Large Matrix Size Research Articles

Efficient Photonic-Crystal Mode Solver: Eigenvalue Rather Than Generalized Eigenvalue Approach

We present an iterative scheme to obtain guided Bloch modes in lossy and dispersive photonic crystals. The formulation is based on the concept of sources and transforms the quadratic generalized eigenvalue problem of modes into simpler eigenvalue counterpart for iterations. This feature makes it particularly useful in computations with large matrix sizes. Fewer memories and less computation time than those of conventional methods are required in these cases. The robustness of iterations is rooted in the reciprocity theorem for periodic Bloch parts. Through repeated estimations of propagation constants, Bloch modes of periodic structures converge quickly. Using this method, we take sinusoidal metal gratings as examples and demonstrate advantages of the scheme.

Development of a symmetric echo planar imaging framework for clinical translation of rapid dynamic hyperpolarized 13 C imaging.

To develop symmetric echo planar imaging (EPI) and a reference scan framework for hyperpolarized 13 C metabolic imaging. Symmetric, ramp-sampled EPI with partial Fourier reconstruction was implemented on a 3T scanner. The framework for acquiring a reference scan on the 1 H channel and applied to 13 C data was described and validated in both phantoms and in vivo metabolism of [1-13 C]pyruvate. Ramp-sampled, symmetric EPI provided a substantial increase in the signal-to-noise ratio of the phantom experiments. The reference scan acquired on the 1 H channel yielded 13 C phantom images that varied in mean signal intensity <2%, compared with 13 C images reconstructed with a reference scan directly measured on the 13 C channel. The structural similarity index and dynamic time course from in vivo 13 C data further support the application of a 1 H reference scan to 13 C data to mitigate Nyquist ghost artifacts. Ramp-sampled, symmetric EPI with spectral-spatial excitation of a single metabolite provides a fast, robust, and clinically efficacious approach to acquire hyperpolarized 13 C dynamic molecular imaging data. The gains of this efficient sampling, combined with partial Fourier methods, enables large matrix sizes required for human studies. Magn Reson Med 77:826-832, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

Lipid suppression via double inversion recovery with symmetric frequency sweep for robust 2D-GRAPPA-accelerated MRSI of the brain at 7 T.

This work presents a new approach for high‐resolution MRSI of the brain at 7 T in clinically feasible measurement times. Two major problems of MRSI are the long scan times for large matrix sizes and the possible spectral contamination by the transcranial lipid signal. We propose a combination of free induction decay (FID)‐MRSI with a short acquisition delay and acceleration via in‐plane two‐dimensional generalised autocalibrating partially parallel acquisition (2D‐GRAPPA) with adiabatic double inversion recovery (IR)‐based lipid suppression to allow robust high‐resolution MRSI. We performed Bloch simulations to evaluate the magnetisation pathways of lipids and metabolites, and compared the results with phantom measurements. Acceleration factors in the range 2–25 were tested in a phantom. Five volunteers were scanned to verify the value of our MRSI method in vivo. GRAPPA artefacts that cause fold‐in of transcranial lipids were suppressed via double IR, with a non‐selective symmetric frequency sweep. The use of long, low‐power inversion pulses (100 ms) reduced specific absorption rate requirements. The symmetric frequency sweep over both pulses provided good lipid suppression (>90%), in addition to a reduced loss in metabolite signal‐to‐noise ratio (SNR), compared with conventional IR suppression (52–70%). The metabolic mapping over the whole brain slice was not limited to a rectangular region of interest. 2D‐GRAPPA provided acceleration up to a factor of nine for in vivo FID‐MRSI without a substantial increase in g‐factors (<1.1). A 64 × 64 matrix can be acquired with a common repetition time of ~1.3 s in only 8 min without lipid artefacts caused by acceleration. Overall, we present a fast and robust MRSI method, using combined double IR fat suppression and 2D‐GRAPPA acceleration, which may be used in (pre)clinical studies of the brain at 7 T. © 2015 The Authors. NMR in Biomedicine published by John Wiley & Sons Ltd.

A unified fluctuation formula for one-cut β-ensembles of random matrices

Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut β-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. This allows us to derive a closed-form expression for the limiting covariances of an arbitrary one-cut β-ensemble. As particular cases of the main result we consider the classical β-Gaussian, β-Wishart and β-Jacobi ensembles, for which we derive previously available results as well as new ones within a unified simple framework. We also discuss the connections between the problem of trace fluctuations for the Gaussian unitary ensemble and the enumeration of planar maps.

