Inversion Of Large Matrix Research Articles

Similarity-Guided and <inline-formula> <tex-math notation="LaTeX">$\ell_p$</tex-math></inline-formula>-Regularized Sparse Unmixing of Hyperspectral Data

In this letter, we propose a novel sparse unmixing model combined with two effective regularization terms: one is a similarity-weighting constraint, and the other is the $\ell_p$ $(0 norm sparse regularization. The former utilizes the spatial structural correlation, which is presented in the hyperspectral data, to guide the abundance estimation. When compared with the existing graph Laplacian regularization, our similarity-weighting constraint avoids large matrix inversion, and thus, it can be efficiently solved. As for the $\ell_p$ -norm, it has numerical advantages over the convex $\ell_1$ -norm and better approximates the $\ell_0$ -norm theoretically. Moreover, the $\ell_p$ -norm regularizer can simultaneously promote sparsity and enforce the abundance sum-to-one constraint. Therefore, this term yields more desirable results in practice. Experimental results on both simulated and real data demonstrate the effectiveness of the proposed model.

Calibrating a large computer experiment simulating radiative shock hydrodynamics

We consider adapting a canonical computer model calibration apparatus, involving coupled Gaussian process (GP) emulators, to a computer experiment simulating radiative shock hydrodynamics that is orders of magnitude larger than what can typically be accommodated. The conventional approach calls for thousands of large matrix inverses to evaluate the likelihood in an MCMC scheme. Our approach replaces that costly ideal with a thrifty take on essential ingredients, synergizing three modern ideas in emulation, calibration and optimization: local approximate GP regression, modularization, and mesh adaptive direct search. The new methodology is motivated both by necessity - considering our particular application - and by recent trends in the supercomputer simulation literature. A synthetic data application allows us to explore the merits of several variations in a controlled environment and, together with results on our motivating real-data experiment, lead to noteworthy insights into the dynamics of radiative shocks as well as the limitations of the calibration enterprise generally.

An efficient radial basis function neural network for hyperspectral remote sensing image classification

A very simple radial basis function neural network (RBFNN) is investigated for hyperspectral remote sensing image classification. Its training can be analytically solved with a closed-form equation, and no parameter needs to be manually tuned. Its computational cost is much lower than the popular support vector machine (SVM). Surprisingly, such an RBFNN can achieve the performance that is similar to or even better than the SVM. By incorporating a simple spatial averaging filter or a Gaussian lowpass filter with negligible additional computational cost, classification accuracy can be further improved. Considering the large matrix inversion operation in the RBFNN when the number of training samples being very large, we also propose a parallel processing method to reduce computing time in matrix inversion.

Reduced-order modeling for mistuned centrifugal impellers with crack damages

An efficient method for nonlinear vibration analysis of mistuned centrifugal impellers with crack damages is presented. The main objective is to investigate the effects of mistuning and cracks on the vibration features of centrifugal impellers and to explore effective techniques for crack detection. Firstly, in order to reduce the input information needed for component mode synthesis (CMS), the whole model of an impeller is obtained by rotation transformation based on the finite element model of a sector model. Then, a hybrid-interface method of CMS is employed to generate a reduced-order model (ROM) for the cracked impeller. The degrees of freedom on the crack surfaces are retained in the ROM to simulate the crack breathing effects. A novel approach for computing the inversion of large sparse matrix is proposed to save memory space during model order reduction by partitioning the matrix into many smaller blocks. Moreover, to investigate the effects of mistuning and cracks on the resonant frequencies, the bilinear frequency approximation is used to estimate the resonant frequencies of the mistuned impeller with a crack. Additionally, statistical analysis is performed using the Monte Carlo simulation to study the statistical characteristics of the resonant frequencies versus crack length at different mistuning levels. The results show that the most significant effect of mistuning and cracks on the vibration response is the shift and split of the two resonant frequencies with the same nodal diameters. Finally, potential quantitative indicators for detection of crack of centrifugal impellers are discussed.

