, we have to exploit other properties of the diffusion process to recover a spectral property concerning the "second eigenvalue" of the resulting Laplacian. Moreover, we show that higher order spectral properties cannot hold in general using the current framework. We consider a stochastic diffusion process, in which each vertex also experiences Brownian noise from outside the system. We show a relationship between the second eigenvalue and the convergence behavior of the process. We show that various hypergraph parameters like multi-way expansion and diameter can be bounded using this operator's spectral properties. Since higher order spectral properties do not hold for the Laplacian operator, we instead use the concept of procedural minimizers to consider higher order Cheeger-like inequalities. For any positive integer $k$, we give a polynomial time algorithm to compute an $O(\log r)$-approximation to the $k$-th procedural minimizer, where $r$ is the maximum cardinality of a hyperedge. We show that this approximation factor is optimal under the SSE hypothesis for constant values of $k$.

We consider two problems that arise in machine learning applications: the problem of recovering a planted sparse vector in a random linear subspace and the problem of decomposing a random low-rank overcomplete 3-tensor. For both problems, the best known guarantees are based on the sum-of-squares method. We develop new algorithms inspired by analyses of the sum-of-squares method. Our algorithms achieve the same or similar guarantees as sum-of-squares for these problems but the running time is significantly faster. For the planted sparse vector problem, we give an algorithm with running time nearly linear in the input size that approximately recovers a planted sparse vector with up to constant relative sparsity in a random subspace of $\mathbb R^n$ of dimension up to $\tilde \Omega(\sqrt n)$. These recovery guarantees match the best known ones of Barak, Kelner, and Steurer (STOC 2014) up to logarithmic factors. For tensor decomposition, we give an algorithm with running time close to linear in the input size (with exponent $\approx 1.086$) that approximately recovers a component of a random 3-tensor over $\mathbb R^n$ of rank up to $\tilde \Omega(n^{4/3})$. The best previous algorithm for this problem due to Ge and Ma (RANDOM 2015) works up to rank $\tilde \Omega(n^{3/2})$ but requires quasipolynomial time.

Various procedures for effecting a rational spectral model of a stationary time series are presented. In particular, the new high performance ARMA modeling method is detailed and its superior spectral estimation performance relative to the AR maximum entropy and Box-Jenkins ARMA method is demonstrated. The major contribution of this report is the development of a super fast adaptive algorithm for implementing the high performance ARMA algorithm thereby making real time estimates feasible. (Author)

The Case 2 chlorophyll a algorithm is based on a semi-analytical, bio-optical model of remote sensing reflectance, R(sub rs)(lambda), where R(sub rs)(lambda) is defined as the water-leaving radiance, L(sub w)(lambda), divided by the downwelling irradiance just above the sea surface, E(sub d)(lambda,0(+)). The R(sub rs)(lambda) model (Section 3) has two free variables, the absorption coefficient due to phytoplankton at 675 nm, a(sub phi)(675), and the absorption coefficient due to colored dissolved organic matter (CDOM) or gelbstoff at 400 nm, a(sub g)(400). The R(rs) model has several parameters that are fixed or can be specified based on the region and season of the MODIS scene. These control the spectral shapes of the optical constituents of the model. R(sub rs)(lambda(sub i)) values from the MODIS data processing system are placed into the model, the model is inverted, and a(sub phi)(675), a(sub g)(400) (MOD24), and chlorophyll a (MOD21, Chlor_a_3) are computed. Algorithm development is initially focused on tropical, subtropical, and summer temperate environments, and the model is parameterized in Section 4 for three different bio-optical domains: (1) high ratios of photoprotective pigments to chlorophyll and low self-shading, which for brevity, we designate as 'unpackaged'; (2) low ratios and high self-shading, which we designate as 'packaged'; and (3) a transitional or global-average type. These domains can be identified from space by comparing sea-surface temperature to nitrogen-depletion temperatures for each domain (Section 5). Algorithm errors of more than 45% are reduced to errors of less than 30% with this approach, with the greatest effect occurring at the eastern and polar boundaries of the basins. Section 6 provides an expansion of bio-optical domains into high-latitude waters. The 'fully packaged' pigment domain is introduced in this section along with a revised strategy for implementing these variable packaging domains. Chlor_a_3 values derived semi-analytically and Chlor_a_2 values derived empirically using the O Reilly et al. OC3M algorithm from MODIS Terra radiances are compared to field chlorophyll-a concentrations in Sections 7 and 8.