Commuting quantum matrix models

We study a quantum system of $p$ commuting matrices and find that such a quantum system requires an explicit curvature dependent potential in its Lagrangian for the system to have a finite energy ground state. In contrast it is possible to avoid such curvature dependence in the Hamiltonian. We study the eigenvalue distribution for such systems in the large matrix size limit. A critical r\^ole is played by $p=4$. For $p\ge4$ the competition between eigenvalue repulsion and the attractive potential forces the eigenvalues to form a sharp spherical shell.

Recent Exact and Asymptotic Results for Products of Independent Random Matrices

In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from ensembles of independent real, complex, or quaternionic Ginibre matrices, or truncated unitary matrices. Additional mixing within one ensemble between matrices and their inverses is also covered. Exact determinantal and Pfaffian expressions are given in terms of the respective kernels of orthogonal polynomials or functions. Here we list all known cases and some straightforward generalisations. The asymptotic results for large matrix size include new microscopic universality classes at the origin and a generalisation of weak non-unitarity close to the unit circle. So far in all other parts of the spectrum the known standard universality classes have been identified. In the limit of infinite products the Lyapunov and stability exponents share the same normal distribution. To leading order they both follow a permanental point processes. Our focus is on presenting recent developments in this rapidly evolving area of research.

The Need and Initial Practice of Parallel Imaging and Compressed Sensing in Hyperpolarized 13C MRI in vivo.

The Need and Initial Practice of Parallel Imaging and Compressed Sensing in Hyperpolarized 13C MRI in vivo.

Harmonic/Percussive Sound Separation Based on Anisotropic Smoothness of Spectrograms

This paper describes a method to separate a monaural music signal into harmonic components e.g., a guitar and percussive components, e.g., a snare drum. Separation of these two components is a useful preprocessing for many music information retrieval applications, and in addition, it can be used as a new kind of music equalizer in itself, which enables a music listener to adjust the ratio of the volume of the guitar and the drum freely by themselves. Because of these potential applications, there have been many attempts to develop such a technique, especially in the last decade. However, some of the state-of-the-art techniques have a drawback that they are based on costly operations, such as the multiplications of large-sized matrix, Monte Carlo method, etc., which may constitute barriers to the practical use on some small computers such as smart phones. In this paper, an efficient method that does not depend on these costly operations is described. In formulating the methods, the authors basically assumed only the anisotropic smoothness of music spectrogram, which can be one of the minimalistic model that reflects the natures of these instruments. To be specific, the authors just assumed that harmonic instruments are smooth in time, while the percussive instruments are smooth in frequency on a music spectrogram. In this paper, on the basis of the assumption, source separation methods are formulated as optimization problems that optimize the anisotropic smoothness under some conditions. Because of the simplicity of the model, the derived algorithms are quite simple. Experimental results show that the methods were effective compared to a state-of-the-art technique, and the computation time was much shorter than an existing method; specifically, it can process a three-minute song in around 4-20 seconds on a laptop PC.

A subspace‐based channel calibration algorithm for geosynchronous satellite‐airborne bistatic multi‐channel radars

Ground moving target indication based on geosynchronous (GEO) satellite-airborne (SA) bistatic radar can keep a certain area under continuous surveillance, and thus has important military application values. However, a channel error will affect the clutter suppression performance of the bistatic multi-channel moving target indication radar. And it is found that the channel calibration algorithm used in the long sequence monostatic side-looking synthetic aperture radar, is not applicable to the long sequence translational variant bistatic radar. An improved subspace-based channel calibration algorithm is proposed in this study to deal with channel errors of the GEO SA bistatic multi-channel radar. The algorithm utilises the clutter as calibrated sources, and estimates channel error parameters by minimising the projections of the calibrated sources onto the noise subspace. As the channel errors are estimated from the data, the proposed algorithm does not need any extra calibration sources. Also, the proposed algorithm does not have the limitation of the traditional array self-calibration algorithms, which may have non-unique estimations under the condition of linear arrays. Besides, it avoids finding the inversion of large-size matrix and calculating with any iteration, and hence it is efficient. Moreover, the proposed algorithm can estimate channel errors that vary with azimuth. Simulations are performed to validate the proposed algorithm.

Numerical solution of special matrix games

A method is proposed for solving large-sized matrix games (zero-sum games) of special form for which there is a fast algorithm of searching for the best pure strategy of a player given any mixed strategy of the opponent. Examples of problems leading to such games are given. The method proposed is numerically compared with the Brown-Robinson iterative method.

