# Packages to map 4-dimensional space with constraints on independent variables

I hope this is not a trivial or insolvable problem and it’s not too much OT. In case it is, apologies in advance for my poor math skills and feel free to hide the question.

I have a complicated function f (continuous, differentiable) of 4 independent parameters, say f(x,y,z,t) all bounded between 0 and 1: x,y,z,t \in [0,1].

Are there packages able to map the 4-dimensional space to find n sets of \{\hat{x},\hat{y},\hat{z},\hat{t}\}_n where the f(\hat{x}_i,\hat{y}_i,\hat{z}_i,\hat{t}_i) \geq 1? I am not sure if this problem can be recast into a constrained optimisation one as I’m interested in the condition f\geq 1 so I suspect that it might be a different type that I’m not aware of and I hope there might be a library able to help me with it.

## Update

After putting some more thought in it, I think that I can follow this high-level algorithm.

1. start with a “Monte Carlo”-like approach sampling the independent variables from uniform distributions
2. automatically select the closest points to f=1 to identify the frontier
3. use NumDiff to explore the numerical derivatives around the frontier to understand where the area f\geq 1 is located with respect to the independent variables.

It might work, although I feel that a more “educated” approach might be available.

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Maybe for several time slices, you can do marching cubes to find the f(x,y,z,t_i)=1 isosurface? I’m not an expert on computational geometry, but that was the first thing that came to mind. For this kind of question, maybe the computational science stackexchange (scicomp.stackexchange.com) is another place to try searching/asking.

Edit

Apparently a marching hypercubes algorithm was published earlier this year. Have not read it, but the discussion/references might be useful to you, even if the algorithm ends up not being what you want.

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“Monte Carlo” methods can be very slow. Are you trying to find an equal potential set of curves like a weather plot?