Tensor Basis Research Articles

Efficient treatment of the nonlinear features in algebraic Reynolds-stress and heat-flux models for stratified and convective flows

This work discusses a new and efficient method for treating the nonlinearity of algebraic turbulence models in the case of stratified and convective flows, for which the equations for the Reynolds stresses and turbulent heat flux are strongly coupled. In such cases, one finds a quasi-linear set of equations, which can be solved through an appropriate linear expansion in basis tensors and vectors, as discussed in earlier work. However, finding a consistent and truly explicit algebraic turbulence model requires solving an additional equation for the production-to-dissipation ratio (P+G)/ε of turbulent kinetic energy. Due to the nonlinear nature of the problem, the equation for (P+G)/ε is a higher-order polynomial equation for which no analytical solution can be found. Here we provide a new method to approximate the solution of this polynomial equation through an analysis of two special limits (shear-dominated and buoyancy-dominated), in which exact solutions are obtainable. The final result is a model that appropriately combines the two limits in more general cases. The method is tested for turbulent channel flow, both with stable and unstable stratification, and the atmospheric boundary layer with periodic and rapid changes between stable and unstable stratification. In all cases, the model is shown to give consistent results, close to the exact solution of (P+G)/ε. This new method greatly increases the range of applicability of explicit algebraic models, which otherwise would rely on the numerical solution of the polynomial equation.

Simultaneous Invariants of Strain and Rotation Rate Tensors and Their Admitted Region

The purpose of this paper is to establish the admitted region for five simultaneous, functionally independent invariants of the strain rate tensorSand rotation rate tensorΩand calculate some simultaneous invariants of these tensors which are encountered in the theory of constitutive relations for turbulent flows. Such a problem, as far as we know, has not yet been considered, though it is obviously an integral part of any problem in which scalar functions of the tensorsSandΩare studied. The theory provided inside this paper is the building block for a derivation of new algebraic constitutive relations for three-dimensional turbulent flows in the form of expansions of the Reynolds-stress tensor in a tensorial basis formed by the tensorsSandΩ, in which the scalar coefficients depend on simultaneous invariants of these tensors.

Two-Sided Projection Methods for Nonlinear Model Order Reduction

In this paper, we investigate a recently introduced approach for nonlinear model order reduction based on generalized moment matching. Using basic tensor calculus, we propose a computationally efficient way of computing reduced-order models. We further extend the idea of two-sided interpolation methods to this more general setting by employing the tensor structure of the Hessian. We investigate the use of oblique projections in order to preserve important system properties such as stability. We test one-sided and two-sided projection methods for different semi-discretized nonlinear partial differential equations and show their competitiveness when compared to proper orthogonal decomposition (POD).

Liquid film condensation along a vertical surface in a thin porous medium with large anisotropic permeability.

The problems of steady film condensation on a vertical surface embedded in a thin porous medium with anisotropic permeability filled with pure saturated vapour are studied analytically by using the Brinkman-Darcy flow model. The principal axes of anisotropic permeability are oriented in a direction that non-coincident with the gravity force. On the basis of the flow permeability tensor due to the anisotropic properties and the Brinkman-Darcy flow model adopted by considering negligible macroscopic and microscopic inertial terms, boundary-layer approximations in the porous liquid film momentum equation is solved analytically. Scale analysis is applied to predict the order-of-magnitudes involved in the boundary layer regime. The first novel contribution in the mathematics consists in the use of the anisotropic permeability tensor inside the expression of the mathematical formulation of the film condensation problem along a vertical surface embedded in a porous medium. The present analytical study reveals that the anisotropic permeability properties have a strong influence on the liquid film thickness, condensate mass flow rate and surface heat transfer rate. The comparison between thin and thick porous media is also presented.

