II型$K$-弱正则分裂的两阶段迭代格式收敛性及其在Covid-19大流行模型中的应用
Convergence of two-stage iterative scheme for $K$-weak regular splittings of type II with application to Covid-19 pandemic model
AI总结：
本研究填补了真锥场景下两阶段迭代格式收敛理论的空白，提出两类$K$-弱正则分裂的对应收敛理论与比较定理，并将其应用于Covid-19大流行模型的$M$-矩阵，通过数值计算求解了模型中的下一代矩阵。
摘要：
单调矩阵在正则分裂及各类弱正则分裂的收敛理论中发挥着关键作用。若单调性不成立，则很难保证上述矩阵类别的收敛性。在这种情况下，$K$-单调性足以保证$K$-正则分裂与$K$-弱正则分裂的收敛性，其中$K$是$\boldsymbol{\rm R}^n$中的一个真锥。然而，一般真锥场景下两阶段迭代格式的收敛理论是现有文献中的研究空白。尤其是针对II型弱正则分裂的同类研究（即便在标准真锥场景，即$K=\boldsymbol{\rm R}^n_+$的情况下）仍属待解决问题。为此，我们提出了真锥场景下两类$K$-弱正则分裂的两阶段迭代格式收敛理论。我们给出了若干充分条件，以保证由两阶段迭代格式导出的分裂为$K$-正则分裂，随后建立了若干比较定理。我们还研究了II型$K$-弱正则分裂情形下定常两阶段迭代法的$K$-单调收敛理论。本工作最具价值与意义的部分围绕Covid-19大流行模型中出现的$M$-矩阵展开。最后，我们利用所提出的技术开展了数值计算，以求解该大流行模型中涉及的下一代矩阵。
英文摘要：
Monotone matrices play a key role in the convergence theory of regular splittings and different types of weak regular splittings. If monotonicity fails, then it is difficult to guarantee the convergence of the above-mentioned classes of matrices. In such a case, $K$-monotonicity is sufficient for the convergence of $K$-regular and $K$-weak regular splittings, where $K$ is a proper cone in $\mathbb{R}^n$. However, the convergence theory of a two-stage iteration scheme in general proper cone setting is a gap in the literature. Especially, the same study for weak regular splittings of type II (even if in standard proper cone setting, i.e., $K=\mathbb{R}^n_+$), is open. To this end, we propose convergence theory of two-stage iterative scheme for $K$-weak regular splittings of both types in the proper cone setting. We provide some sufficient conditions which guarantee that the induced splitting from a two-stage iterative scheme is a $K$-regular splitting and then establish some comparison theorems. We also study $K$-monotone convergence theory of the stationary two-stage iterative method in case of a $K$-weak regular splitting of type II. The most interesting and important part of this work is on $M$-matrices appearing in the Covid-19 pandemic model. Finally, numerical computations are performed using the proposed technique to compute the next generation matrix involved in the pandemic model.
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