Orthogonal Tensor Research Articles

Modeling of textile composite using analytical network-averaging and gradient damage approach

In this contribution, we present a gradient damage model for anisotropic textile reinforcements including fiber inextensibility and fiber sliding. In contrast to previous works, the gradient damage formulation stems not from a numerical regularization basis but from the thermodynamics of internal variables. It results in a nonlocal term as the internal energy of fiber bending with measurable nonlocal parameter. Furthermore, to guarantee a priori that rotations and reflections determined by orthogonal tensors among the symmetry group do not affect the response function of the anisotropic constitutive law, a novel mesoscopic kinematic measure for the representative volume element of the fabric is defined on the basis of the analytical network-averaging concept. Such kinematic measure is of crucial importance for material modeling of damage-elastoplasticity in anisotropic textile reinforcements, and allows for analytical descriptions of inter- and intra-ply sliding of fibers. A mixed finite element formulation is then presented for textile reinforcements taking into account fiber inextensibility. The predictive capability of the computational model is demonstrated by comparing with multiple experimental datasets of dry textile fabrics.

Sparse Regularization Based on Orthogonal Tensor Dictionary Learning for Inverse Problems

In seismic data processing, data recovery including reconstruction of the missing trace and removal of noise from the recorded data are the key steps in improving the signal-to-noise ratio (SNR). The reconstruction of seismic data and removal of noise becomes a sparse optimization problem that can be solved by using sparse regularization. Sparse regularization is a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed inverses feasible. It deals with ill-posedness by replacing an ill-posed inverse problem with a well-posed problem that has a solution close to the true solution. In the last 2 decades, interest has shifted from linear toward nonlinear regularization methods even for linear inverse problems. In inverse problems, regularizations serve as stabilizing the solution of ill-posed inverse problems and give a solution that adequately fits measurements without producing unjustifiably complex artifacts. In this paper, we present a novel sparse regularization based on a tensor-based dictionary method for inverse problems (seismic data interpolation and denoising). This regularization avoids the vectorization step for sparse representation of seismic data during the reconstruction process. The key step in sparsifying signals is the choice of sparsity-promoting dictionary learning. The learning-based approach can adaptively sparsify datasets but has high computational complexity and involves no prior-constraint pattern information for the dataset. Many existing dictionary learning methods would be computationally infeasible for the high dimensional seismic data processing. These methods also destroy the essential information as well as it reduces the discriminability and expressibility of the signal, since they deal with vectorization. In this paper, the orthogonal tensor dictionary learning that learns a dictionary from the input data by employing orthogonality and separability is proposed as sparse regularization for the inverse problems. The performance of the proposed method was validated in seismic data interpolation and denoising individually as well as simultaneously. Numerical examples of synthetic and real seismic datasets demonstrate the validity of the proposed method. The SNR of the recovered data confirms that the proposed method is the most effective method than K-singular value decomposition and orthogonal dictionary learning methods.

Regularity for a geometrically nonlinear flat Cosserat micropolar membrane shell with curvature

We consider the rigorously derived thin shell membrane \Gamma -limit of a three-dimensional isotropic geometrically nonlinear Cosserat micropolar model and deduce full interior regularity of both the midsurface deformation m\colon \omega\subset \mathbb{R}^{2}\to \mathbb{R}^{3} and the orthogonal microrotation tensor field R\colon \omega\subset \mathbb{R}^{2}\to \operatorname{SO}(3) . The only further structural assumption is that the curvature energy depends solely on the uni-constant isotropic Dirichlet-type energy term |\mathrm{D}R|^{2} . We use Rivière’s regularity techniques of harmonic-map-type systems for our system which couples harmonic maps to \operatorname{SO}(3) with a linear equation for m . The additional coupling term in the harmonic map equation is of critical integrability and can only be handled because of its special structure.

Orthogonal Tensor Recovery Based on Non-Convex Regularization and Rank Estimation

Orthogonal Tensor Recovery Based on Non-Convex Regularization and Rank Estimation

Duality of O(N) and Sp(N) random tensor models: tensors with symmetries

In a recent series of papers, a duality between orthogonal and symplectic random tensor models has been proven, first for quartic models and then for models with interactions of arbitrary order. However, the tensor models considered so far in the literature had no symmetry under permutation of the indices. In this paper, we generalize these results for tensors models with interactions of arbitrary order which further have non-trivial symmetry under the permutation of the indices. Totally symmetric and anti-symmetric tensors are thus treated as a particular case of our result.

