This is a short follow up on my last post where I wrote about the sweet spot of the stepsize of the Douglas-Rachford iteration. For the case $\beta$-Lipschitz + $\mu$-strongly monotone, the iteration with stepsize $t$ converges linear with rate

$\displaystyle r(t) = \tfrac{1}{2(1+t\mu)}\left(\sqrt{2t^{2}\mu^{2}+2t\mu + 1 +2(1 - \tfrac{1}{(1+t\beta)^{2}} - \tfrac1{1+t^{2}\beta^{2}})t\mu(1+t\mu)} + 1\right)$

Here is animated plot of this contraction factor depending on $\beta$ and $\mu$ and $t$ acts as time variable:

What is interesting is, that this factor has increasing or decreasing in $t$ depending on the values of $\beta$ and $\mu$.

For each pair $(\beta,\mu)$ there is a best $t^*$ and also a smallest contraction factor $r(t^*)$. Here are plots of these quantities:

Comparing the plot of te optimal contraction factor to the animated plot above, you see that the right choice of the stepsize matters a lot.