Parallel Communication-Avoiding Algorithm for Triangular Matrix Inversion on Homogeneous and Heterogeneous Platforms

We address in this paper the parallelization of a recursive algorithm for large scale triangular matrix inversion based on the `Divide and Conquer' (D&C) paradigm. A set of different versions of an original sequential algorithm are first presented. A theoretical performance study permits to establish an accurate comparison between the designed algorithms. Afterwards, we develop in the second part of the paper, an optimal parallel avoiding-communication algorithm for a given number of available homogeneous and heterogeneous processors. To reach this target, we use a so called `non equitable and incomplete' version of the D&C paradigm consisting in recursively decomposing the original problem into two sub-problems of non equal sizes, then decomposing only one sub-problem in the same previous manner. The theoretical study is validated by a series of experiments achieved on three target platforms, namely an 8-core shared memory machine, a distributed memory cluster and a heterogeneous CPU-GPU cluster. The obtained results permit to illustrate the interest of the contribution.

The Study on a New Approximate Method of Computing Electromagnetic Scattering from Multilayer Rough Sea Surfaces

At high sea states strong winds make the sea surface broken, and become a multilayer-rough sea surface made of a large number of foams and droplets. Similarly, if the sea surface covered by oil and other dirts will also belong to the mutilayer rough sea surfaces of various medium properties. In this paper, applying the Kirchhoff Approximation (KA) and the electromagnetic theory of stratified media, electromagnetic scattering characteristics from a multilayer rough medium surface are calculated. Firstly, a detailed analysis of electromagnetic reflection from multilayer parallel interfaces under different incident angles is carried out. Secondly, combining the preceding two results and courses, electromagnetic scattering from the multilayer random rough surfaces is studied. The computed results are in good agreement with those by using the method of moments (MOM) and reported by some experts. Finally, the random rough sea surface covered by spilling oil or droplets and foams is calculated in detail. Compared with MOM, the new approximate analysis method in the paper can avoid a large matrix inversion, and thus greatly reduce the computation time and improve the computational efficiency.

An efficient hierarchical identification method for general dual-rate sampled-data systems

For the lifted input–output representation of general dual-rate sampled-data systems, this paper presents a decomposition based recursive least squares (D-LS) identification algorithm using the hierarchical identification principle. Compared with the recursive least squares (RLS) algorithm, the proposed D-LS algorithm does not require computing the covariance matrices with large sizes and matrix inverses in each recursion step, and thus has a higher computational efficiency than the RLS algorithm. The performance analysis of the D-LS algorithm indicates that the parameter estimates can converge to their true values. A simulation example is given to confirm the convergence results.

Backstepping experimentally applied to an antagonistically driven finger with flexible tendons

The Hand Arm System of DLR is a complex mechatronic system built to approach the human in terms of dynamics and dexterity. Its fingers are antagonistically actuated by flexible tendons, resulting in a nonlinear flexible joint. The advantages of such a design are the high dynamics but more importantly the enhanced robustness. Nonlinear control methods have been developed in the community that are, at least in principle, applicable to such systems. However, because of the complexity of some methods, most works are focusing on simulations and do not consider the practical issues arising with hardware. Such issues are, for example, the need for derivatives or the use of large matrix inversions. In this paper, we adapt the backstepping method to the specific case of an antagonistic actuation. The modeling of the mechanism is followed by the design of the controller and the work is concluded by a set of experimental results on the hand of the Hand Arm System, the ”Awiwi hand”.

PRIMORDIAL GRAVITATIONAL WAVE DETECTABILITY WITH DEEP SMALL-SKY COSMIC MICROWAVE BACKGROUND EXPERIMENTS

We use the Bayesian estimation on direct T - Q - U cosmic microwave background (CMB) polarization maps to forecast errors on the tensor-to-scalar power ratio r, and hence on primordial gravitational waves, as a function of sky coverage f sky. This map-based likelihood filters the information in the pixel-pixel space into the optimal combinations needed for r detection for cut skies, providing enhanced information over a first-step linear separation into a combination of E, B, and mixed modes, and ignoring the latter. With current computational power and for typical resolutions appropriate for r detection, the large matrix inversions required are accurate and fast. Our simulations explore two classes of experiments, with differing bolometric detector numbers, sensitivities, and observational strategies. One is motivated by a long duration balloon experiment like Spider, with pixel noise for a specified observing period. This analysis also applies to ground-based array experiments. We find that, in the absence of systematic effects and foregrounds, an experiment with Spider-like noise concentrating on f sky ~ 0.02-0.2 could place a 2σ r 0.014 boundary (~95% confidence level), which rises to 0.02 with an l-dependent foreground residual left over from an assumed efficient component separation. We contrast this with a Planck-like fixed instrumental noise as f sky varies, which gives a Galaxy-masked (f sky = 0.75) 2σ r 0.015, rising to 0.05 with the foreground residuals. Using as the figure of merit the (marginalized) one-dimensional Shannon entropy of r, taken relative to the first 2003 WMAP CMB-only constraint, gives –2.7 bits from the 2012 WMAP9+ACT+SPT+LSS data, and forecasts of –6 bits from Spider (+ Planck); this compares with up to –11 bits for CMBPol, COrE, and PIXIE post-Planck satellites and –13 bits for a perfectly noiseless cosmic variance limited experiment. We thus confirm the wisdom of the current strategy for r detection of deeply probed patches covering the f sky minimum-error trough with balloon and ground experiments.

