Solvability of the Sylvester tensor equation

Some iterative approaches for Sylvester tensor equations, Part I: A tensor format of truncated Loose Simpler GMRES

On applying deflation and flexible preconditioning to the adaptive Simpler GMRES method for Sylvester tensor equations

Sylvester tensor equation has widely applications in many fields, thus it is meaningful to construct effective methods to solve it. In this paper, we design two new gradient-based iterative-like algorithms for solving the Sylvester tensor equations to further improve computational efficiencies of some existing gradient-based iterative-like ones. By replacing the system matrices in mode products in the modified gradient-based iterative algorithm (Chen, Z. and Lu, L.-Z. [2013] “A gradient based iterative solutions for Sylvester tensor equations,” Math. Probl. Eng. 2013, 151–164) by their diagonal parts, we construct the accelerated modified gradient-based iterative algorithm for the Sylvester tensor equations, which requires less computational load and is more efficient than the modified gradient-based one. Besides, we apply a new updated strategy to the modified gradient-based one and develop an improved modified gradient-based iterative algorithm for the Sylvester tensor equations. Compared with the modified gradient-based one, the improved modified gradient-based iterative algorithm can make more full use of computed results and have better numerical performances. We establish the convergence conditions and convergence intervals of the proposed algorithms based on the spectral radius and matrix spectral norm. Finally, some numerical examples are performed to show that the proposed algorithms are efficient, and outperform several existing gradient-based iterative-like ones in terms of the number of iterations and computational time.

This paper is concerned with the numerical solution of the Sylvester tensor equation, which includes the Sylvester matrix equation as special case. By applying hierarchical identification principle proposed by Ding and Chen, 2005, and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective.

How can we identify the same or similar users from a collection of social network platforms (e.g., Facebook, Twitter, LinkedIn, etc.)? Which restaurant shall we recommend to a given user at the right time at the right location? Given a disease, which genes and drugs are most relevant? Multi-way association, which identifies strongly correlated node sets from multiple input networks, is the key to answering these questions. Despite its importance, very few multi-way association methods exist due to its high complexity. In this paper, we formulate multi-way association as a convex optimization problem, whose optimal solution can be obtained by a Sylvester tensor equation. Furthermore, we propose two fast algorithms to solve the Sylvester tensor equation, with a linear time and space complexity. We further provide theoretic analysis in terms of the sensitivity of the Sylvester tensor equation solution. Empirical evaluations demonstrate the efficacy of the proposed method.

Motivated by the effectiveness of Krylov projection methods and the CP decomposition of tensors, which is a low rank decomposition, we propose Arnoldi-based methods (block and global) to solve Sylvester tensor equation with low rank right-hand sides. We apply a standard Krylov subspace method to each coefficient matrix, in order to reduce the main problem to a projected Sylvester tensor equation, which can be solved by a global iterative scheme. We show how to extract approximate solutions via matrix Krylov subspaces basis. Several theoretical results such as expressions of residual and its norm are presented. To show the performance of the proposed approaches, some numerical experiments are given.

On global iterative schemes based on Hessenberg process for (ill-posed) Sylvester tensor equations

In this article, we investigate the backward error and perturbation bounds for the high order Sylvester tensor equation (STE). The bounds of the backward error and three types of upper bounds for the perturbed STE with or without dropping the second order terms are presented. The classic perturbation results for the Sylvester equation are extended to the high order case.

Developing iterative algorithms to solve Sylvester tensor equations

A tensor format for the generalized Hessenberg method for solving Sylvester tensor equations

Some iterative approaches for Sylvester tensor equations, Part II: A tensor format of Simpler variant of GCRO-based methods

Gradient-based iterative algorithms for the tensor nearness problems associated with Sylvester tensor equations

The restarted GMRES based on tensor format (GMRES_BTF(m)) is a tensor Krylov subspace method that can solve Sylvester tensor equations. When the GMRES_BTF procedure is restarted, the current search space is thrown away at each restart. This leads to slow convergence. In this paper, we present a technique for accelerating the convergence of the restarted GMRES_BTF method by retaining some information from previous cycles. We take inspiration from the heavy ball method in optimization to derive the algorithm. A numerical example is given to show the efficiency of the proposed method.

SummaryThis paper deals with studying some of well‐known iterative methods in their tensor forms to solve a Sylvester tensor equation. More precisely, the tensor form of the Arnoldi process and full orthogonalization method are derived by using a product between two tensors. Then tensor forms of the conjugate gradient and nested conjugate gradient algorithms are also presented. Rough estimation of the required number of operations for the tensor form of the Arnoldi process is obtained, which reveals the advantage of handling the algorithms based on tensor format over their classical forms in general. Some numerical experiments are examined, which confirm the feasibility and applicability of the proposed algorithms in practice. Copyright © 2016 John Wiley & Sons, Ltd.

The preconditioned iterative solvers for solving Sylvester tensor equations are considered in this paper. By fully exploiting the structure of the tensor equation, we propose a projection method based on the tensor format, which needs less flops and storage than the standard projection method. The structure of the coefficient matrices of the tensor equation is used to design the nearest Kronecker product (NKP) preconditioner, which is easy to construct and is able to accelerate the convergence of the iterative solver. Numerical experiments are presented to show good performance of the approaches.