On the representation and implicit integration of general isotropic elastoplasticity based on a set of mutually orthogonal unit basis tensors

SUMMARYA simple and compact representation framework and the corresponding efficient numerical integration algorithm are developed for constitutive equations of isotropic elastoplasticity. Central to this work is the utilization of a set of mutually orthogonal unit tensor bases and the corresponding invariants. The set of bases can be regarded equivalently as a local cylindrical coordinate system in the three‐dimensional coaxial tensor subspace, namely, the principal space. The base tensors are given in the global coordinate system. Similar to the principal space approach, the proposed method reduces the problem dimension from six to three. In contrast to the conventional approach, the transformation procedure between the principal space and the general space is avoided and explicit computation of the principal axes is bypassed. With the proposed technique, the matrices, which need to be inverted during iteration, take a simple form for the great majority of constitutive equations in use. The tangent operator consistent with the proposed algorithm can be decomposed into the direct sum of two linear maps over the coaxial tensor subspace and the subspace orthogonal to it. Consequently, its closed form is derived in an extremely simple manner. Finally, numerical examples demonstrate the high quality performances of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.

Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains

The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to “lift” this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sub-linearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging -discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the “basis” of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of magnitude storage savings over direct approaches.

Калибровочно-инвариантные тензоры 4-многообразия конформной связности без кручения

We calculated basic gauge-invariant tensors algebraically expressed through the matrix of conformal curvature. In particular, decomposition of the main tensor into gaugeinvariant irreducible summands consists of 4 terms, one of which is determined by only one scalar. First, this scalar enters the Einstein’s equations with cosmological term as a cosmological scalar. Second, metric being multiplied by this scalar becomes gauge invariant. Third, the geometric point, which is not gauge-invariant, after multiplying by the square root of this scalar becomes gauge-invariant object - a material point. Fourth, the equations of motion of the material point are exactly the same as in the general relativity, which allows us to identify the square root of this scalar with mass. Thus, we obtained an unexpected result: the cosmological scalar coincides with the square of the mass. Fifth, the cosmological scalar allows us to introduce the gauge-invariant 4measure on the manifold. Using this measure, we introduce a new variational principle for the Einstein equations with cosmological term. The matrix of conformal curvature except the components of the main tensor contains other components. We found all basic gauge-invariant tensors, expressed in terms of these components. They are 1- or 3-valent. Einstein’s equations are equivalent to the gauge invariance of one of these covectors. Therefore the conformal connection manifold, where Einstein’s equations are satisfied, can be divided into 4 types according to the type of this covector: timelike, spacelike, light-like or zero.

New Cartan’s tensors and pseudotensors in a generalized Finsler space

In this work we defined a generalized Finsler space (GFN) as 2N-dimensional differentiable manifold with a non-symmetric basic tensor gij(x,x?), which applies that gij_?|m(x,x?)=0; ?=1,2. Based on non-symmetry of basic tensor, we obtained ten Ricci type identities, comparing to two kinds of covariant derivative of a tensor in Rund?s sense. There appear two new curvature tensors and fifteen magnitudes, we called ?curvature pseudotensors?.

A Bayesian Approach to Distinguishing Interdigitated Muscles in the Tongue from Limited Diffusion Weighted Imaging.

Fiber tracking in crossing regions is a well known issue in diffusion tensor imaging (DTI). Multi-tensor models have been proposed to cope with the issue. However, in cases where only a limited number of gradient directions can be acquired, for example in the tongue, the multi-tensor models fail to resolve the crossing correctly due to insufficient information. In this work, we address this challenge by using a fixed tensor basis and incorporating prior directional knowledge. Within a maximum a posteriori (MAP) framework, sparsity of the basis and prior directional knowledge are incorporated in the prior distribution, and data fidelity is encoded in the likelihood term. An objective function can then be obtained and solved using a noise-aware weighted ℓ1-norm minimization. Experiments on a digital phantom and in vivo tongue diffusion data demonstrate that the proposed method is able to resolve crossing fibers with limited gradient directions.

Properties of point correspondences between three multidimensional surfaces of projective spaces

For a correspondence in question, we establish a sequence of fundamental geometric objects, construct invariant normalizations of the surfaces, and single out basic tensors. We also establish a connection between the geometry of correspondences in question and the theory of multidimensional three-webs.