Structure‐preserving invariant interpolation schemes for invertible second‐order tensors

AbstractTensor interpolation is an essential step for tensor data analysis in various fields of application and scientific disciplines. In the present work, novel interpolation schemes for general, that is, symmetric or non‐symmetric, invertible square tensors are proposed. Critically, the proposed schemes rely on a combined polar and spectral decomposition of the tensor data , followed by an individual interpolation of the two rotation tensors and and the positive definite diagonal eigenvalue tensor resulting from this decomposition. Two different schemes are considered for a consistent rotation interpolation within the special orthogonal group , either based on relative rotation vectors or quaternions. For eigenvalue interpolation, three different schemes, either based on the logarithmic weighted average, moving least squares or logarithmic moving least squares, are considered. It is demonstrated that the proposed interpolation procedure preserves the structure of a tensor, that is, and remain orthogonal tensors and remains a positive definite diagonal tensor during interpolation, as well as scaling and rotational invariance (objectivity). Based on selected numerical examples considering the interpolation of either symmetric or non‐symmetric tensors, the proposed schemes are compared to existing approaches such as Euclidean, Log‐Euclidean, Cholesky and Log‐Cholesky interpolation. In contrast to these existing methods, the proposed interpolation schemes result in smooth and monotonic evolutions of tensor invariants such as determinant, trace, fractional anisotropy (FA), and Hilbert's anisotropy (HA). Moreover, a consistent spatial convergence behavior is confirmed for first‐ and second‐order realizations of the proposed schemes. The present work is mainly motivated by the frequently occurring necessity for remeshing or mesh adaptivity when applying the finite element method to complex problems of nonlinear continuum mechanics with inelastic constitutive behavior, which requires the consistent interpolation of tensor‐valued history data for the transfer between different meshes. However, the proposed schemes are very general in nature and suitable for the interpolation of general invertible second‐order square tensors independent of the specific application.

Duality of orthogonal and symplectic random tensor models

The groups \mathrm{O}(N) and \mathrm{Sp}(N) are related by an analytic continuation to negative values of N , \mathrm{O}(-N)\simeq\mathrm{Sp}(N) . This duality has been studied for vector models, \mathrm{SO}(N) and \mathrm{Sp}(N) gauge theories, as well as some random matrix ensembles. We extend this duality to real random tensor models of arbitrary order D with no symmetry under permutation of the indices and with quartic interactions. The N to -N duality is shown to hold graph by graph to all orders in perturbation theory for the partition function, the free energy and the connected two-point function.

Duality of orthogonal and symplectic random tensor models: general invariants

In Gurau and Keppler 2022 (arxiv:2207.01993), a relation between orthogonal and symplectic tensor models with quartic interactions was proven. In this paper, we provide an alternative proof that extends to polynomial interactions of arbitrary order. We consider tensor models of order D with no symmetry under permutation of the indices that transform in the tensor product of D fundamental representations of O(N) and Sp(N). We explicitly show that the models obey the N to -N duality graph by graph in perturbation theory.

Half-quadratic alternating direction method of multipliers for robust orthogonal tensor approximation

Half-quadratic alternating direction method of multipliers for robust orthogonal tensor approximation

A Semiblind Receiver for a Two-Way UAV-Aided PIC System Based on the PARAFAC Model

In this paper, we consider a two-way communication system using the unmanned aerial vehicle (UAV) as a relay (UAV-aided). This system eliminates impulse interference through an adaptive filter based on the least mean square (LMS) and uses the received signal transmitted by the UAVs to construct a parallel factor (PARAFAC) model. Based on the identifiability condition of the PARAFAC model, a pulse interference cancellation orthogonal pilot tensor (PIC-OPT) receiver without iteration is proposed. Our algorithm is also used in millimeter-wave to achieve the acquisition of channel information. Compared with the least squares method, the simulation results demonstrate the superiority of the proposed semiblind receiver in terms of the relative mean square error and bit error rate.

