Computing symmetric rank for symmetric tensors

We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for 2×?×2 tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetric rank strata for some secant varieties of Veronese varieties.     

Pathline glyphs

Visualization of pathlines is common and highly relevant for the analysis of unsteady flow. However, pathlines can intersect, leading to visual clutter and perceptual issues. This makes it intrinsically difficult to provide expressive visualizations of the entire domain by an arrangement of multiple pathlines, in contrast to well‐established streamline placement techniques. We present an approach to reduce these problems. It is inspired by glyph‐based visualization and small multiples: we partition the domain into cells, each corresponding to a downscaled version of the entire domain. Inside these cells, a single downscaled pathline is drawn. On the overview scale, our pathline glyphs lead to emergent visual patterns that provide insight into time‐dependent flow behavior. Zooming‐in allows us to analyze individual pathlines in detail and compare neighboring lines. The overall approach is complemented with a context‐preserving zoom lens and interactive pathline‐based exploration. While we primarily target the visualization of 2D flow, we also address the extension to 3D. Our evaluation includes several examples, comparison to other flow visualization techniques, and a user study with domain experts.     

Symmetric rank-1 approximation of symmetric high-order tensors

ABSTRACT

Finding the symmetric rank-1 approximation to a given symmetric tensor is an important problem due to its wide applications and its close relationship to the Z-eigenpair of a tensor. In this paper, we propose a method based on the proximal alternating linearized minimization to directly solve the optimization problem. Global convergence of our algorithm is established. Numerical experiments show that our algorithm is very competitive in speed, accuracy and robustness compared to other state-of-the-art methods.     

Superellipsoid-based, real symmetric traceless tensor glyphs motivated by nematic liquid crystal alignment visualization

A glyph-based method for visualizing the nematic liquid crystal alignment tensor is introduced. Unlike previous approaches, the glyph is based upon physically-linked metrics, not offsets of the eigenvalues. These metrics, combined with a set of superellipsoid shapes, communicate both the strength of the crystal's uniaxial alignment and the amount of biaxiality. With small modifications, our approach can visualize any real symmetric traceless tensor.     

Information rich glyphs for software management data

Three high-dimensional glyphs for viewing software project management data assist in perhaps the most challenging engineering task in modern times managing large software projects. In data exploration, glyphs refer to graphical objects or symbols that represent data through visual parameters that are either spatial (positions x or y), retinal (color and size), or temporal. Common examples of graphical objects include the bars in a bar chart or the points within a scatter plot. We focus on glyphs for visualizing software project management data. Any large-scale project will have many different classes of resources (lab equipment, staff time, machine cycles, disk resources, interim deliverables, and customer commitments) that must be scheduled and tracked. Inevitably, problems will arise and solutions must be found. To support the management process, information systems collect and maintain large status databases. We aim to support and improve the understanding of this information through visualization. Our glyphs are designed to expose patterns among sets of software artifacts and to help identify differences between items     

A finite element to interpolate symmetric tensors with divergence inL 2

Inner approximations of the space of second order symmetric tensors with square-integrable «divergence» over a bounded domain inR ² are built up by means of a family of affine elements, analogous to the family defined by Raviart and Thomas in the case of vectors. As an application, we solve the Dirichlet problem for the biharmonic operator by a method of «equilibrium» type.     

Faster Calculation of Superquadric Shapes

Nonparametric methods of calculating points on the curve produce the recently introduced superquadric objects at great savings in time.     

New optimized symmetric and symplectic trigonometrically fitted RKN methods for second-order oscillatory differential equations

Symmetric, symplectic and trigonometrically fitted Runge–Kutta–Nyström (SSTFRKN) methods for second-order differential equations with oscillatory solutions are investigated. Symmetry, symplecticity and trigonometric fitting conditions for modified Runge–Kutta–Nyström (RKN) methods are presented. Order conditions for modified RKN methods are derived via the special Nyström tree theory. Two explicit SSTFRKN methods with variable nodes are derived. The two new methods are zero-dissipative due to symplecticity. Their dispersion orders are analysed and their periodicity regions are obtained. The results of numerical experiments show the robustness and competence of the new SSTFRKN methods compared with some highly efficient codes in the recent literature.     

Time-aware tensor decomposition for sparse tensors

Machine Learning - Given a sparse time-evolving tensor, how can we effectively factorize it to accurately discover latent patterns? Tensor decomposition has been extensively utilized for analyzing...     

