Splitting methods for tensor equations

Abstract

Summary The Jacobi, Gauss‐Seidel and successive over‐relaxation methods are well‐known basic iterative methods for solving system of linear equations. In this paper, we extend those basic methods to solve the tensor equation scriptAxm−1−b=0, where scriptA is an mth‐order n−dimensional symmetric tensor and b is an n‐dimensional vector. Under appropriate conditions, we show that the proposed methods are globally convergent and locally r‐linearly convergent. Taking into account the special structure of the Newton method for the problem, we propose a Newton‐Gauss‐Seidel method, which is expected to converge faster than the above methods. The proposed methods can be extended to solve a general symmetric tensor equations. Our preliminary numerical results show the effectiveness of the proposed methods.

Generalized tensor equations with leading structured tensors

“…Since most of the recent numerical algorithms are devoted to (1.1) with M-tensors, we employ an efficient Levenberg-Marquardt algorithm to find numerical solutions of the generalized tensors equations (1.3). The computational results demonstrate that the proposed Levenberg-Marquardt algorithm is competitive to the state-of-art algorithms in [12,13,17] when dealing with (1.1)…”

“…Besides, we compare the proposed LMA with three benchmark algorithms, including the homotopy method [12] (HM for short), the Newton-Gauss-Seidel method with one-step Gauss-Seidel iteration [17] (NGSM for short), and the quadratically convergent algorithm [13] (QCA for short). Then, we consider the generic model The code of the HM proposed by Han [12] was downloaded from Han's homepage 1 .…”

“…Finally, we employ a Levenberg-Marquardt algorithm to find a solution to GTEs and report some preliminary numerical results.Keywords Generalized tensor equations · Z + -tensor · P-tensor · error bound · Levenberg-Marquardt algorithm.Recently, it has been well-documented that the system of tensor equations (1.1) arises in a number of applications such as data mining [19], numerical partial differential equations [7], and tensor complementarity problems [30,32]. Therefore, the system of tensor equations (1.1) has received much considerable attention in the recent literature, e.g., see [7,12,13,17,20,21,25,33] and references therein. Especially, in [7], Ding and Wei proved that, if the coefficient tensor A in (1.1) is a nonsingular M-tensor [6,38], then the problem (1.1) has a unique positive solution for any given positive vector b (i.e., each component of b is positive) in R n , in addition to generalizing the Jacobi and Gauss-Seidel methods to find the unique solution.…”

“…Especially, in [7], Ding and Wei proved that, if the coefficient tensor A in (1.1) is a nonsingular M-tensor [6,38], then the problem (1.1) has a unique positive solution for any given positive vector b (i.e., each component of b is positive) in R n , in addition to generalizing the Jacobi and Gauss-Seidel methods to find the unique solution. Since solving tensor equations system plays an instrumental role in engineering and scientific computing, many numerical methods have been developed to solve (1.1) with M-tensors, e.g., see [12,13,17,21,33]. However, the coefficient tensor A of (1.1) arising from many real-world problems, such as data mining [19], tensor complementarity problems [30,32] and high dimensional interpolations in the reproducing kernel Banach spaces [35], is often not a nonsingular M-tensor.Moreover, we observe that (1.1) is a system of homogenous polynomial equations, but some applications usually lack of the underlying homogeneousness emerging in (1.1), for example the high-order Markov chains [18] and multilinear PageRank problems [11].…”

Further study on tensor absolute value equations

“…Obviously, TAVEs (1.3) becomes the system of AVEs when both tensors A and B reduce to matrices, and in particular, TAVEs reduces to the multilinear system (i.e., by taking B|x| q−1 = 0 ) studied in recent work [8,12,16,41], which has found many important applications in data mining and numerical partial differential equations (e.g., see [8,17]), to name just a few. Most recently, Du et al [9] considered another special case of (1.3) with the setting of B being a negative p-th order n-dimensional unit tensor (i.e., B|x| q−1 reduces to −|x| [p−1] ), which is equivalent to a generalized tensor complementarity problem.…”