Algorithm 5.1.

Optimizing the particle allocation.

Given the particles and weights ${\left({\xi }_0^i,{\omega }_0^i\right)}^{i=1,\dots ,N_{\text{init}}}$ at t = 0 or ${\left({\xi }_t^i,{\omega }_t^i\right)}^{i=1,\dots ,N}$ at t ≥ 1:

• Define the following approximate solution to (4.3):

$${\tilde{N}}_t(u)=\frac{N\sqrt{{\omega }_t(u){\sum }_{i:{\xi }_t^i\in u}{\omega }_t^i{\left[\tilde{K}{\tilde{h}}^2-{(\tilde{K}\tilde{h})}^2\right]}_{p\left({\xi }_t^i\right)}}}{\sum _{u\in \mathcal{B}}\sqrt{{\omega }_t(u){\sum }_{i:{\xi }_t^i\in u}{\omega }_t^i{\left[\tilde{K}{\tilde{h}}^2-{(\tilde{K}\tilde{h})}^2\right]}_{p\left({\xi }_t^i\right)}}},$$ (5.3)

where $p\left({\xi }_t^i\right)\in \mathcal{M}\mathcal{B}$ is the microbin containing ${\xi }_t^i$.

• Let $\tilde{N}$ count the occupied bins,

$$\tilde{N}=\sum _{u\in \mathcal{B}}{\mathbb{1}}_{{\omega }_t(u)>0}.$$

• Let $R_t{(u)}^{u\in \mathcal{B}}$ be $N-\tilde{N}$ samples from the distribution $\left\{{\tilde{N}}_t(u)/N:u\in \mathcal{B}\right\}$.

• In Algorithm 3.1, define the particle allocation as

$$N_t(u)={\mathbb{1}}_{{\omega }_t(u)>0}+R_t(u).$$

If the denominator of (5.3) is 0, we set $N_t(u)=\#\left\{i:{\xi }_t^i\in u\right\}$.