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The paper introduces a multiway tensor generalization of the bigraphical lasso which uses a two-way sparse Kronecker sum multivariate normal model for the precision matrix to model parsimoniously conditional dependence relationships of matrix variate data based on the Cartesian product of graphs. We call this tensor graphical lasso generalization TeraLasso. We demonstrate by using theory and examples that the TeraLasso model can be accurately and scalably estimated from very limited data samples of high dimensional variables with multiway co-ordinates such as space, time and replicates. Statistical consistency and statistical rates of convergence are established for both the bigraphical lasso and TeraLasso estimators of the precision matrix and estimators of its support (non-sparsity) set respectively. We propose a scalable composite gradient descent algorithm and analyse the computational convergence rate, showing that the composite gradient descent algorithm is guaranteed to converge at a geometric rate to the global minimizer of the TeraLasso objective function. Finally, we illustrate TeraLasso by using both simulation and experimental data from a meteorological data set, showing that we can accurately estimate precision matrices and recover meaningful conditional dependence graphs from high dimensional complex data sets.

In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension d by means of tensor product formulas. This is useful, for example, in the context of multivariate interpolation, multidimensional function approximation using pseudospectral expansions and solution of stiff differential equations on tensor product domains. The key point to obtain an efficient-to-implement BLAS formulation consists in the suitable usage of the μ -mode product (also known as tensor-matrix product or mode- n product) and related operations, such as the Tucker operator. Their MathWorks MATLAB ® /GNU Octave implementations are discussed in the paper, and collected in the package KronPACK. We present numerical results on experiments up to dimension six from different fields of numerical analysis, which show the effectiveness of the approach.

We present a method for computing actions of the exponential-like φ -functions for a Kronecker sum K of d arbitrary matrices A μ . It is based on the approximation of the integral representation of the φ -functions by Gaussian quadrature formulas combined with a scaling and squaring technique. The resulting algorithm, which we call phiks , evaluates the required actions by means of μ -mode products involving exponentials of the small sized matrices  A μ , without forming the large sized matrix  K itself. phiks , which profits from the highly efficient level 3 BLAS, is designed to compute different φ -functions applied on the same vector or a linear combination of actions of φ -functions applied on different vectors. In addition, thanks to the underlying scaling and squaring techniques, the desired quantities are available simultaneously at suitable time scales. All these features allow the effective usage of phiks in the exponential integration context. In fact, our newly designed method has been tested in popular exponential Runge–Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of φ -functions. The numerical experiments with discretized semilinear evolutionary 2D or 3D advection–diffusion–reaction, Allen–Cahn, and Brusselator equations show the superiority of the proposed μ -mode approach.

A new procedure is presented for determining the kernel and the offspring hypersurfaces for general linear time invariant (LTI) dynamics with multiple delays. These hypersurfaces, as they have very recently been introduced in a concept paper [R. Sipahi and N. Olgac, Automatica, 41 (2005), pp. 1413-1422], form the basis of the overriding paradigm which is called the cluster treatment of characteristic roots (CTCR). In fact, these two sets of hypersurfaces exhaustively represent the locations in the domain of the delays where the system possesses at least one pair of imaginary characteristic roots. To determine the kernel and offspring we use the extraordinary features of the \"extended Kronecker summation\" operation in this paper. The end result is that the infinite-dimensional problem reduces to a finite-dimensional one (and preferably into an eigenvalue problem). Following the procedure described in this paper, we are able to shorten the computational time considerably in determining these hypersurfaces. We demonstrate these concepts via some example case studies. One of the examples treats a 3-delay system. For this case another interesting perspective, called the \"building block,\" is also utilized to display the kernel in three-dimensional space in the domain of \"spectral delays.\"

A new computational strategy produces independent samples from the joint posterior distribution for a broad class of Bayesian spatial and spatiotemporal conditional autoregressive models. The method is based on reparameterization and marginalization of the posterior distribution and massive parallelization of rejection sampling using graphical processing units (GPUs) or other accelerators. It enables very fast sampling for small to moderate-sized datasets (up to approximately 10,000 observations) and feasible sampling for much larger datasets. Even using a mid-range GPU and a high-end CPU, the GPU-based implementation is up to 30 times faster than the same algorithm run serially on a single CPU, and the numbers of effective samples per second are orders of magnitude higher than those obtained with popular Markov chain Monte Carlo software. The method has been implemented in the R package CARrampsOcl. This work provides both a practical computing strategy for fitting a popular class of Bayesian models and a proof of concept that GPU acceleration can make independent sampling from Bayesian joint posterior densities feasible.