Tensor Body: Real-Time Reconstruction of the Human Body and Avatar Synthesis From RGB-D

Real-time 3-D reconstruction of the human body has many applications in anthropometry, telecommunications, gaming, fashion, and other areas of human-computer interaction. In this paper, a novel framework is presented for reconstructing the 3-D model of the human body from a sequence of RGB-D frames. The reconstruction is performed in real time while the human subject moves arbitrarily in front of the camera. The method employs a novel parameterization of cylindrical-type objects using Cartesian tensor and b-spline bases along the radial and longitudinal dimension respectively. The proposed model, dubbed tensor body, is fitted to the input data using a multistep framework that involves segmentation of the different body regions, robust filtering of the data via a dynamic histogram, and energy-based optimization with positive-definite constraints. A Riemannian metric on the space of positive-definite tensor splines is analytically defined and employed in this framework. The efficacy of the presented methods is demonstrated in several real-data experiments using the Microsoft Kinect sensor.

ChemInform Abstract: Tensors and Rotations in NMR

AbstractReview: 26 refs.

Piecewise tensor product wavelet bases by extensions and approximation rates

DIn this chapter, we present some of the major results that have been achieved in the context of the DFG-SPP project Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces and Applications to Nonlinear Inverse Problems. This project has been concerned with (nonlinear) elliptic and parabolic operator equations on nontrivial domains as well as with related inverse parameter identification problems. One crucial step has been the design of an efficient forward solver. We employed a spatially adaptive wavelet Rothe scheme. The resulting elliptic subproblems have been solved by adaptive wavelet Galerkin schemes based on generalized tensor wavelets that realize dimension-independent approximation rates. In this chapter, we present the construction of these new tensor bases and discuss some numerical experiments.

A polynomial approach for extracting the extrema of a spherical function and its application in diffusion MRI

Antipodally symmetric spherical functions play a pivotal role in diffusion MRI in representing sub-voxel-resolution microstructural information of the underlying tissue. This information is described by the geometry of the spherical function. In this paper we propose a method to automatically compute all the extrema of a spherical function. We then classify the extrema as maxima, minima and saddle-points to identify the maxima. We take advantage of the fact that a spherical function can be described equivalently in the spherical harmonic (SH) basis, in the symmetric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. We extract the extrema of the spherical function by computing the stationary points of its constrained HP representation. Instead of using traditional optimization approaches, which are inherently local and require exhaustive search or re-initializations to locate multiple extrema, we use a novel polynomial system solver which analytically brackets all the extrema and refines them numerically, thus missing none and achieving high precision. To illustrate our approach we consider the Orientation Distribution Function (ODF). In diffusion MRI the ODF is a spherical function which represents a state-of-the-art reconstruction algorithm whose maxima are aligned with the dominant fiber bundles. It is, therefore, vital to correctly compute these maxima to detect the fiber bundle directions. To demonstrate the potential of the proposed polynomial approach we compute the extrema of the ODF to extract all its maxima. This polynomial approach is, however, not dependent on the ODF and the framework presented in this paper can be applied to any spherical function described in either the SH basis, ST basis or the HP basis.

Covariant vacuum polarizations on de Sitter background

We derive covariant expressions for the one-loop vacuum polarization induced by a charged scalar on de Sitter background. Two forms are employed: one in which two covariant derivatives act on de Sitter-invariant basis tensors multiplied by scalar structure functions, and the other in which four covariant derivatives act on a single basis tensor times a structure function. The second representation permits the correction to dynamical photons to be expressed as a surface integral, which raises the important question of what sorts of effects can be absorbed into corrections of the initial state. Results are obtained for charged, minimally coupled scalars which are either massless or else light. The former show de Sitter breaking whereas the latter are de Sitter invariant. However, the de Sitter-invariant formulation does not seem useful, even when it is possible. Our work has two important physical consequences: (1) an upcoming computation of the vacuum polarization from inflationary gravitons should be expressed using the old, noncovariant formalism, and (2) one should re-examine the conclusion of a previous study of the effect of inflationary scalars on gravitons.