This paper develops local signal detection strategies for spectral resolution of frequencies of nearby tones. The problem of interest is to decide whether a received noise-corrupted and discrete signal is a single-frequency sinusoid or a double-frequency sinusoid. This paper presents an extension to M. Shahram and P. Milanfar (On the resolvability of sinusoids with nearby frequencies in the presence of noise, IEEE Trans. Signal Process., to appear, available at http://www.soe.ucsc.edu/milanfar) the case where the noise variance is unknown. A general signal model is considered where the frequencies, amplitudes, phases and also the level of the noise variance is unknown to the detector. We derive a fundamental trade-off between SNR and the minimum detectable difference between the frequencies of two tones, for any desired decision error rate. We also demonstrate that the algorithm, when implemented in a practical scenario, yields significantly better performance compared to the standard subspace-based methods like MUSIC. It is also observed that the performance for the case where the noise variance is unknown, is very close to that when the noise variance is known to the detector.

Various procedures for effecting a rational spectral model of a stationary time series are presented. In particular, the new high performance ARMA modeling method is detailed and its superior spectral estimation performance relative to the AR maximum entropy and Box-Jenkins ARMA method is demonstrated. The major contribution of this report is the development of a super fast adaptive algorithm for implementing the high performance ARMA algorithm thereby making real time estimates feasible. (Author)

Algorithms to estimate the power spectrum using Maximum Entropy Methods were developed. These algorithms were coded in FORTRAN 77 and were implemented on the VAX 780. The important considerations in this analysis are: (1) resolution, i.e., how close in frequency two spectral components can be spaced and still be identified; (2) dynamic range, i.e., how small a spectral peak can be, relative to the largest, and still be observed in the spectra; and (3) variance, i.e., how accurate the estimate of the spectra is to the actual spectra. The application of the algorithms based on Maximum Entropy Methods to a variety of data shows that these criteria are met quite well. Additional work in this direction would help confirm the findings. All of the software developed was turned over to the technical monitor. A copy of a typical program is included. Some of the actual data and graphs used on this data are also included.

In this paper we introduce a notion of spectral approximation for directed graphs. While there are many potential ways one might define approximation for directed graphs, most of them are too strong to allow sparse approximations in general. In contrast, we prove that for our notion of approximation, such sparsifiers do exist, and we show how to compute them in almost linear time. Using this notion of approximation, we provide a general framework for solving asymmetric linear systems that is broadly inspired by the work of [Peng-Spielman, STOC`14]. Applying this framework in conjunction with our sparsification algorithm, we obtain an almost linear time algorithm for solving directed Laplacian systems associated with Eulerian Graphs. Using this solver in the recent framework of [Cohen-Kelner-Peebles-Peng-Sidford-Vladu, FOCS`16], we obtain almost linear time algorithms for solving a directed Laplacian linear system, computing the stationary distribution of a Markov chain, computing expected commute times in a directed graph, and more. For each of these problems, our algorithms improves the previous best running times of $O((nm^{3/4} + n^{2/3} m) \log^{O(1)} (n \kappa \epsilon^{-1}))$ to $O((m + n2^{O(\sqrt{\log{n}\log\log{n}})}) \log^{O(1)} (n \kappa \epsilon^{-1}))$ where $n$ is the number of vertices in the graph, $m$ is the number of edges, $\kappa$ is a natural condition number associated with the problem, and $\epsilon$ is the desired accuracy. We hope these results open the door for further studies into directed spectral graph theory, and will serve as a stepping stone for designing a new generation of fast algorithms for directed graphs.

Various procedures for effecting a rational spectral model of a stationary time series are presented. In particular, the new high performance ARMA modeling method is detailed and its superior spectral estimation performance relative to the AR maximum entropy and Box-Jenkins ARMA method is demonstrated. The major contribution of this report is the development of a super fast adaptive algorithm for implementing the high performance ARMA algorithm thereby making real time estimates feasible. (Author)

