Assume that numerical global sensitivity analysis of a given function :
$y = f(X), f : [0,1]^n \rightarrow R$
To my understanding (and I might be wrong), common global sensitivity analysis methods, e.g. Morris and Sobol indices, assume independent non-correlated inputs.
- How far does correlation among inputs $x_i , i \in \{1,2,..,n\}$ impact the reliability of such indices?
- What could be the recommended alternative methods?
Your understanding is correct, these methods DO require independent inputs.
Ad 1: This is difficult to answer, since it is not clear what kind of "reliability" you require. If the assumptions are false, the conclusions do not follow (in general). So for example: Sobol indices may not add up to the total, they may not be positive, hence not interpretable as components of total variance and the list goes on...
Ad 2: The only alternative, I am aware of, is to use the Shapley algorithm for allocation, which has its own set of issues. Good discussions can be found in this paper by Owen and this one Iooss and Prieur.