Dear Mohamed Salem,

I suggest you to see links and attached files on topic.

-Stochastic Model Updating Using Perturbation Methods Combined ...

https://pdfs.semanticscholar.org/.../aaa1909158d03b4161d0a77e73...

-Monte Carlo Sampling-Based Methods for Stochastic Optimization

www.optimization-online.org/DB_FILE/2013/06/3920.pdf

-gradient-based stochastic optimization methods in ... - DSpace@MIT

https://dspace.mit.edu/openaccess-disseminate/1721.1/96975

-A simulation technique for estimation in perturbed stochastic activity ...

citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.903...

-Statistical inverse analysis and stochastic modeling of transition

https://web.stanford.edu/group/ctr/Summer/.../03.01_marxen.pdf

Best regards

Dear Mohamed Salem , assuming that linear response theory holds in your system (this will fail, for instance, in the vicinity of a critical point) all you need to do is compute a response function (the convolution of the response function with the perturbation, will give you directly the difference between the value of the conjugated observable in the pertubed case and in the unperturbed one, to first order in the perturbation, which is fine if the perturbation is small).

There are indeed methods to compute response functions without perturbing the original system, which is helpful because as you say, the time needed to sample important differences between the perturbed and unperturbed system is large if the perturbation is very small. Please find a relevant reference in the context of Montecarlo simulations of spin glasses in the paper below.

Best regards,

Reinaldo.

http://chimera.roma1.infn.it/FEDERICO/Publications_files/2003_PRE_68_065104.pdf

Mohamed-Mourad Lafifi thanks, for your suggestion

Reinaldo Garcia-Garcia

Thanks , but even the perturbation would be small, I don't want to neglect it , that the main idea, as I am thinking for a correction scale to multiphase volume, which might cause small perturbations to become a serious error/changes in the end

and thanks for the publication, it is really interested

Dear Mohamed Salem,

You're welcome. I must say that what you're now saying is by no means what I interpreted from your original question, so I apologize if my answer was off-topic.

With kind regards,

Reinaldo.