  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.