Developed is an algorithmic approach for wide field of view interferometric spatial-spectral image synthesis. The data collected from the interferometer consists of a set of double-Fourier image data cubes, one cube per baseline. These cubes are each three-dimensional consisting of arrays of two-dimensional detector counts versus delay line position. For each baseline a moving delay line allows collection of a large set of interferograms over the 2D wide field detector grid; one sampled interferogram per detector pixel per baseline. This aggregate set of interferograms, is algorithmically processed to construct a single spatial-spectral cube with angular resolution approaching the ratio of the wavelength to longest baseline. The wide field imaging is accomplished by insuring that the range of motion of the delay line encompasses the zero optical path difference fringe for each detector pixel in the desired field-of-view. Each baseline cube is incoherent relative to all other baseline cubes and thus has only phase information relative to itself. This lost phase information is recovered by having point, or otherwise known, sources within the field-of-view. The reference source phase is known and utilized as a constraint to recover the coherent phase relation between the baseline cubes and is key to the image synthesis. Described will be the mathematical formalism, with phase referencing and results will be shown using data collected from NASA/GSFC Wide-Field Imaging Interferometry Testbed (WIIT).

Various procedures for effecting a rational spectral model of a stationary time series are presented. In particular, the new high performance ARMA modeling method is detailed and its superior spectral estimation performance relative to the AR maximum entropy and Box-Jenkins ARMA method is demonstrated. The major contribution of this report is the development of a super fast adaptive algorithm for implementing the high performance ARMA algorithm thereby making real time estimates feasible. (Author)

Various procedures for effecting a rational spectral model of a stationary time series are presented. In particular, the new high performance ARMA modeling method is detailed and its superior spectral estimation performance relative to the AR maximum entropy and Box-Jenkins ARMA method is demonstrated. The major contribution of this report is the development of a super fast adaptive algorithm for implementing the high performance ARMA algorithm thereby making real time estimates feasible. (Author)

Simulation results of Ising systems for several update rules, observables, and dimensions are analyzed. The lattice-size dependence is discussed for the autocorrelation times and for the weights of eigenvalues, giving fit results in the case of power laws. Implications of spectral properties are pointed out and the behavior of a particular observable not governed by detailed balance is explained.

Clustering is the problem of separating a set of objects into groups (called clusters) so that objects within the same cluster are more similar to each other than to those in different clusters. Spectral clustering is a now well-known method for clustering which utilizes the spectrum of the data similarity matrix to perform this separation. Since the method relies on solving an eigenvector problem, it is computationally expensive for large datasets. To overcome this constraint, approximation methods have been developed which aim to reduce running time while maintaining accurate classification. In this article, we summarize and experimentally evaluate several approximation methods for spectral clustering. From an applications standpoint, we employ spectral clustering to solve the so-called attrition problem, where one aims to identify from a set of employees those who are likely to voluntarily leave the company from those who are not. Our study sheds light on the empirical performance of existing approximate spectral clustering methods and shows the applicability of these methods in an important business optimization related problem.

We give a new algorithm for Unique Games which is based on purely {\em spectral} techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our algorithm is able to recover a good assignment. The approximation guarantee depends only on the completeness of the game, and not on the alphabet size, while the running time depends on spectral properties of the {\em Label-Extended} graph associated with the instance of Unique Games. We further show that on input the integrality gap instance of Khot and Vishnoi, our algorithm runs in quasi-polynomial time and decides that the instance if highly unsatisfiable. Notably, when run on this instance, the standard SDP relaxation of Unique Games {\em fails}. As a special case, we also re-derive a polynomial time algorithm for Unique Games on expander constraint graphs. The main ingredient of our algorithm is a technique to effectively use the full spectrum of the underlying graph instead of just the second eigenvalue, which is of independent interest. The question of how to take advantage of the full spectrum of a graph in the design of algorithms has been often studied, but no significant progress was made prior to this work.

The data augmentation (DA) algorithm is a widely used Markov chain Monte Carlo algorithm that is easy to implement but often suffers from slow convergence. The sandwich algorithm is an alternative that can converge much faster while requiring roughly the same computational effort per iteration. Theoretically, the sandwich algorithm always converges at least as fast as the corresponding DA algorithm in the sense that $\Vert {K^*}\Vert \le \Vert {K}\Vert$, where $K$ and $K^*$ are the Markov operators associated with the DA and sandwich algorithms, respectively, and $\Vert\cdot\Vert$ denotes operator norm. In this paper, a substantial refinement of this operator norm inequality is developed. In particular, under regularity conditions implying that $K$ is a trace-class operator, it is shown that $K^*$ is also a positive, trace-class operator, and that the spectrum of $K^*$ dominates that of $K$ in the sense that the ordered elements of the former are all less than or equal to the corresponding elements of the latter. Furthermore, if the sandwich algorithm is constructed using a group action, as described by Liu and Wu [J. Amer. Statist. Assoc. 94 (1999) 1264--1274] and Hobert and Marchev [Ann. Statist. 36 (2008) 532--554], then there is strict inequality between at least one pair of eigenvalues. These results are applied to a new DA algorithm for Bayesian quantile regression introduced by Kozumi and Kobayashi [J. Stat. Comput. Simul. 81 (2011) 1565--1578].

