A class of splitting alternating algorithms is proposed for finding the sparse solution of linear systems with concatenated orthogonal matrices. Depending on the number of matrices concatenated, the proposed algorithms are classified into the two-block splitting alternating algorithm (TSAA) and the multi-block splitting alternating algorithm (MSAA). These algorithms aim to decompose a large-scale linear system into two or more coupled subsystems, each significantly smaller than the original system, and then combine the solutions of these subsystems to produce the sparse solution of the original system. The proposed algorithms only involve matrix-vector products and reduced orthogonal projections. It turns out that the proposed algorithms are globally convergent to the sparse solution of a linear system if the matrix (along with the sparsity level of the solution) satisfies a coherence-type condition. Numerical experiments indicate that the proposed algorithms are very promising and can quickly and accurately locate the sparse solution of a linear system with significantly fewer iterations than several mainstream iterative methods.

翻译：本文提出了一类分裂交替算法，用于求解具有拼接正交矩阵的线性系统的稀疏解。根据所拼接矩阵的数量，所提出的算法分为双块分裂交替算法（TSAA）和多块分裂交替算法（MSAA）。这些算法旨在将大规模线性系统分解为两个或多个耦合的子系统，每个子系统都显著小于原始系统，然后结合这些子系统的解以产生原始系统的稀疏解。所提出的算法仅涉及矩阵-向量乘积和降维正交投影。结果表明，如果矩阵（以及解的稀疏度水平）满足一种相干性条件，所提出的算法能够全局收敛到线性系统的稀疏解。数值实验表明，所提出的算法非常有前景，能够以比几种主流迭代方法少得多的迭代次数，快速且准确地定位线性系统的稀疏解。