Dense noise face recognition based on sparse representation and algorithm optimization

To improve the speed and anti-noise performance of face recognition based on sparse representation,the Cross-And-Bouquet(CAB) model and Compressed Sensing(CS) reconstruction algorithm were studied.Concerning the large matrix inversion of reconstruction algorithm,a Fast Orthogonal Matching Pursuit(FOMP) algorithm was proposed.The proposed algorithm could convert the high complexity operations of matrix inversion into the lightweight operation of vector-matrix computation.To increase the amount of effective information in dense noise pictures,several practical and efficient methods were put forward.The experimental results verify that these methods can effectively improve the face recognition rate in dense noise cases,and identifiable noise ratio can reach up to 75%.These methods are of practical values.

Outsourcing Large Matrix Inversion Computation to A Public Cloud

Cloud computing enables resource-constrained clients to economically outsource their huge computation workloads to a cloud server with massive computational power. This promising computing paradigm inevitably brings in new security concerns and challenges, such as input/output privacy and result verifiability. Since matrix inversion computation (MIC) is a quite common scientific and engineering computational task, we are motivated to design a protocol to enable secure, robust cheating resistant, and efficient outsourcing of MIC to a malicious cloud in this paper. The main idea to protect the privacy is employing some transformations on the original matrix to get a encrypted matrix which is sent to the cloud; and then transforming the result returned from the cloud to get the correct inversion of the original matrix. Next, a randomized Monte Carlo verification algorithm with one-sided error is employed to successfully handle result verification. In this paper, the superiority of this novel technique in designing inexpensive result verification algorithm for secure outsourcing is well demonstrated. We analytically show that the proposed protocol simultaneously fulfills the goals of correctness, security, robust cheating resistance, and high-efficiency. Extensive theoretical analysis and experimental evaluation also show its high-efficiency and immediate practicability.

Successive interference cancelation and MAP decoding for mobile MIMO OFDM systems and their convergence behavior

Turbo equalization schemes based on minimum mean square error criteria available in the literature for multiple-input multiple-output (MIMO) systems are computationally expensive, as they require a relatively large matrix inversion. In this article, we propose a suboptimal, successive interference cancelation (SIC)-based maximum a posteriori (MAP) decoding in doubly dispersive channels for orthogonal frequency division multiplexing (OFDM) MIMO systems (SIC-MAP-MIMO). SIC-MAP-MIMO leverages on the soft feedback symbol estimate to remove the intercarrier interference and coantenna interference from the received data thus making the subsequent MAP decoding simple. Extrinsic information transfer chart analysis supplemented with numerical simulation results show that SIC-MAP-MIMO achieves comparable BER performance to similar equalization schemes but with significant computational savings.

Comparing implementations of penalized weighted least‐squares sinogram restoration

