I have a radial basis function (RBF) from sampled data points from a function. The RBF looks like,
$$ h(x) = \sum_{i=1}^{N} w_i \exp(-\gamma ||x-m_i||^2) $$
where $\gamma,w_i$ are the result of RBF interpolation, and $m_1,..,m_N$ are the selected basis points.
My question is how can I retrieve the $x$ values that belong to a level set $h(x)=c$, or a contour at $c$ for this RBF function? What methods would be necessary? Root finding (multivariate)? Or turning the RBF function into a probability distribution and sample from it? In essence I need the $x$ values that will make this equation true,
$$ w_1 \exp(-\gamma ||x-m_1||^2) + ... + w_n \exp(-\gamma ||x-m_N||^2) = c $$
where $c$ is a scalar constant, like 0, 1, or 2, .. etc. If RBF form is too complex a cubic based RBF would be fine too,
$$ w_1 ||x-m_1||^3 + ... + w_n ||x-m_N||^3 = c $$
Is there a clean analytical way to retrieve this result?
My ultimate goal is to calculate the area within this contour, so maybe there are more shortcuts available in that case.
Thanks,