The article alone is not enough to implement the algorithm (in my point of view). As usual, you have to look to the references.
The method is based on a change of variable from [$]R^d[$] to [$] [0~1]^d [$] (thus the OT theory) and the use of sampling set of the underlying process as a "grid" to solve the adjoint PDE.
It is said that the PDE is solved by a meshfree method (Fasshauer notes)
but it is not explicited which one is used.
PS1: The goal of a meshfree method and quantization is to solve the curse of dimentionality.....
PS2: Have a look to Jameson or Löhner papers on meshfree methods. They are more accurate.