During these first months in my PhD, I realized how my computational problems can be drastically reduced to *one* single problem:



> 
> Find an efficient way to sample from a Gibbs measure.
> 
> 
> 


Let me elaborate: if $H$ is a Hilbert space, $\mu$ a gaussian measure on it, then I need to numerically approximate a probability measure of the form $\mu\_1(x) = Z^{-1} e^{-G(x)} \mu(dx)$ for a "potential" $G$ and a normalization constant $Z$.


I do not have a fixed class for $G$, sometimes $H$ is just $\mathbb{R}^d$ for $d >> 1$, and generally the hypothesis on it might vary a lot (say, $G$ might come from a Neural Network, or from a PDE discretization).


It would be absolutely fantastic to study how the properties on $G$ afflict $\mu\_1$, and how they might be exploited e.g. to project numerical methods (in my case, e.g., a good approximation of $\mu\_1$ might be used for image reconstruction).


By searching on the web I realized how they are commonly called "Gibbs measure". I was very excited and tried to understand more, but on the other hand I found "only" material concerning the discrete case (lattices). This is certainly a starting point, but I was a bit confused and the approaches were different. Therefore I ask:



> 
> Does a systematic theory for the (generic) problem above exist?
>  Are there classics books that you recommend to study?
>  Can you suggest some literature / papers?
> 
> 
> 


(Theoretical investigations are welcome, as well as numerics-oriented results)


Thanks in advance.


Ps: my background is in mathematics, but I am in a Computer Science department. People around me tried to help, and my question arouse after having successfully discussed with them - we are all very curious!



Is there a systematic theory for Gibbs measures (better if on Hilbert spaces)?
Any probability measure $\mu\_1$ absolutely continuous with respect to $\mu\_1$ can be written as a Gibbs measure if you allow $G$ to take values $\pm \infty$. If the density is bounded above and below, $G$ will be bounded. So you're basically asking about how to sample from
a probability measure. This is a big field of study.  

Markov chain Monte Carlo ([MCMC](https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo)) methods are commonly used, but they
may run into difficulties, especially when the energy landscape has
deep valleys separated by high barriers. 


