It is well-known that a multilinear system with a nonsingular M-tensor and a positive right-hand side has a unique positive solution. Tensor splitting methods generalizing the classical iterative methods for linear systems have been proposed for finding the unique positive solution. The Alternating Anderson-Richardson (AAR) method is an effective method to accelerate the classical iterative methods. In this study, we apply the idea of AAR for finding the unique positive solution quickly. We first present a tensor Richardson method based on tensor regular splittings, then apply Anderson acceleration to the tensor Richardson method and derive a tensor Anderson-Richardson method, finally, we periodically employ the tensor Anderson-Richardson method within the tensor Richardson method and propose a tensor AAR method. Numerical experiments show that the proposed method is effective in accelerating tensor splitting methods.

翻译：众所周知，具有非奇异M-张量和正右端项的多重线性系统存在唯一正解。为求解该唯一正解，学者们提出了张量分裂方法，推广了经典线性系统迭代方法。交替Anderson-Richardson（AAR）方法是加速经典迭代方法的有效手段。本研究利用AAR思想快速求解唯一正解：首先提出基于张量正则分裂的张量Richardson方法，进而对张量Richardson方法应用Anderson加速推导出张量Anderson-Richardson方法，最后在张量Richardson方法中周期性地引入张量Anderson-Richardson方法，提出了张量AAR方法。数值实验表明，所提方法能有效加速张量分裂方法。