In this report, the performance characteristics of the LMS Gradient Algorithm, two Adaptive Fixed-Point Iteration Algorithms and of a non-iterative method based on Levinson's Algorithm are considered for the case where an adaptive algorithm is used to determine the unit sample response for a system which attempts to discriminate between the signal and noise processes on the basis of bandwidth. Such a system is often referred to as a Spectral Line Enhancer. Theoretical bounds on the mean-square error as a function of the time index n are derived for each of the three iterative methods. A comparison of these bounds is made for the case where the input data to the Spectral Line Enhancer is composed of a single sinusoid of random phase in the presence of an additive autoregressive noise sequence. The results of extensive computer simulations of the adaptive algorithms considered are used to determine the usefulness of the theoretical bounds and to make comparisons of the performance of the four methods.

Despite computational superiorities, some traditional conjugate gradient algorithms such as Polak-Ribiére-Polyak and Hestenes-Stiefel methods generally fail to guarantee the descent condition. Here, in a matrix viewpoint, spectral versions of such methods are developed which fulfill the descent condition. The convergence of the given spectral algorithms is argued briefly. Afterwards, we propose an improved version of the nonnegative matrix factorization problem by adding penalty terms to the model, for controlling the condition number of one of the factorization elements. Finally, the computational merits of the method are examined using a set of CUTEr test problems as well as some random nonnegative matrix factorization models. The results typically agree with our analytical spectrum. 

We consider the problem of community detection in the Stochastic Block Model with a finite number $K$ of communities of sizes linearly growing with the network size $n$. This model consists in a random graph such that each pair of vertices is connected independently with probability $p$ within communities and $q$ across communities. One observes a realization of this random graph, and the objective is to reconstruct the communities from this observation. We show that under spectral algorithms, the number of misclassified vertices does not exceed $s$ with high probability as $n$ grows large, whenever $pn=\omega(1)$, $s=o(n)$ and \begin{equation*} \lim\inf_{n\to\infty} {n(\alpha_1 p+\alpha_2 q-(\alpha_1 + \alpha_2)p^{\frac{\alpha_1}{\alpha_1 + \alpha_2}}q^{\frac{\alpha_2}{\alpha_1 + \alpha_2}})\over \log (\frac{n}{s})} >1,\quad\quad(1) \end{equation*} where $\alpha_1$ and $\alpha_2$ denote the (fixed) proportions of vertices in the two smallest communities. In view of recent work by Abbe et al. and Mossel et al., this establishes that the proposed spectral algorithms are able to exactly recover communities whenever this is at all possible in the case of networks with two communities with equal sizes. We conjecture that condition (1) is actually necessary to obtain less than $s$ misclassified vertices asymptotically, which would establish the optimality of spectral method in more general scenarios.

The sparsest cut problem consists of identifying a small set of edges that breaks the graph into balanced sets of vertices. The normalized cut problem balances the total degree, instead of the size, of the resulting sets. Applications of graph cuts include community detection and computer vision. However, cut problems were originally proposed for static graphs, an assumption that does not hold in many modern applications where graphs are highly dynamic. In this paper, we introduce the sparsest and normalized cut problems in temporal graphs, which generalize their standard definitions by enforcing the smoothness of cuts over time. We propose novel formulations and algorithms for computing temporal cuts using spectral graph theory, multiplex graphs, divide-and-conquer and low-rank matrix approximation. Furthermore, we extend our formulation to dynamic graph signals, where cuts also capture node values, as graph wavelets. Experiments show that our solutions are accurate and scalable, enabling the discovery of dynamic communities and the analysis of dynamic graph processes.

We describe a simple algorithm for spectral graph sparsification, based on iterative computations of weighted spanners and uniform sampling. Leveraging the algorithms of Baswana and Sen for computing spanners, we obtain the first distributed spectral sparsification algorithm. We also obtain a parallel algorithm with improved work and time guarantees. Combining this algorithm with the parallel framework of Peng and Spielman for solving symmetric diagonally dominant linear systems, we get a parallel solver which is much closer to being practical and significantly more efficient in terms of the total work.

