A convergent numerical algorithm for the stochastic growth-fragmentation problem

TL;DR

The paper tackles the forward problem of a stochastic growth-fragmentation process by introducing a finite-volume discretization that truncates the state space and renders the Markovian GF chain finite-dimensional. It proves that the numerical kernel $\mathcal{P}_{a,h}$ converges to the true kernel $\mathcal{P}$ with a quantified rate $\|\mathcal{P}-\mathcal{P}_{a,h}\|_V \le C(h+ah+e^{-k a^{\alpha}})$ under polynomial growth and smoothness assumptions on $S(x)=B(x)/g(x)$, leading to convergence of the invariant measure $\pi_{a,h}$ to the true invariant measure $\pi$. A triangle-inequality framework ties kernel convergence to invariant-measure convergence, and ergodicity results (notably $V$-uniform ergodicity) underpin existence and uniqueness of $\pi$ and stability of the numerical method. Numerical tests corroborate first-order convergence in the mesh size $h$ with a suitably chosen truncation $a$, illustrating practical applicability to forward simulations and to downstream inverse problems, including Bayesian or maximum-likelihood approaches for GF systems.

Abstract

The stochastic growth-fragmentation model describes the temporal evolution of a structured cell population through a discrete-time and continuous-state Markov chain. The simulations of this stochastic process and its invariant measure are of interest. In this paper, we propose a numerical scheme for both the simulation of the process and the computation of the invariant measure, and show that under appropriate assumptions, the numerical chain converges to the continuous growth-fragmentation chain with an explicit error bound. With a triangle inequality argument, we are also able to quantitatively estimate the distance between the invariant measures of these two Markov chains.

Figures (8)

• Figure 1: Grid points and units
• Figure 2: Illustration of Propostion \ref{['fv-on-half-grid']}. The density function of $\tilde{\xi}^{n+1}|\tilde{\xi}^n$ compared with that of $\xi^{n+1}|\xi^n$, both starting at $x_i$.
• Figure 3: Illustration of the numerical scheme in three steps. Top left: the function values of $S(x)$ drawn as vertical bars; top right: the values $Q_{i,k}$ whose bars are drawn to the left of point $x_k$; bottom right: distribution of $\tilde{\eta} \approx 2\xi^{n+1}$ by taking the difference of $Q_k$; bottom left: distribution of $\tilde{\xi}^{n+1}$ through dividing $\tilde{\eta}$ by 2 and merging adjacent odd and even intervals, the same hue means the same amount of probability.
• Figure 4: Numerical invariant measure in Example \ref{['example1']}. Left: different $a$'s and fixed $h=0.02$; right: different $h$'s and fixed $a=5$.
• Figure 5: Relation between $\|\pi_{a,h}-\pi_{a,2h}\|_{\rm TV}$ and $h$ in Example \ref{['example1']} on log-log scale.

Theorems & Definitions (33)

• Definition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Remark 1
• Remark 2
• Theorem 4