Branching processes are a class of continuous-time Markov chains (CTMCs) prevalent for modeling stochastic population dynamics in ecology, biology, epidemiology, and many other fields. The transient or finite-time behavior of these systems is fully characterized by their transition probabilities. However, computing them requires marginalizing over all paths between endpoint-conditioned values, which often poses a computational bottleneck. Leveraging recent results that connect generating function methods to a compressed sensing framework, we recast this task from the lens of sparse optimization. We propose a new solution method using variable splitting; in particular, we derive closed form updates in a highly efficient ADMM algorithm. Notably, no matrix products -- let alone inversions -- are required at any step. This reduces computational cost by orders of magnitude over existing methods, and the resulting algorithm is easily parallelizable and fairly insensitive to tuning parameters. A comparison to prior work is carried out in two applications to models of blood cell production and transposon evolution, showing that the proposed method is orders of magnitudes more scalable than existing work.

翻译：分支过程是一类连续时间马尔可夫链（CTMC），广泛应用于生态学、生物学、流行病学及许多其他领域中对随机种群动态的建模。此类系统的瞬态或有限时间行为完全由其转移概率刻画。然而，计算这些概率需要对端点条件值之间的所有路径进行边际化，这常构成计算瓶颈。利用近期将生成函数方法与压缩感知框架相联系的研究成果，我们从稀疏优化的视角重新构建了这一任务。我们提出了一种基于变量分裂的新求解方法；具体而言，我们在一个高效ADMM算法中推导出了闭式更新公式。值得注意的是，任何步骤均无需矩阵乘积——更无需矩阵求逆。这使得计算成本相比现有方法降低了数个数量级，且所得算法易于并行化，对调参不敏感。在血细胞生成与转座子演化两个模型的应用中，我们与先前工作进行了对比，结果表明所提方法在扩展性上优于现有方法数个数量级。