Experience with the maximum entropy method of spectral analysis suggests that it can produce inaccurate frequency estimates of short sample sinusoidal data, and it sometimes produces calculated values for the filter coefficients that are unduly contaminated by rounding errors. Consequently, this report develops an algorithm for solving the underlying least-squares problem directly, without forcing a Toeplitz structure on the model. This approach leads to more accurate frequency determination for short sample harmonic processes, and our algorithm is computationally efficient and numerically stable. The algorithm can also be applied to two other versions of the linear prediction problem. A FORTRAN program is supplied.

Prototype. An integrated sensor suite and local data fusion algorithms are being designed as the Volume Sensor Detection Suite (VSDS). As an example of potential inter-component data fusion for the VSDS, the Long Wavelength Video Detection (LWVD) Component Luminosity was investigated as a substitute for the near-infrared (NIR) sensor element in the Spectral-Based Volume Sensor (SBVS) component of the VSDS. The LWVD Luminosity data stream can be used to produce an effective 3-component SBVS configuration for the VSDS with only the UV and IR sensor elements from the SBVS Component Prototype. The current recommendation for the SBVS configuration of the VSDS is a UV/IR, an NIR/UV/IR, or an LWVD/UV/IR configuration, depending on the VSDS data fusion component's tolerance for increased probability of false alarm (Pfa) with increased probability of detection (Pd) in different event categories and on the specific intended applications.

Prototype. An integrated sensor suite and local data fusion algorithms are being designed as the Volume Sensor Detection Suite (VSDS). As an example of potential inter-component data fusion for the VSDS, the Long Wavelength Video Detection (LWVD) Component Luminosity was investigated as a substitute for the near-infrared (NIR) sensor element in the Spectral-Based Volume Sensor (SBVS) component of the VSDS. The LWVD Luminosity data stream can be used to produce an effective 3-component SBVS configuration for the VSDS with only the UV and IR sensor elements from the SBVS Component Prototype. The current recommendation for the SBVS configuration of the VSDS is a UV/IR, an NIR/UV/IR, or an LWVD/UV/IR configuration, depending on the VSDS data fusion component's tolerance for increased probability of false alarm (Pfa) with increased probability of detection (Pd) in different event categories and on the specific intended applications.

Quantum Algorithms for spectral sums by Alessandro Luongo @QTMLConference

Book Source: Digital Library of India Item 2015.188989 dc.contributor.author: Madhusudhana H S dc.date.accessioned: 2015-07-07T22:36:29Z dc.date.available: 2015-07-07T22:36:29Z dc.date.digitalpublicationdate: 2005-09-27 dc.identifier.barcode: 1990010089690 dc.identifier.origpath: /rawdataupload/upload/0089/690 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/188989 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 230 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: I I T Kanpur dc.rights: Out_of_copyright dc.source.library: I I T Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Electrical Engineering dc.title: New Decoding Algorithms For Reed Muller And Cyclic Codes Based On Spectral Domain

Various procedures for effecting a rational spectral model of a stationary time series are presented. In particular, the new high performance ARMA modeling method is detailed and its superior spectral estimation performance relative to the AR maximum entropy and Box-Jenkins ARMA method is demonstrated. The major contribution of this report is the development of a super fast adaptive algorithm for implementing the high performance ARMA algorithm thereby making real time estimates feasible. (Author)

There is concern that high power sources of electromagnetic radiation may cause physical harm to an exposed individual even when the frequencies are below those of X rays. Specifically concerns arise in the use of microwave equipment, radars, lasers, active imaging devices and transmitters. Early efforts to address this question considered total absorbed power and then local internal temperature increases; Computer algorithms to make these predictions were developed by USAF/SAM. Recently, as-equipment with small duty cycles and very high peak power have been developed, concerns over the effect of sweeping fields and electromagnetic transients oh biological-structures have arisen. The algorithm developed in this report has as its purpose a highly accurate benchmark code which will predict the response of an N layer bianisotropic spherical structure to multiple plane waves with different amplitudes, frequencies, polarizations, and directions of travel and full wave solutions involving all of the pm sub n. Bioelectromagnetics, hazard assessment, cancer therapy, bianisotropy, temperature prediction, full wave solutions, Mie like solution.

