analysis of function landscape

Hi, first I must let know I'm not a mathematician, so I might use the wrong terms and ask trivial questions. There is a knowledge gap I have, hoping to fill it thanks to you.

Right, the problem I have is coming from engineering, specifically from naval architecture. In engineering we often deal with so called 'black box' functions of which analytical expressions, in closed form, and hence the characters are unknown. We can however employ statistical methods to sample the domain and do curve fitting or simply visualise the dependency in terms of scatter plots. So, it is usually possible to find a good fit, thus having a closed form formula at hand. This is a case in the actual situation. So, I have a multivariate function and my objective is to study the geometry or the character of its landscape. Specifically, I'm interested in how many extrema this function has, whether it is convex/concave globally or not, how those extrema are distributed and whether there is one global extremum or not, and stuff like that.

I'm wondering whether there is a well established methodology for this sort of analysis, so that this can be done efficiently. I'd appreciate any hint or a reference to it.

Thanks,

Romas