Stress sensitivity of multiscale elastic properties under hydrostatic loading conditions of organic shales at different thermal maturity stages

Shales serve essential functions as seals for conventional hydrocarbon reservoirs, waste storage repositories and unconventional plays. Exploration and evaluation of shales rely on the characterization of their multiscale elastic properties. However, there are limited studies on the geologic controls of elastic stress sensitivity in organic shales. This study investigated dry Lokpanta, Bakken, and Niobrara shales of different thermal maturity stages. First, we used a pulse-transmission ultrasonic setup to conduct multidirectional (0o, 45o and 90o) hydrostatic-stress sensitivity (0–27.58 MPa) tests to measure elastic waves and derive other elastic properties. Furthermore, we inverted microscopic parameters from the elastic measurements based on effective medium theories such as damage parameters (αij and βijkl), crack density (ρ), normal to tangential microcrack compliance (B), specific tangential crack compliance (snBT) and microcracks orientation anisotropy (η). Finally, we studied the effects of organic matter maturity and clay diagenesis on the multi-scaled elastic stress sensitivity. Under applied hydrostatic stress, the non-linear elastic response in the shales is consistent with the closure of aligned microcracks. Microcrack closure decreased the elastic anisotropy at lower stress (<10 MPa), while aligned clays and organic matter control the stress-insensitive anisotropy at higher stress. As organic matter matures and smectite transforms to illite, orthogonal velocity stress sensitivity increases and extrinsic contribution to elastic anisotropy increases. There is also a corresponding increase in orthogonal damage second-rank tensor (α33) and a decrease in orthogonal damage fourth-rank tensor (β3333) with organic matter-clay thermal maturity. These relationships are due to the thermally-induced development of new and/or extensional roughly-textured microcracks parallel to the laminated microfabric, which is of higher compliance in the shear than compression. Microcrack closure also decreases ρ under applied pressure. Most of the samples had B < 0.5, suggesting the presence of non-idealized roughly textured microcracks. Generally, snBT broadly decreases and η increases with clay diagenesis and organic matter maturity also because of the generation and/or enhancement of aligned microcracks resulting in higher tangential stiffness.

Convergence of a Jacobi-type method for the approximate orthogonal tensor diagonalization

Convergence of a Jacobi-type method for the approximate orthogonal tensor diagonalization

Recovering orthogonal tensors under arbitrarily strong, but locally correlated, noise

SummaryWe consider the problem of recovering an orthogonally decomposable tensor with a subset of elements distorted by noise with arbitrarily large magnitude. We focus on the particular case where each mode in the decomposition is corrupted by noise vectors with components that are correlated locally, that is, with nearby components. We show that this deterministic tensor completion problem has the unusual property that it can be solved in polynomial time if the rank of the tensor is sufficiently large. This is the polar opposite of the low‐rank assumptions of typical low‐rank tensor and matrix completion settings. We show that our problem can be solved through a system of coupled Sylvester‐like equations and show how to accelerate their solution by an alternating solver. This enables recovery even with a substantial number of missing entries, for instance for ‐dimensional tensors of rank with up to missing entries.

Rank Properties and Computational Methods for Orthogonal Tensor Decompositions

The orthogonal decomposition factorizes a tensor into a sum of an orthogonal list of rank-one tensors. The corresponding rank is called orthogonal rank. We present several properties of orthogonal rank, which are different from those of tensor rank in many aspects. For instance, a subtensor may have a larger orthogonal rank than the whole tensor. To fit the orthogonal decomposition, we propose an algorithm based on the augmented Lagrangian method. The gradient of the objective function has a nice structure, inspiring us to use gradient-based optimization methods to solve it. We guarantee the orthogonality by a novel orthogonalization process. Numerical experiments show that the proposed method has a great advantage over the existing methods for strongly orthogonal decompositions in terms of the approximation error.

New kinematics description of the rigid-body fixed-point rotation in a tensor frame

<p indent="0mm">The current theoretical mechanics does not give a complete kinematic description of the fixed-point rotation of a rigid body. For example, the rotation angle comprising multiple fixed-point rotations is underdetermined, only the angular velocity vector about Euler angles is defined, and the angular acceleration vector composed of two rotations is an open problem. This paper demonstrates that the rotation angle, angle velocity, and angular acceleration of the rigid-body fixed-point rotation are essentially not vectors but second-order antisymmetric tensors. The latter can be expressed by the corresponding pseudovectors. The composition of ordered rotations is established to be the ordered inner product of the corresponding orthogonal tensors in the absolute reference system. General composition rules of the angular velocity and angle acceleration tensors and their vectors are formulated. Euler angles describe the attitude of a rigid body through three orderly fixed-axis rotations that are equivalent to a fixed-point rotation. The angle velocity and angular acceleration vectors in terms of Euler angles are presented on the basis of the general composition rule, and their concise form is given in the Resal coordinate system.

