Is there any formal way to generate samples of physical fields (e.g. electromagnetic field, fluid flow) conditional on observations? The samples would need to satisfy conditions like being continuous, globally divergence free, boundary conditions etc.
I could generate something with numerical simulation that met the criteria, but this feels messy and I'm not really sure it would be sampling the right distribution.
From reading around, I have the feeling that I should be able to put something together using Lagrangians and generating functions. But I'm not really sure where to start.
I will try to be more specific:
Suppose I have a vector field representing a fluid flow. I know some the field is continuous, divergence free, and have partial information on boundary conditions (i.e. the flow does not cross some boundaries).
Now suppose I observe a version of the field that has been blurred, or averaged over. What I would like to do is sample from the distribution of realistic fields (satisfying the conditions above) which are also consistent with my partial observations.