Various procedures for effecting a rational spectral model of a stationary time series are presented. In particular, the new high performance ARMA modeling method is detailed and its superior spectral estimation performance relative to the AR maximum entropy and Box-Jenkins ARMA method is demonstrated. The major contribution of this report is the development of a super fast adaptive algorithm for implementing the high performance ARMA algorithm thereby making real time estimates feasible. (Author)

Various procedures for effecting a rational spectral model of a stationary time series are presented. In particular, the new high performance ARMA modeling method is detailed and its superior spectral estimation performance relative to the AR maximum entropy and Box-Jenkins ARMA method is demonstrated. The major contribution of this report is the development of a super fast adaptive algorithm for implementing the high performance ARMA algorithm thereby making real time estimates feasible. (Author)

In this paper we describe spectral transformation algorithms for the computation of eigenvalues with positive real part of sparse nonsymmetric matrix pencils $(J,L)$, where $L$ is of the form $\pmatrix{M&0\cr 0&0}$. For this we define a different extension of M\"obius transforms to pencils that inhibits the effect on iterations of the spurious eigenvalue at infinity. These algorithms use a technique of preconditioning the initial vectors by M\"obius transforms which together with shift-invert iterations accelerate the convergence to the desired eigenvalues. Also, we see that M\"obius transforms can be successfully used in inhibiting the convergence to a known eigenvalue. Moreover, the procedure has a computational cost similar to power or shift-invert iterations with M\"obius transforms: neither is more expensive than the usual shift-invert iterations with pencils. Results from tests with a concrete transient stability model of an interconnected power system whose Jacobian matrix has order 3156 are also reported here.

We study algorithms for spectral graph sparsification. The input is a graph $G$ with $n$ vertices and $m$ edges, and the output is a sparse graph $\tilde{G}$ that approximates $G$ in an algebraic sense. Concretely, for all vectors $x$ and any $\epsilon>0$, $\tilde{G}$ satisfies $$ (1-\epsilon) x^T L_G x \leq x^T L_{\tilde{G}} x \leq (1+\epsilon) x^T L_G x, $$ where $L_G$ and $L_{\tilde{G}}$ are the Laplacians of $G$ and $\tilde{G}$ respectively. We show that the fastest known algorithm for computing a sparsifier with $O(n\log n/\epsilon^2)$ edges can actually run in $\tilde{O}(m\log^2 n)$ time, an $O(\log n)$ factor faster than before. We also present faster sparsification algorithms for slightly dense graphs. Specifically, we give an algorithm that runs in $\tilde{O}(m\log n)$ time and generates a sparsifier with $\tilde{O}(n\log^3{n}/\epsilon^2)$ edges. This implies that a sparsifier with $O(n\log n/\epsilon^2)$ edges can be computed in $\tilde{O}(m\log n)$ time for graphs with more than $O(n\log^4 n)$ edges. We also give an $\tilde{O}(m)$ time algorithm for graphs with more than $n\log^5 n (\log \log n)^3$ edges of polynomially bounded weights, and an $O(m)$ algorithm for unweighted graphs with more than $n\log^8 n (\log \log n)^3 $ edges and $n\log^{10} n (\log \log n)^5$ edges in the weighted case. The improved sparsification algorithms are employed to accelerate linear system solvers and algorithms for computing fundamental eigenvectors of slightly dense SDD matrices.

Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1

Spectral element methods are high-order weighted residual techniques for partial differential equations that combine the geometric flexibility of finite element methods with the rapid convergence of spectral techniques. Spectral element methods are described for the simulation of incompressible fluid flows, with special emphasis on implementation of spectral element techniques on medium-grained parallel processors. Two parallel architectures are considered: the first, a commercially available message-passing hypercube system; the second, a developmental reconfigurable architecture based on Geometry-Defining Processors. High parallel efficiency is obtained in hypercube spectral element computations, indicating that load balancing and communication issues can be successfully addressed by a high-order technique/medium-grained processor algorithm-architecture coupling.

Various procedures for effecting a rational spectral model of a stationary time series are presented. In particular, the new high performance ARMA modeling method is detailed and its superior spectral estimation performance relative to the AR maximum entropy and Box-Jenkins ARMA method is demonstrated. The major contribution of this report is the development of a super fast adaptive algorithm for implementing the high performance ARMA algorithm thereby making real time estimates feasible. (Author)

Despite computational superiorities, some traditional conjugate gradient algorithms such as Polak–Ribiére–Polyak and Hestenes–Stiefel methods generally fail to guarantee the descent condition. Here, in a matrix viewpoint, spectral versions of such methods are developed which fulfill the descent condition. The convergence of the given spectral algorithms is argued briefly. Afterwards, we propose an improved version of the nonnegative matrix factorization problem by adding penalty terms to the model, for controlling the condition number of one of the factorization elements. Finally, the computational merits of the method are examined using a set of CUTEr test problems as well as some random nonnegative matrix factorization models. The results typically agree with our analytical spectrum.