Linear convergence of an alternating polar decomposition method for low rank orthogonal tensor approximations

Low rank orthogonal tensor approximation (LROTA) is an important problem in tensor computations and their applications. A classical and widely used algorithm is the alternating polar decomposition method (APD). In this paper, an improved version iAPD of the classical APD is proposed. For the first time, all of the following four fundamental properties are established for iAPD: (i) the algorithm converges globally and the whole sequence converges to a KKT point without any assumption; (ii) it exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual O(1/k) for first order methods in optimization; (iii) more importantly, it converges R-linearly for a generic tensor without any assumption; (iv) for almost all LROTA problems, iAPD reduces to APD after finitely many iterations if it converges to a local minimizer.

On Approximation Algorithm for Orthogonal Low-Rank Tensor Approximation

This work studies solution methods for approximating a given tensor by a sum of R rank-1 tensors with one or more of the latent factors being orthonormal. Such a problem arises from applications such as image processing, joint singular value decomposition, and independent component analysis. Most existing algorithms are of the iterative type, while algorithms of the approximation type are limited. By exploring the multilinearity and orthogonality of the problem, we introduce an approximation algorithm in this work. Depending on the computation of several key subproblems, the proposed approximation algorithm can be either deterministic or randomized. The approximation lower bound is established, both in the deterministic and the expected senses. The approximation ratio depends on the size of the tensor, the number of rank-1 terms, and is independent of the problem data. When reduced to the rank-1 approximation case, the approximation bound coincides with those in the literature. Moreover, the presented results fill a gap left in Yang (SIAM J Matrix Anal Appl 41:1797–1825, 2020), where the approximation bound of that approximation algorithm was established when there is only one orthonormal factor. Numerical studies show the usefulness of the proposed algorithm.

The Set of Orthogonal Tensor Trains

The Set of Orthogonal Tensor Trains

Development of nonlinear Reynolds average turbulent [formula omitted] models

Reynolds Average Navier Stokes (RANS) models are among the most employed models to solve turbulent flows, due to their low computational cost. The majority of RANS models use the Boussinesq approximation, based on a linear relation between the deviatoric part of Reynolds stress tensor and the rate of strain tensor, with the turbulent viscosity as the positive proportionality parameter. However, these models fail in several situations, and a great deal of effort has been made by the scientific community aiming to improve models prediction through the development of nonlinear models. Analysis of higher-order models employing objective orthogonal tensors has shown to be very promising to improve the prediction of the normal components of the Reynolds stress. In this work, nonlinear models based on combinations of the square of the rate-strain tensor and non-persistence tensor are examined for a range of friction Reynolds number from 395 to 5200. Two sets of models employing different tensor basis are proposed, both employing the turbulent kinetic energy as the turbulent characteristic velocity. With respect to the turbulent characteristic length, the first set of models employs the dissipation rate of the turbulent kinetic energy, while the second set employs the intensity of the rate of strain tensor. The models prediction for a channel flow are compared with DNS data. As expected, the nonlinear models present a better adherence to the DNS data. In addition, the new approach with the norm of the deformation replacing the dissipation of turbulent kinetic energy lead to more accurate predictions with respect to the DNS.

Further results on eigenvalues of symmetric decomposable tensors from multilinear dynamical systems

Eigenvalues of tensors play a crucial role in many practical problems. In this paper, we present several new properties on eigenvalues of symmetric decomposable tensors from multilinear dynamical systems. Under orthonormal conditions, we provide a new proof for a conjecture such that the spectral radius of an orthogonal decomposable tensor is included in its coefficients. Moreover, if there is a generating vector orthogonal to other generating vectors for a symmetric decomposable tensor, it is proved that the vector is an eigenvector of the tensor. As an application of the conjecture, a sufficient region is given to guarantee the asymptotically stability of the multilinear system and numerical example verifies its performance.