On the perturbation of rank‐one symmetric tensors

The problem of symmetric rank‐one approximation of symmetric tensors is important in independent components analysis, also known as blind source separation, as well as polynomial optimization. We derive several perturbative results that are relevant to the well‐posedness of recovering rank‐one structure from approximately‐rank‐one symmetric tensors. We also specialize the analysis of the shifted symmetric higher‐order power method, an algorithm for computing symmetric tensor eigenvectors, to approximately‐rank‐one symmetric tensors. Copyright © 2012 John Wiley & Sons, Ltd.     

A unitary joint diagonalization algorithm for nonsymmetric higher‐order tensors based on Givens‐like rotations

Based on Givens‐like rotations, we present a unitary joint diagonalization algorithm for a set of nonsymmetric higher‐order tensors. Each unitary rotation matrix only depends on one unknown parameter which can be analytically obtained in an independent way following a reasonable assumption and a complex derivative technique. It can serve for the canonical polyadic decomposition of a higher‐order tensor with orthogonal factors. Furthermore, based on cross‐high‐order cumulants of observed signals, we show that the proposed algorithm can be applied to solve the joint blind source separation problem. The simulation results reveal that the proposed algorithm has a competitive performance compared with those of several existing related methods.     

A constructive arbitrary‐degree Kronecker product decomposition of tensors

We generalize the matrix Kronecker product to tensors and propose the tensor Kronecker product singular value decomposition that decomposes a real k‐way tensor into a linear combination of tensor Kronecker products with an arbitrary number of d factors. We show how to construct , where each factor is also a k‐way tensor, thus including matrices (k=2) as a special case. This problem is readily solved by reshaping and permuting into a d‐way tensor, followed by a orthogonal polyadic decomposition. Moreover, we introduce the new notion of general symmetric tensors (encompassing symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors, etc.) and prove that when is structured then its factors will also inherit this structure.     

Subtracting a best rank‐1 approximation from p × p × 2(p≥2) tensors

We introduce one special form of the ptimesp × 2 (p≥2) tensors by multilinear orthonormal transformations, and present some interesting properties of the special form. Through discussing on the special form, we provide a solution to one conjecture proposed by Stegeman and Comon in a conference paper (Proceedings of the EUSIPCO 2009 Conference, Glasgow, Scotland, 2009), and reveal an important conclusion about subtracting a best rank‐1 approximations from p × p × 2 tensors of the special form. All of these confirm that consecutively subtracting the best rank‐1 approximations may not lead to a best low rank approximation of a tensor. Numerical examples show the correctness of our theory. Copyright © 2011 John Wiley & Sons, Ltd.     

Asymptotically self‐similar global solutions for a higher‐order semilinear parabolic equation

In this paper, we study the higher‐order semilinear parabolic equation where m, p>1 and $a\,\in\,\mathbb{R}$. For p>1+2m/N, we prove that the global existence of mild solutions for small initial data with respect to some norm. Some of those solutions are proved to be asymptotic self‐similar. Copyright © 2009 John Wiley & Sons, Ltd.     

On a second‐order expansion of the truncated singular subspace decomposition

We present a second‐order expansion for singular subspace decomposition in the context of real matrices. Furthermore, we show that, when some particular assumptions are considered, the obtained results reduce to existing ones. Some numerical examples are provided to confirm the theoretical developments of this study. Copyright © 2016 John Wiley & Sons, Ltd.     

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher‐order finite‐element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse‐grid correction is assumed to be an ideal operator that annihilates the low‐frequency error components and leaves the high‐frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two‐grid convergence factor. While two‐grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is “between” two‐grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher‐order finite element discretizations, including for Q₂‐Q₁ (Taylor‐Hood) elements for the Stokes equations. Finally, we present two‐grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two‐grid and multigrid convergence factors.     