Various procedures for effecting a rational spectral model of a stationary time series are presented. In particular, the new high performance ARMA modeling method is detailed and its superior spectral estimation performance relative to the AR maximum entropy and Box-Jenkins ARMA method is demonstrated. The major contribution of this report is the development of a super fast adaptive algorithm for implementing the high performance ARMA algorithm thereby making real time estimates feasible. (Author)

The development of high-order methods (order of accuracy 2nd order) on unstructured grids is widely viewed as a major pacing item in computational fluid dynamics (CFD). Efficient high-order methods capable of handling complex geometries are required to compute vortex dominated flows, and to perform large eddy simulation and direct numerical simulation with complex configurations, and to predict aeroacoustic noise generation and propagation. The primary objective of the present research is to develop implicit, multigrid and adaptive solution algorithms for a promising high-order method, the spectral difference (SD) method. Several major activities were carried out: 1. The development of a hp-adaptation capability to capture both discontinuous and smooth flow features with high efficiency; 2. Development of a accuracy-preserving limiter for discontinuity-capturing for the high-order SD Navier-Stokes solver; 3. Demonstration of the developed code for real world unsteady vortex-dominated flow problems.

In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial-time algorithms. Among the latter, the best statistical guarantees have been proved, for third-order tensors, using the sixth level of the sum-of-squares (SOS) semidefinite programming hierarchy (Barak and Moitra, 2014). However, the SOS approach does not scale well to large problem instances. By contrast, spectral methods --- based on unfolding or matricizing the tensor --- are attractive for their low complexity, but have been believed to require a much larger sample size. This paper presents two main contributions. First, we propose a new unfolding-based method, which outperforms naive ones for symmetric $k$-th order tensors of rank $r$. For this result we make a study of singular space estimation for partially revealed matrices of large aspect ratio, which may be of independent interest. For third-order tensors, our algorithm matches the SOS method in terms of sample size (requiring about $rd^{3/2}$ revealed entries), subject to a worse rank condition ($r\ll d^{3/4}$ rather than $r\ll d^{3/2}$). We complement this result with a different spectral algorithm for third-order tensors in the overcomplete ($r\ge d$) regime. Under a random model, this second approach succeeds in estimating tensors of rank $d\le r \ll d^{3/2}$ from about $rd^{3/2}$ revealed entries.

We study the problem of selecting a subset of k random variables from a large set, in order to obtain the best linear prediction of another variable of interest. This problem can be viewed in the context of both feature selection and sparse approximation. We analyze the performance of widely used greedy heuristics, using insights from the maximization of submodular functions and spectral analysis. We introduce the submodularity ratio as a key quantity to help understand why greedy algorithms perform well even when the variables are highly correlated. Using our techniques, we obtain the strongest known approximation guarantees for this problem, both in terms of the submodularity ratio and the smallest k-sparse eigenvalue of the covariance matrix. We further demonstrate the wide applicability of our techniques by analyzing greedy algorithms for the dictionary selection problem, and significantly improve the previously known guarantees. Our theoretical analysis is complemented by experiments on real-world and synthetic data sets; the experiments show that the submodularity ratio is a stronger predictor of the performance of greedy algorithms than other spectral parameters.

Various procedures for effecting a rational spectral model of a stationary time series are presented. In particular, the new high performance ARMA modeling method is detailed and its superior spectral estimation performance relative to the AR maximum entropy and Box-Jenkins ARMA method is demonstrated. The major contribution of this report is the development of a super fast adaptive algorithm for implementing the high performance ARMA algorithm thereby making real time estimates feasible. (Author)

Various procedures for effecting a rational spectral model of a stationary time series are presented. In particular, the new high performance ARMA modeling method is detailed and its superior spectral estimation performance relative to the AR maximum entropy and Box-Jenkins ARMA method is demonstrated. The major contribution of this report is the development of a super fast adaptive algorithm for implementing the high performance ARMA algorithm thereby making real time estimates feasible. (Author)

