I am reading about the Douglas-Rachford Splitting algorithm and in that algorithm, the two operators resolvent and reflected resolvent are used extensively.
I wanted to know if these operators have any graphical interpretations. e.g. What is the reflected resolvent of an operator? What does it reflect and etc.?
There is a nice graphical interpretation. I highly recommend the video by Jonathan Eckstein introducing the concept at a DIMACS workshop: https://youtu.be/T-JUf23khZU?si=PCCHCFH7wwMp4XQ8 - start around minute 12.
Here's my version: because of Minty's representation lemma, given a maximal monotone operator $T$, for any point $t$, we can find a pair $(x, u)$ where $u \in T(x)$, such that $t = x + u$. For $x$ in one dimension, this is the same as tracing along the line of slope -1 from the location $t$ on the x-axis until intersecting the graph of $T$.
The reflection operator then returns $x-u$, moving from that point $(x,u)$ along the line with slope 1 until intersecting the x-axis again.
This movement along the x-axis will not take $t$ further from the zero (i.e. it's non-expansive). It is not guaranteed to be a contraction, though. That's why an averaged non-expansive operator is used in a convergent fixed point algorithm.
With a vertical line extending through the zero from $-u$ to $u$, we can get cycling.