Asymptotic behavior of higher‐order Navier‐Stokes‐Cahn‐Hilliard systems

Our aim in this paper is to study the asymptotic behavior, in terms of finite‐dimensional attractors, for higher‐order Navier‐Stokes‐Cahn‐Hilliard systems. Such equations describe the evolution of a mixture of 2 immiscible incompressible fluids. We also give several numerical simulations.     

Rational Chebyshev collocation method for solving higher‐order linear ordinary differential equations

A collocation method to find an approximate solution of higher‐order linear ordinary differential equation with variable coefficients under the mixed conditions is proposed. This method is based on the rational Chebyshev (RC) Tau method and Taylor‐Chebyshev collocation methods. The solution is obtained in terms of RC functions. Also, illustrative examples are included to demonstrate the validity and applicability of the technique, and performed on the computer using a program written in maple9. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1130–1142, 2011     

Numerical simulation of microstructures via higher‐order rank‐one relaxation

Calculation of inelastic material behaviour by means of relaxation algorithms provides an effective way to deal with fine‐scale microstructures resulting from non‐quasiconvex potentials. Especially the rank‐one relaxation admits a straight forward algorithmical realization and results in valuable information that enables the reconstruction of the underlying microstructure. This article is concerned with the application of higher–order rank–one relaxation to a single–slip plasticity model and focusses especially on the numerical algorithm. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relaxation methods for solving the tensor equation arising from the higher‐order Markov chains

In this paper, we propose several relaxation algorithms for solving the tensor equation arising from the higher‐order Markov chain and the multilinear PageRank. The semi‐symmetrization technique on the original equation is also employed to modify the proposed algorithms. The convergence analysis is given for the proposed algorithms. It is shown that the new algorithms are more efficient than the existing ones by some numerical experiments when relaxation parameters are chosen suitably.     

New results on the existence of periodic solutions for a higher‐order Li énard type p‐Laplacian differential equation

In this paper, by using the continuation theorem of coincidence degree theory, we consider the higher‐order Li énard type p‐Laplacian differential equation as follows Some new results on the existence of periodic solutions for the previous equation are obtained, which generalize and enrich some known results to some extent from the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

A geometric‐based algebraic multigrid method for higher‐order finite element equations in two‐dimensional linear elasticity

In this paper, we will discuss the geometric‐based algebraic multigrid (AMG) method for two‐dimensional linear elasticity problems discretized using quadratic and cubic elements. First, a two‐level method is proposed by analyzing the relationship between the linear finite element space and higher‐order finite element space. And then a geometric‐based AMG method is obtained with the existing solver used as a solver on the first coarse level. The resulting AMG method is applied to some typical elasticity problems including the plane strain problem with jumps in Young's modulus. The results of various numerical experiments show that the proposed AMG method is much more robust and efficient than a classical AMG solver that is applied directly to the high‐order systems alone. Moreover, we present the corresponding theoretical analysis for the convergence of the proposed AMG algorithms. These theoretical results are also confirmed by some numerical tests. Copyright © 2008 John Wiley & Sons, Ltd.     

Convergence of a transition probability tensor of a higher‐order Markov chain to the stationary probability vector

In this paper, first we introduce a new tensor product for a transition probability tensor originating from a higher‐order Markov chain. Subsequently, some properties of the new tensor product are explained, and its relationship with the stationary probability vector is studied. Also, similarity between results obtained by this new product and the first‐order case is shown. Furthermore, we prove the convergence of a transition probability tensor to the stationary probability vector. Finally, we show how to achieve a stationary probability vector with some numerical examples and make some comparison between the proposed method and another existing method for obtaining stationary probability vectors. Copyright © 2016 John Wiley & Sons, Ltd.     

