# Finding nodal lines numerically in a volume

Dear all,

I’m trying to the solve the following problem. Consider some volume V \in \mathbb{R}^3 with an almost everywhere smooth function f(x) \in \mathbb{R} \leq 0 with x \in V . For symmetry reasons, all zero manifolds f(x) = 0 of that function are one-dimensional.

1. How can I find them numerically? Do you know julia packages fitting to that problem? Right now I’m just calculating a large grid over all of the volume, and make it finer within some threshold of f(x). However, that is very inefficient. I wonder if there is a way to follow the lines? Could I parametrize these lines by splines (instead of saving millions of trajectory levels)?

2. I wondered if there was maybe a very julian way to solve this problem. Particularly, I wondered if I could use AD to follow the zero-level sets.

Disclaimer: I do not have an optimization background. That is why maybe this question might sound knowledgeable and/or badly-posed to you.

Thanks,

v.

I’m not sure what is your exact question: are you maximising/minimising f(x) over V?

Some packages that might be relevant:

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You could reach one maximum, then explore the submanifold taking steps in the direction of the eigenvector of the Hessian with 0 eigenvalue.

thanks, looks good, I’ll have a look