A fast derivation of Karhunen-Loeve transform kernel for first-order autoregressive discrete process

Karhunen-Loève Transform (KLT), also called principal component analysis (PCA) or factor analysis, based signal processing methods have been successfully used in applications spanning from eigenfiltering to recommending systems. KLT is a signal dependent transform and comprised of three major steps where each has its own computational requirement. Namely, statistical measurement of random data is performed to populate its covariance matrix. Then, eigenvectors (eigenmatrix) and eigenvalues are calculated for the given covariance matrix. Last, incoming random data vector is mapped onto the eigenspace (subspace) by using the calculated eigenmatrix. The recently developed method by Torun and Akansu offers an efficient derivation of the explicit eigenmatrix for the covariance matrix of first-order autoregressive, AR(1), discrete stochastic process. It is the second step of the eigenanalysis implementation as summarized in the paper. Its computational complexity is investigated and compared with the currently used techniques. It is shown that the new method significantly outperforms the others, in particular, for very large matrix sizes that are common in big data applications.

Mitigation of nitrification inhibition by silver nanoparticles using cell entrapment technique

Effects of silver nanoparticles (AgNPs) on nitrification activities of free and entrapped nitrifying activated sludge (NAS) were investigated using a respirometric assay. Initial concentrations of ammonia and AgNPs, and entrapment materials and matrix sizes were the variables. Scanning and transmission electron microscopic analyses of the microbial cells and entrapment matrices were also performed. Results showed that AgNPs (0.05–5.00 mg/L) decreased nitrification activity down to only 2 % compared to the control (free NAS without AgNP exposure). Calcium alginate and polyvinyl alcohol-entrapped cells could mitigate inhibition of nitrification activity up to 100 % (no inhibition observed) and 88 % compared to the controls (entrapped NAS without AgNP exposure), respectively. Cells entrapped in a larger matrix size (both with and without AgNP exposure) performed nitrification better. Silver nanoparticles caused damages to cell membrane and cytoplasm, which likely led to a decrease of nitrification activity. The entrapment matrices successfully reduced the adverse effects of AgNPs on nitrification activity.

Automatic image segmentation using constraint learning and propagation

In this paper, we propose automatic image segmentation using constraint learning and propagation. Recently, kernel learning is receiving much attention because a learned kernel can fit the given data better than a predefined kernel. To effectively learn the constraints generated by initial seeds for image segmentation, we employ kernel propagation (KP) based on kernel learning. The key idea of KP is first to learn a small-sized seed-kernel matrix and then propagate it into a large-sized full-kernel matrix. By applying KP to automatic image segmentation, we design a novel segmentation method to achieve high performance. First, we generate pairwise constraints, i.e., must-link and cannot-link, from initially selected seeds to make the seed-kernel matrix. To select the optimal initial seeds, we utilize global k-means clustering (GKM) and self-tuning spectral clustering (SSC). Next, we propagate the seed-kernel matrix into the full-kernel matrix of the entire image, and thus image segmentation results are obtained. We test our method on the Berkeley segmentation database, and the experimental results demonstrate that the proposed method is very effective in automatic image segmentation.

Performance modeling and analysis of parallel Gaussian elimination on multi-core computers

Gaussian elimination is used in many applications and in particular in the solution of systems of linear equations. This paper presents mathematical performance models and analysis of four parallel Gaussian Elimination methods (precisely the Original method and the new Meet in the Middle –MiM– algorithms and their variants with SIMD vectorization) on multi-core systems. Analytical performance models of the four methods are formulated and presented followed by evaluations of these models with modern multi-core systems’ operation latencies. Our results reveal that the four methods generally exhibit good performance scaling with increasing matrix size and number of cores. SIMD vectorization only makes a large difference in performance for low number of cores. For a large matrix size (n⩾16K), the performance difference between the MiM and Original methods falls from 16× with four cores to 4× with 16K cores. The efficiencies of all four methods are low with 1K cores or more stressing a major problem of multi-core systems where the network-on-chip and memory latencies are too high in relation to basic arithmetic operations. Thus Gaussian Elimination can greatly benefit from the resources of multi-core systems, but higher performance gains can be achieved if multi-core systems can be designed with lower memory operation, synchronization, and interconnect communication latencies, requirements of utmost importance and challenge in the exascale computing age.

Distribution of the Ratio of Consecutive Level Spacings in Random Matrix Ensembles

We derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices. This ratio distribution was recently introduced to study spectral properties of many-body problems, as, contrary to the standard level spacing distributions, it does not depend on the local density of states. Our Wigner-like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit. Quantitative improvements are found through a polynomial expansion. Examples from a quantum many-body lattice model and from zeros of the Riemann zeta function are presented.