用粒子群优化算法重构超二次曲面三维模型

针对在三维重构过程中用L—M（Levenberg—Marquardt）方法求解超二次曲面参数拟合问题的不足，提出了用粒子群优化算法来进行超二次曲面参数拟合的新方法．本文详细阐述了超二次曲面的三维表示特性，L—M算法拟合超二次曲面参数模型的分析，以及用粒子群优化算法拟合超二次曲面参数模型的原理、实现方法和实验结果．用粒子群优化算法对超二次曲面进行参数拟合，克服了L—M方法的缺陷，取了满意的效果．     

用协同演化并行PSO重构扩展的超二次曲面模型

针对用传统方法难以求解的扩展的超二次曲面三维模型参数拟合问题,提出了用协同演化的并行粒子群优化算法求解的新方法。通过对扩展的超二次曲面三维表示特性的研究,设计和实现了基于岛屿群体模型的并行粒子群优化算法,并用协同演化的思想,将约束非线性优化转化为极小极大问题进行求解。实验结果表明用协同演化的并行粒子群优化算法重构扩展的超二次曲面三维模型,扩大了模型表示能力,建模精确且效率高。     

Fisher tensors for classifying human epithelial cells

Analyzing and classifying Human Epithelial type 2 (HEp-2) cells using Indirect Immunofluorescence protocol has been the golden standard for detecting connective tissue diseases such as Rheumatoid Arthritis. However, this suffers from numerous shortcomings such as being subjective as well as time and labor intensive. Recently, several studies explore the advantages of artificial systems to automate the process, not only to reduce the test turn-around time but also to deliver more consistent results. In this paper, we extend the conventional bag of word models from Euclidean space to non-Euclidean Riemannian manifolds and utilize them to classify the HEp-2 cells. The main motivation comes from the observation that HEp-2 cells can be efficiently described by symmetric positive definite matrices which lie on a Riemannian manifold. With this motivation, we first discuss an intrinsic bag of Riemannian words model. We then propose Fisher tensors which can in turn encode additional information about the distribution of the signatures in a bag of word model. Experiments on two challenging HEp-2 images datasets, namely ICPRContest and SNPHEp-2 show that the proposed methods obtain notable improvements in discrimination accuracy, in comparison to baseline and several state-of-the-art methods. The proposed framework, while hand-crafted towards cell classification, is a generic framework for object recognition. This is supported by assessing the performance of our proposal on a challenging texture classification task.     

Model order reduction of second-order systems with nonlinear stiffness using Krylov subspace methods and their symmetric transfer functions

In this investigation, Model Order Reduction (MOR) of second-order systems having cubic nonlinearity in stiffness is developed for the first time using Krylov subspace methods and the associated symmetric transfer functions. In doing so, new second-order Krylov subspaces will be defined for MOR procedure which avoids the need to transform the second-order system to its state space form and thus the main characteristics of the second-order system such as symmetry and positive definiteness of mass and stiffness matrices will be preserved. To show the efficacy of the presented method, three examples will be considered as practical case studies. The first example is a nonlinear shear-beam building model subjected to a seismic disturbance. The second and third examples are nonlinear longitudinal vibration of a rod and vibration of a cantilever beam resting on a nonlinear elastic foundation, respectively. Simulation results in all cases show good accuracy of the vibrational response of the reduced order models when compared with the original ones while reducing the computational load.     

Multi-point algorithms for second-order coupled equations

A general method has been derived to obtain all the possible algorithms for second-order equations involving the function at two points and the values of the second derivatives at one or more points. Both predictor and corrector algorithms may be obtained, together with their truncation errors, and some examples are given. These algorithms have been tested for sinusoidal and exponential solutions, and compared with the Baylis-Peel algorithm. A predictor-corrector algorithm of outstanding accuracy has been derived, and the source of its small errors for sinusoidal solutions is discussed. Some suggestions are made for the improvement of existing coupled channels codes, using these algorithms.     

Orthogonal canonical forms for second-order systems

It is shown that a linear second-order system with arbitrary damping cannot be reduced to Hessenberg-triangular form by means of orthogonal transformations. However, it is also shown that such an orthogonal reduction is always possible for the modal damping commonly assumed for models of flexible structures. It is shown that modally damped models can be orthogonally reduced to a new triangular second-order Schur form