The accelerated overrelaxation splitting method for solving symmetric tensor equations
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The accelerated overrelaxation splitting method for solving symmetric tensor equations Xin-Fang Zhang1 · Qing-Wen Wang1
· Tao Li1
Received: 31 January 2020 / Revised: 14 April 2020 / Accepted: 30 April 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This paper is concerned with solving the multilinear systems A xm−1 = b whose coefficient tensors are mth-order and n-dimensional symmetric tensors. We first extend the accelerated overrelaxation (AOR) splitting method to solve the tensor equation. To improve the convergence, we develop a Newton-AOR (NAOR) method that hybridizes the Newton method and the accelerated overrelaxation scheme. Convergence analysis shows that the proposed methods converge under appropriate assumptions. Finally, some numerical examples are provided to show the effectiveness of the methods proposed. Keywords Symmetric tensors · Multilinear systems · Splitting methods · Convergence analysis Mathematics Subject Classification 15A69 · 65F10
1 Introduction In recent years, the study of tensor equations with various products (such as Einstein product, n-mode product, and general tensor product) has attracted extensive attention and interest, since the work of Qi (2005). As was in Qi (2005), Qi and Luo (2017), Qi et al. (2018), and Song and Qi (2015), an order m dimension n 1 × n 2 × · · · × n m tensor A = (ai1 i2 ...im ) with 1 ≤ i j ≤ n j , 1 ≤ j ≤ m is a multi-dimensional array consisting of n 1 . . . n m entries. Especially, A is called an mth-order n-dimensional tensor if n 1 = · · · = n m = n. Let R[m,n] be the set of all these real tensors. A tensor A ∈ R[m,n] is called symmetric if its entries are invariant under any permutation of their indices. Let SR[m,n] be the set of all these real symmetric tensors.
Communicated by Jinyun Yuan. Research is supported by National Natural Science Foundation of China [Grant numbers 11971294 and 11571220].
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Qing-Wen Wang [email protected]; [email protected]; [email protected] Department of Mathematics, Shanghai University, Shanghai, People’s Republic of China 0123456789().: V,-vol
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The main purpose of this paper is to consider the numerical solution of the following tensor equation: A xm−1 = b, (1.1) where A ∈ SR[m,n] , b ∈ Rn is a vector, and A xm−1 ∈ Rn is a vector whose ith component is defined by: (A xm−1 )i =
n
...
i 2 =1
n
aii2 ,...,im xi2 . . . xim .
i m =1
This equation is sometimes called a multilinear system in literature, since it is linear over x for each mode of the tensor A . It was not merely applied in signal processing (Regalia and Kofidis 2003), numerical partial differential equation (Ding and Wei 2016; He et al. 2018; Liu et al. 2018), noncooperative game (Huang and Qi 2017), documents analysis (Cai et al. 2006), also extensively penetrated in tensor complementarity (Luo et al. 2017), and higher order web link analysis (Kolda and Bader 2006, 2009). For example, we consider the following Klein–Gordon equation with Dirichlet’s boundary condi
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