Preparation of TiC Hard Metal by Presssureless Infiltration Method

The steel matrix composites pressureless infiltration technology was studied in this paper. It is necessary to develop a reasonable process, select the appropriate gel system and additives, and prepare the complex shape of the large-size steel matrix composites by using gel-casting. For reactive melt infiltration technique, the porous skeleton was prepared by gelcasting preparation of the titanium powder and ceramic particles mixed, and the C-containing liquid metal infiltration of the porous skeleton. To improve the wettability between the molten metal and the ceramic phase, it is used for the process Ti-C in situ reaction to accelerate the impregnation process. Finally, the steel matrix composites with higher density were prepared.

Moments of a Single Entry of Circular Orthogonal Ensembles and Weingarten Calculus

Consider a symmetric unitary random matrix V = (v ij )1 ≤ i, j ≤ N from a circular orthogonal ensemble. In this paper, we study moments of a single entry v ij . For a diagonal entry v ii , we give the explicit values of the moments, and for an off-diagonal entry v ij , we give leading and subleading terms in the asymptotic expansion with respect to a large matrix size N. Our technique is to apply the Weingarten calculus for a Haar-distributed unitary matrix.

32-Bit NxN Matrix Multiplication: Performance Evaluation for Altera FPGA, i5 Clarkdale and Atom Pineview-D Intel General Purpose Processors

Nowadays mobile devices represent a significant portion of the market for embedded systems, and are continuously demanded in daily life. From the end-user perspective size, weight, features are the key quality criteria. These benchmarks criteria became the usual design constraints in the embedded systems design process and put a high impact on the power consumption. This paper survey and explore different low power design techniques for FPGA and processors. We compare, evaluate, and analyze, the power and energy consumption in three different designs namely, Altera FPGA Cyclone II which has a systolic array matrix multiplication implemented, i5 Clarkdale, and Atom Pineview-D Intel general purpose processors, which multiply two nxn 32-bit matrices and produce a 64-bit matrix as an output. We concluded that FPGA is a more power and energy efficient on low matrix size. However, general purpose processor performance is close to FPGA on larger matrix size as the larger cache size in general purpose processor help in reducing latency. We also concluded that the performance of FPGA can be improved in terms of latency if more systolic array processing elements are implemented in parallel to allow more concurrency.

A Superfast Structured Solver for Toeplitz Linear Systems via Randomized Sampling

We propose a superfast solver for Toeplitz linear systems based on rank structured matrix methods and randomized sampling. The solver uses displacement equations to transform a Toeplitz matrix $T$ into a Cauchy-like matrix $\mathcal{C}$, which is known to have low-numerical-rank off-diagonal blocks. Thus, we design a fast scheme for constructing a hierarchically semiseparable (HSS) matrix approximation to $\mathcal{C}$, where the HSS generators have internal structures. Unlike classical HSS methods, our solver employs randomized sampling techniques together with fast Toeplitz matrix-vector multiplications, and thus converts the direct compression of the off-diagonal blocks of $\mathcal{C}$ into the compression of much smaller blocks. A strong rank-revealing QR factorization method is used to generate/preserve certain special structures, and also to ensure stability. A fast ULV HSS factorization scheme is provided to take advantage of the special structures. We also propose a precomputation procedure for the HSS construction so as to further improve the efficiency. The complexity of these methods is significantly lower than some similar Toeplitz solvers for large matrix size $n$. Detailed flop counts are given, with the aid of a rank relaxation technique. The total cost of our methods includes $O(n)$ flops for HSS operations and $O(n\log^{2} n)$ flops for matrix multiplications via FFTs, where $n$ is the order of $T$. Various numerical tests on classical examples, including ill-conditioned ones, demonstrate the efficiency, and also indicate that the methods are stable in practice. This work shows a practical way of using randomized sampling in the development of fast rank structured methods.

The Modal Substructuring Method: An Efficient Technique for Large-Size Numerical Simulations

This article presents an extension of the branch Eigenmodes reduction method (BERM) to substructuring models. One of the method's interests lies in its capacity to build a reduced modal model for complex geometries which are characterized by very large matrix sizes. This article treats a 3-D nonlinear application. Such a substructuring reduced model permits us to obtain precise results over the whole domain, with an important gain of CPU time. In the case considered, a global reduced branch model can also be built. A comparison between the two reduced models shows the interest of the substructuring technique.