A steady barycentric Lagrange interpolation method for the 2D higher‐order time‐fractional telegraph equation with nonlocal boundary condition with error analysis

In this paper, we consider the numerical solution of the time‐fractional telegraph equation with a nonlocal boundary condition. A novel barycentric Lagrange interpolation collocation method is developed to solve this equation. Two difficulties have been sorted: the singularity of the integration and the higher accuracy. At the same, we put forward a steady barycentric Lagrange interpolation technique to overcome the new “Runge” phenomenon in computation. Error estimates of the barycentric Lagrange interpolation and the time‐fractional telegraph system for the present method are presented in Sobolev spaces. High convergence rates of the proposed method are obtained and are consisted with the numerical values. Especially in the time dimension, we get the error bound, for h‐refinement and for n_t‐density in the L₂ norms. The numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time‐fractional telegraph equation. Experiments demonstrate the high convergence rates of the proposed method are consisted with the theoretical values.     

The tensor t‐function: A definition for functions of third‐order tensors

A definition for functions of multidimensional arrays is presented. The definition is valid for third‐order tensors in the tensor t‐product formalism, which regards third‐order tensors as block circulant matrices. The tensor function definition is shown to have similar properties as standard matrix function definitions in fundamental scenarios. To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third‐order tensors and computed for a small‐scale example via block Krylov subspace methods for matrix functions. A complexity analysis for these methods in the context of tensors is also provided.     

Decompositions of third‐order tensors: HOSVD,T‐SVD,and Beyond

The higher order singular value decomposition, which is regarded as a generalization of the matrix singular value decomposition (SVD), has a long history and is well established, while the T‐SVD is relatively new and lacks systematic analysis. Because of the unusual tensor‐tensor product that the T‐SVD is based on, the form of the T‐SVD may be difficult to comprehend. The main aim of this article is to establish a connection between these two decompositions. By converting the form of the T‐SVD into the sum of outer product terms, we compare the forms of the two decompositions. Moreover, from establishing the connection, a new decomposition which has a specific nonzero pattern, is proposed and developed. Numerical examples are given to demonstrate the useful ability of the new decomposition for approximation and data compression.     

Low‐rank approximation of tensors via sparse optimization

The goal of this paper is to find a low‐rank approximation for a given nth tensor. Specifically, we give a computable strategy on calculating the rank of a given tensor, based on approximating the solution to an NP‐hard problem. In this paper, we formulate a sparse optimization problem via an l₁‐regularization to find a low‐rank approximation of tensors. To solve this sparse optimization problem, we propose a rescaling algorithm of the proximal alternating minimization and study the theoretical convergence of this algorithm. Furthermore, we discuss the probabilistic consistency of the sparsity result and suggest a way to choose the regularization parameter for practical computation. In the simulation experiments, the performance of our algorithm supports that our method provides an efficient estimate on the number of rank‐one tensor components in a given tensor. Moreover, this algorithm is also applied to surveillance videos for low‐rank approximation.     

Square function estimates on layer potentials for higher‐order elliptic equations

In this paper we establish square‐function estimates on the double and single layer potentials for divergence form elliptic operators, of arbitrary even order 2m , with variable t‐independent coefficients in the upper half‐space. This generalizes known results for variable‐coefficient second‐order operators, and also for constant‐coefficient higher‐order operators.     

A new higher‐order accurate numerical method for solving heat conduction in a double‐layered film with the neumann boundary condition

Heat conduction in multilayered films with the Neumann (or insulated) boundary condition is often encountered in engineering applications, such as laser process in a gold thin‐layer padding on a chromium thin‐layer for micromachining and patterning. Predicting the temperature distribution in a multilayered thin film is essential for precision of laser process. This article presents an accurate finite difference (FD) scheme for solving heat conduction in a double‐layered thin film with the Neumann boundary condition. In particular, the heat conduction equation is discretized using a fourth‐order accurate compact FD method in space coupled with the Crank–Nicolson method in time, where the Neumann boundary condition and the interfacial condition are approximated using a third‐order accurate compact FD method. The overall scheme is proved to be convergent and hence unconditionally stable. Furthermore, the overall scheme can be written into a tridiagonal linear system so that the Thomas algorithm can be easily used. Numerical errors and convergence rates of the solution are tested by an example. Numerical results coincide with the theoretical analysis. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1291–1314, 2014