This paper presents a comparison of cloud-top altitude retrieval methods applied to S-HIS (Scanning High Resolution Interferometer Sounder) measurements. Included in this comparison is an improvement to the traditional CO2 Slicing method. The new method, CO2 Sorting, determines optimal channel pairs to apply the CO2 Slicing. Measurements from collocated samples of the Cloud Physics Lidar (CPL) and Modis Airborne Simulator (MAS) instruments assist in the comparison. For optically thick clouds good correlation between the S-HIS and lidar cloud-top retrievals are found. For tenuous ice clouds there can be large differences between lidar (CPL) and S-HIS retrieved cloud-tops. It is found that CO2 Sorting significantly reduces the cloud height biases for the optically thin cloud (total optical depths less then 1.0). For geometrically thick but optically thin cirrus clouds large differences between the S-HIS infrared cloud top retrievals and the CPL detected cloud top where found. For these cases the cloud height retrieved by the S-HIS cloud retrievals correlated closely with the level the CPL integrated cloud optical depth was approximately 1.0.

Quantum algorithms are well-suited to calculate estimates of the energy spectra for spin lattice systems. These algorithms are based on the efficient calculation of the discrete Fourier components of the density of states. The efficiency of these algorithms in calculating the free energy per spin of general spin lattices to bounded error is examined. We find that the number of Fourier components required to bound the error in the free energy due to the broadening of the density of states scales polynomially with the number of spins in the lattice. However, the precision with which the Fourier components must be calculated is found to be an exponential function of the system size.

We focus on \emph{row sampling} based approximations for matrix algorithms, in particular matrix multipication, sparse matrix reconstruction, and \math{\ell_2} regression. For \math{\matA\in\R^{m\times d}} (\math{m} points in \math{d\ll m} dimensions), and appropriate row-sampling probabilities, which typically depend on the norms of the rows of the \math{m\times d} left singular matrix of \math{\matA} (the \emph{leverage scores}), we give row-sampling algorithms with linear (up to polylog factors) dependence on the stable rank of \math{\matA}. This result is achieved through the application of non-commutative Bernstein bounds. Keywords: row-sampling; matrix multiplication; matrix reconstruction; estimating spectral norm; linear regression; randomized

The largest eigenvalue of the adjacency matrix of a network (referred to as the spectral radius) is an important metric in its own right. Further, for several models of epidemic spread on networks (e.g., the `flu-like' SIS model), it has been shown that an epidemic dies out quickly if the spectral radius of the graph is below a certain threshold that depends on the model parameters. This motivates a strategy to control epidemic spread by reducing the spectral radius of the underlying network. In this paper, we develop a suite of provable approximation algorithms for reducing the spectral radius by removing the minimum cost set of edges (modeling quarantining) or nodes (modeling vaccinations), with different time and quality tradeoffs. Our main algorithm, \textsc{GreedyWalk}, is based on the idea of hitting closed walks of a given length, and gives an $O(\log^2{n})$-approximation, where $n$ denotes the number of nodes; it also performs much better in practice compared to all prior heuristics proposed for this problem. We further present a novel sparsification method to improve its running time. In addition, we give a new primal-dual based algorithm with an even better approximation guarantee ($O(\log n)$), albeit with slower running time. We also give lower bounds on the worst-case performance of some of the popular heuristics. Finally we demonstrate the applicability of our algorithms and the properties of our solutions via extensive experiments on multiple synthetic and real networks.

Community detection has been one of the central problems in network studies and directed network is particularly challenging due to asymmetry among its links. In this paper, we found that incorporating the direction of links reveals new perspectives on communities regarding to two different roles, source and terminal, that a node plays in each community. Intriguingly, such communities appear to be connected with unique spectral property of the graph Laplacian of the adjacency matrix and we exploit this connection by using regularized SVD methods. We propose harvesting algorithms, coupled with regularized SVDs, that are linearly scalable for efficient identification of communities in huge directed networks. The proposed algorithm shows great performance and scalability on benchmark networks in simulations and successfully recovers communities in real network applications.

In this effort we acquired and registered a multi-spectral dynamic image test set with the intent of using the imagery to assess the operational effectiveness of static and dynamic image fusion techniques for a range of relevant military tasks. This paper describes the image acquisition methodology, the planned human visual performance task approach, the lessons learned during image acquisition and the plans for a future, improved image set, resolution assessment methodology and human visual performance task.