Many scientific and engineering applications necessitate computing the minimum norm solution of a sparse underdetermined linear system of equations. The minimum 2-norm solution of such systems can be obtained by a recent parallel algorithm, whose numerical effectiveness and parallel scalability are validated in both shared- and distributed-memory architectures. This parallel algorithm assumes the coefficient matrix in a block-diagonal column-overlapped (BDCO) form, which is a variant of the block-diagonal form where the successive diagonal blocks may overlap along their columns. The total overlap size of the BDCO form is an important metric in the performance of the subject parallel algorithm since it determines the size of the reduced system, solution of which is a bottleneck operation in the parallel algorithm. In this work, we propose a hypergraph partitioning model for reordering sparse matrices into BDCO form with the objective of minimizing the total overlap size and the constraint of maintaining balance on the number of nonzeros of the diagonal blocks. Our model makes use of existing partitioning tools that support fixed vertices in the recursive bipartitioning paradigm. Experimental results validate the use of our model as it achieves small overlap size and balanced diagonal blocks.

许多科学和工程应用都需要计算稀疏欠定线性方程组的最小范数解。可以通过最近的并行算法获得此类系统的最小2范数解，该算法的数值有效性和并行可伸缩性在共享内存和分布式内存体系结构中均得到了验证。该并行算法假定系数矩阵为块对角列重叠（BDCO）形式，这是块对角形式的变体，其中连续对角块可以沿其列重叠。BDCO形式的总重叠大小是主题并行算法性能的重要指标，因为它确定简化系统的大小，其解决方案是并行算法中的瓶颈操作。在这项工作中 我们提出了一种超图分割模型，用于将稀疏矩阵重新排序为BDCO形式，目的是最大程度地减小总重叠大小以及对角线块非零数目保持平衡的约束。我们的模型利用了现有的分区工具，这些工具支持递归双向划分范例中的固定顶点。实验结果验证了我们模型的使用，因为它实现了小的重叠大小和平衡的对角线块。