Algorithm 3.1.

Weighted ensemble.

Pick initial parents and weights ${\left({\xi }_0^i,{\omega }_0^i\right)}^{i=1,\dots ,N_{\text{init}}}$ with ${\sum }_{i=1}^{N_{\text{init}}}{\omega }_0^i=1$, choose a collection $\mathcal{B}$ of bins, and define a final time T. Then for t ≥ 0, iterate the following:

• (Selection step) Each parent ${\xi }_t^i$ is assigned a number $C_t^i$ of children, as follows:

In each occupied bin $u\in \mathcal{B}$, conditional on ${\mathcal{F}}_t$, let ${\left(C_t^i\right)}^{i:\mathrm{bin}\left({\xi }_t^i\right)=u}$ be Nt(u) samples from the distribution $\left\{{\omega }_t^i/{\omega }_t(u):{\xi }_t^i\in u\right\}$, where $N_t{(u)}^{u\in \mathcal{B}}$ satisfies (3.2). The children ${\left({\widehat{\xi }}_t^i\right)}^{i=1,\dots ,N}$ are defined by (3.1) with weights

$${\widehat{\omega }}_t^i=\frac{{\omega }_t(u)}{N_t(u)}\text{if}{\widehat{\xi }}_t^i\in u.$$ (3.3)

Selections in distinct bins are conditionally independent.

• (Mutation step) Each child ${\widehat{\xi }}_t^i$ independently evolves one time step as follows.

Conditionally on ${\widehat{\mathcal{F}}}_t$, the children ${\left({\widehat{\xi }}_t^i\right)}^{i=1,\dots ,N}$ evolve independently according to the Markov kernel K, becoming the next parents ${\left({\xi }_{t+1}^i\right)}^{i=1,\dots ,N}$, with weights

$${\omega }_{t+1}^i={\widehat{\omega }}_t^i,i=1,\dots ,N.$$ (3.4)

Then time advances, t ← t + 1. Stop if t = T, else return to the selection step.

Algorithm 5.1 outlines an optimization for the allocation $N_t{(u)}^{u\in \mathcal{B}}$, and a procedure for choosing the bins $\mathcal{B}$ is in Algorithm 5.2. For the initialization, see Algorithm 5.3.