A CT scanner measures the energy that is deposited in each channel of a detector array by x rays that have been partially absorbed on their way through the object. The measurement process is complex and quantitative measurements are always and inevitably associated with errors, so CT data must be preprocessed prior to reconstruction. In recent years, the authors have formulated CT sinogram preprocessing as a statistical restoration problem in which the goal is to obtain the best estimate of the line integrals needed for reconstruction from the set of noisy, degraded measurements. The authors have explored both penalized Poisson likelihood (PL) and penalized weighted least-squares (PWLS) objective functions. At low doses, the authors found that the PL approach outperforms PWLS in terms of resolution-noise tradeoffs, but at standard doses they perform similarly. The PWLS objective function, being quadratic, is more amenable to computational acceleration than the PL objective. In this work, the authors develop and compare two different methods for implementing PWLS sinogram restoration with the hope of improving computational performance relative to PL in the standard-dose regime. Sinogram restoration is still significant in the standard-dose regime since it can still outperform standard approaches and it allows for correction of effects that are not usually modeled in standard CT preprocessing. The authors have explored and compared two implementation strategies for PWLS sinogram restoration: (1) A direct matrix-inversion strategy based on the closed-form solution to the PWLS optimization problem and (2) an iterative approach based on the conjugate-gradient algorithm. Obtaining optimal performance from each strategy required modifying the naive off-the-shelf implementations of the algorithms to exploit the particular symmetry and sparseness of the sinogram-restoration problem. For the closed-form approach, the authors subdivided the large matrix inversion into smaller coupled problems and exploited sparseness to minimize matrix operations. For the conjugate-gradient approach, the authors exploited sparseness and preconditioned the problem to speed up convergence. All methods produced qualitatively and quantitatively similar images as measured by resolution-variance tradeoffs and difference images. Despite the acceleration strategies, the direct matrix-inversion approach was found to be uncompetitive with iterative approaches, with a computational burden higher by an order of magnitude or more. The iterative conjugate-gradient approach, however, does appear promising, with computation times half that of the authors' previous penalized-likelihood implementation. Iterative conjugate-gradient based PWLS sinogram restoration with careful matrix optimizations has computational advantages over direct matrix PWLS inversion and over penalized-likelihood sinogram restoration and can be considered a good alternative in standard-dose regimes.

NEW APPROACHES TO PHOTOMETRIC REDSHIFT PREDICTION VIA GAUSSIAN PROCESS REGRESSION IN THE SLOAN DIGITAL SKY SURVEY

Expanding upon the work of Way and Srivastava 2006 we demonstrate how the use of training sets of comparable size continue to make Gaussian process regression (GPR) a competitive approach to that of neural networks and other least-squares fitting methods. This is possible via new large size matrix inversion techniques developed for Gaussian processes (GPs) that do not require that the kernel matrix be sparse. This development, combined with a neural-network kernel function appears to give superior results for this problem. Our best fit results for the Sloan Digital Sky Survey (SDSS) Main Galaxy Sample using u,g,r,i,z filters gives an rms error of 0.0201 while our results for the same filters in the luminous red galaxy sample yield 0.0220. We also demonstrate that there appears to be a minimum number of training-set galaxies needed to obtain the optimal fit when using our GPR rank-reduction methods. We find that morphological information included with many photometric surveys appears, for the most part, to make the photometric redshift evaluation slightly worse rather than better. This would indicate that most morphological information simply adds noise from the GP point of view in the data used herein. In addition, we show that cross-match catalog results involving combinations of the Two Micron All Sky Survey, SDSS, and Galaxy Evolution Explorer have to be evaluated in the context of the resulting cross-match magnitude and redshift distribution. Otherwise one may be misled into overly optimistic conclusions.

Modeling of Isomerization of C 8 Aromatics by Online Least Squares Support Vector Machine

The least squares support vector regression (LS-SVR) is usually used for the modeling of single output system, but it is not well suitable for the actual multi-input-multi-output system. The paper aims at the modeling of multi-output systems by LS-SVR. The multi-output LS-SVR is derived in detail. To avoid the inversion of large matrix, the recursive algorithm of the parameters is given, which makes the online algorithm of LS-SVR practical. Since the computing time increases with the number of training samples, the sparseness is studied based on the projection of online LS-SVR. The residual of projection less than a threshold is omitted, so that a lot of samples are kept out of the training set and the sparseness is obtained. The standard LS-SVR, nonsparse online LS-SVR and sparse online LS-SVR with different threshold are used for modeling the isomerization of C 8 aromatics. The root-mean-square-error (RMSE), number of support vectors and running time of three algorithms are compared and the result indicates that the performance of sparse online LS-SVR is more favorable.

Alignment of the ATLAS silicon tracker

ATLAS is one of the four experiments currently under preparation at Large Hadron Collider. Charged particle track reconstruction in the ATLAS Inner Detector is performed both with silicon and drift-tube-based detectors. The alignment of the ATLAS tracking system is one of the challenges that the experiment must overcome in order to achieve its physics goals. This requires the determination of almost 35 000 degrees of freedom. The precision required for the most sensitive coordinate of the silicon devices is of the order of few microns. This precision will be attained with a combination of two techniques: a hardware system with Frequency Scan Interferometers, and track-based alignment. The latter requires the application of complex alignment algorithms that can be both CPU and memory intensive due to the possible requirement of large matrix inversion or many iterations. The alignment algorithms have been already exercised on several challenges such as a Combined Test Beam, cosmic ray runs and large scale computing simulation of physics samples. This note reports on the methods, their computing requirements and preliminary results.