Dimensionality reduction and topographic mapping of binary tensors

In this paper, a decomposition method for binary tensors, generalized multi-linear model for principal component analysis (GMLPCA) is proposed. To the best of our knowledge at present there is no other principled systematic framework for decomposition or topographic mapping of binary tensors. In the model formulation, we constrain the natural parameters of the Bernoulli distributions for each tensor element to lie in a sub-space spanned by a reduced set of basis (principal) tensors. We evaluate and compare the proposed GMLPCA technique with existing real-valued tensor decomposition methods in two scenarios: (1) in a series of controlled experiments involving synthetic data; (2) on a real-world biological dataset of DNA sub-sequences from different functional regions, with sequences represented by binary tensors. The experiments suggest that the GMLPCA model is better suited for modelling binary tensors than its real-valued counterparts. Furthermore, we extended our GMLPCA model to the semi-supervised setting by forcing the model to search for a natural parameter subspace that represents a user-specified compromise between the modelling quality and the degree of class separation.

Geometric interpretation of curvature and torsion tensors in a generalized Finsler space

We give geometric interpretations of a torsion tensor and curvature tensors in a generalized Finsler space (with an asymmetric basic tensor).

On projective-differential properties of point correspondences between three hypersurfaces

Point correspondences between three hypersurfaces of projective spaces are studied on the base of G. F. Laptev’s invariant methods [1] (Proc. of Moscow Math. Soc., 2, 275–382, 1953). We establish the basic equations and geometrical objects of the point correspondences in question. We construct invariant normalizations of the hypersurfaces, single out the basic tensors of the correspondences, and establish a connection between the geometry of correspondences and the theory of multidimensional 3-webs.

On spaces with non-symmetric affine connection, containing subspaces without torsion

In the works by Minčič et al. (1998, 1999, 2007) [8,9,11] we have studied the problem (started by M. Prvanović): Can a generalized Riemannian space GRN (with non-symmetric basic tensor) contain a Riemannian subspace (possessing a symmetric induced basic tensor)?In the present work the analogous problem is studied for a space LN with non-symmetric affine connection and its subspace LM. In the Section 2 it is assumed that the equations defining submanifold are given and that it is valid (2.4), i.e., induced torsion T∼=0, and the connection on the manifold MN is determined so that T≠0. At the third section we start from connection on the manifold MNT≠0. From the mentioned condition (2.4) we determine the Eq. (1.1) defining the submanifold MM⊂MN, so that induced torsion T∼=0, i.e., we get LMO⊂LN.It is proved that the answer to the question cited above is affirmative and several examples are constructed. The examinations have local character.

Symmetric positive semi-definite Cartesian Tensor fiber orientation distributions (CT-FOD)

A novel method for estimating a field of fiber orientation distribution (FOD) based on signal de-convolution from a given set of diffusion weighted magnetic resonance (DW-MR) images is presented. We model the FOD by higher order Cartesian tensor basis using a parametrization that explicitly enforces the positive semi-definite property to the computed FOD. The computed Cartesian tensors, dubbed Cartesian Tensor-FOD (CT-FOD), are symmetric positive semi-definite tensors whose coefficients can be efficiently estimated by solving a linear system with non-negative constraints. Next, we show how to use our method for converting higher-order diffusion tensors to CT-FODs, which is an essential task since the maxima of higher-order tensors do not correspond to the underlying fiber orientations. Finally, we propose a diffusion anisotropy index computed directly from CT-FODs using higher order tensor distance measures thus consolidating the whole analysis pipeline of diffusion imaging solely using CT-FODs. We evaluate our method qualitatively and quantitatively using simulated DW-MR images, phantom images, and human brain real dataset. The results conclusively demonstrate the superiority of the proposed technique over several existing multi-fiber reconstruction methods.