Among the objectives of the upcoming space missions Tandem-L and BIOMASS, is the 3-D representation of the global forest structure via synthetic aperture radar (SAR) tomography (TomoSAR). To achieve such a goal, modern approaches suggest solving the TomoSAR inverse problems by exploiting polarimetric diversity and structural model properties of the different scattering mechanisms. This way, the related tomographic imaging problems are treated in descriptive regularization settings, applying modern non-parametric spatial spectral analysis (SSA) techniques. Nonetheless, the achievable resolution of the commonly performed SSA-based estimators highly depends on the span of the tomographic aperture; furthermore, irregular sampling and non-uniform constellations sacrifice the attainable resolution, introduce artifacts and increase ambiguity. Overcoming these drawbacks, in this paper, we address a new multi-stage iterative technique for feature-enhanced TomoSAR imaging that aggregates the virtual adaptive beamforming (VAB)-based SSA approach, with the wavelet domain thresholding (WDT) regularization framework, which we refer to as WAVAB (WDT-refined VAB). First, high resolution imagery is recovered applying the descriptive experiment design regularization (DEDR)-inspired reconstructive processing. Next, the additional resolution enhancement with suppression of artifacts is performed, via the WDT-based sparsity promoting refinement in the wavelet transform (WT) domain. Additionally, incorporation of the sum of Kronecker products (SKP) decomposition technique at the pre-processing stage, improves ground and canopy separation and allows for the utilization of different better adapted TomoSAR imaging techniques, on the ground and canopy structural components, separately. The feature enhancing capabilities of the novel robust WAVAB TomoSAR imaging technique are corroborated through the processing of airborne data of the German Aerospace Center (DLR), providing detailed volume height profiles reconstruction, as an alternative to the competing non-parametric SSA-based methods.

Supersaturated designs (SSDs) offer a potentially useful way to investigate many factors with only a few experiments during the preliminary stages of experimentation. While the construction and analysis of symmetrical SSDs have been widely explored, asymmetrical (or mixed-level) SSDs deserve further investigation. Mixed-level SSDs can be judged by various criteria. But, justified by existing results, the χ2 criterion proposed by Yamada and Lin (1999) is adopted here. Optimality results for mixed-level SSDs are provided. A new construction method for χ2-optimal SSDs is proposed, and we discuss properties of the resulting designs. Many new designs are tabulated for practical use.

In the context of processing global navigation satellite system (GNSS) data by least squares adjustment, one may encounter a mathematical problem when inverting a sum of two Kronecker products. As a solution of this problem, we propose to invert this sum in the form of another sum of two Kronecker products. We present and demonstrate two mathematical formulas that enable us to achieve this task. We conclude from the demonstration that there is one condition for each formula to be checked before applying this proposed matrix inversion technique. In fact, these conditions restrict greatly the application of the formulas from being more general to the inversion problems of this kind. However, when applicable, the formulas obviously save computations in general and are very useful for large and fully populated matrices. In addition, this proposed matrix inversion technique shows several benefits when used in the processing of a single baseline with multi-frequency GNSS signals. These benefits are summarized in the following. First, the fully populated variance–covariance matrix of observations is easily inverted. Second, the computation of the normal matrix becomes easier as well since the blocks in both the design and weight matrix are all written in the form of Kronecker products. Third, this proposed matrix inversion technique contributes greatly to the computation of the variance–covariance matrix of estimates.

We consider the waiting time (delay) W in a FCFS c-server queue with arrivals which are either renewal or governed by Neuts' Markovian arrival process, and (possibly heterogeneous) service time distributions of general phase-type Fi, with mi phases for the ith server. The distribution of W is then again phase-type, with m1[sm middot][sm middot][sm middot]mc phases for the general heterogeneous renewal case and ${\\pmatrix{{m+c-1}\\\{c}}}$ phases for the homogeneous case Fi=F, mi=m. We derive the phase-type representation in a form which is explicit up to the solution of a matrix fixed point problem; the key new ingredient is a careful study of the not-all-busy period where some or all servers are idle. Numerical examples are presented as well. [PUBLICATION ABSTRACT]