Suppose we have a nonlinear algebraic equation (by "algebraic" I mean it does not involve derivatives of the unknown) that does not have unique solution, e.g. $$\sin(x) = 0 \qquad x\in [-10, 10] $$ We could also think of more general problems $$\mathcal{N}(x) = 0 \qquad x\in\Omega$$ where $\mathcal{N}: \Omega \rightarrow \mathbb{R}$ is a nonlinear algebraic operator defined on $\Omega$, a compact subset of $\mathbb{R}^d, d \geq 1$. Also suppose $\Omega$ is "well behaved", e.g. is the closure of an open set.
Are there numerical methods to approximate the whole set of solutions of such equations? In other words, are there methods to approximate the kernel $K(\mathcal{N}) = \left\{ x\in\Omega: \mathcal{N}(x)=0\right\}$ in the case this is not a singleton?
What if, in addiction, the operator $\mathcal{N}$ is a "blackbox" (i.e. no explicit formula for $\mathcal{N}$ or its gradient are available)?
The best I can think about at the moment is to turn the problem into a minimization problem with objective function: $$\vert\mathcal{N}(x)\vert \qquad x\in\Omega$$ then run an appropriate optimization algorithm. However this approach is meant to determine one solution rather than a set of solution.
Another aspect of the problem I was considering is the fact that the kernel $K(\mathcal{N})$ may have connected components (that may not be singletons), however I do not know if/how to exploit this information to approach the problem.