Profile information matrix for nonlinear transformation models

For semiparametric models, interval estimation and hypothesis testing based on the information matrix for the full model is a challenge because of potentially unlimited dimension. Use of the profile information matrix for a small set of parameters of interest is an appealing alternative. Existing approaches for the estimation of the profile information matrix are either subject to the curse of dimensionality, or are ad-hoc and approximate and can be unstable and numerically inefficient. We propose a numerically stable and efficient algorithm that delivers an exact observed profile information matrix for regression coefficients for the class of Nonlinear Transformation Models [A. Tsodikov (2003) J R Statist Soc Ser B 65:759-774]. The algorithm deals with the curse of dimensionality and requires neither large matrix inverses nor explicit expressions for the profile surface.

The COMPLOC Earthquake Location Package

This article describes the programs included in the COMPLOC computer program package that are designed to apply the source-specific station term (SSST) method to solve for local earthquake locations using P- and S- wave phase data. These programs can greatly improve the relative location accuracy of nearby events by applying empirical corrections for the biasing effects of three-dimensional velocity structure. They have been tested on data from both the Southern California Seismic Network (SCSN) and Northern California Seismic Network (NCSN) The SSST method (Richards-Dinger and Shearer 2000; Lin and Shearer 2005) works by assigning each station a travel-time correction that varies as a function of source position. This approach provides relative location accuracy comparable to master event or hypocentroidal decomposition (Jordan and Sverdrup 1981) methods within compact event clusters, but is applicable to distributed seismicity. It has some similarities to the double-difference algorithm (Waldhauser and Ellsworth 2000, 2002) and can be shown to give comparable results in tests on synthetic data (Lin and Shearer 2005). However, the SSST approach has computational advantages for big data sets because the location and station term parts of the computation are separate, so that large matrix inversions are not necessary. In addition, our implementation of SSST provides the option to use L1-norm misfit measures, which are more robust than least squares in the case of occasional timing errors or bad phase picks. We implement the SSST approach by selecting nearby events located within a sphere of specified radius r max around the target event. The station term for the target event is then computed as the median (or mean) residual of these events. Different results will be obtained depending upon the size of the cutoff distance r max. If r max is set to a large enough distance, then the SSST …

Strategies for Fitting Large, Geostatistical Data in MCMC Simulation

Models for geostatistical data introduce spatial dependence in the covariance matrix of location-specific random effects. This is usually defined to be a parametric function of the distances between locations. Bayesian formulations of such models overcome asymptotic inference and estimation problems involved in maximum likelihood-based approaches and can be fitted using Markov chain Monte Carlo (MCMC) simulation. The MCMC implementation, however, requires repeated inversions of the covariance matrix which makes the problem computationally intensive, especially for large number of locations. In the present work, we propose to convert the spatial covariance matrix to a sparse matrix and compare a number of numerical algorithms especially suited within the MCMC framework in order to accelerate large matrix inversion. The algorithms are assessed empirically on simulated datasets of different size and sparsity. We conclude that the band solver applied after ordering the distance matrix reduces the computational time in inverting covariance matrices substantially.

Sparse linear regression in unions of bases via Bayesian variable selection

In this letter, we propose an approach for sparse linear regression in unions of bases inspired by Bayesian variable selection. Conditionally upon an indicator variable that is 0 or 1, one expansion coefficient of the signal corresponding to one atom of the dictionary is either set to zero or given a Student t prior. A Gibbs sampler (a standard Markov chain Monte Carlo technique) is used to sample from the posterior distribution of the indicator variables, the expansion coefficients (corresponding to nonzero indicator variables), the hyperparameters of the Student t priors, and the variance of the residual signal. The structure of the dictionary, assumed to be a union of bases, allows for alternate sampling of the indicator variables and the expansion coefficients from each basis and avoids any large matrix inversion. Our method is applied to the denoising problem of a piano sequence, using a dual-resolution union of two modified discrete cosine transform bases