I'm a physicist who works on inverse problems; I'll explain what these are by means of an example. Consider an object whose refractive index is known; then, the problem of computing scattered electromagnetic fields (also called the forward problem) has a unique solution (whose complexity is essentially that of solving a system of linear equations). The inverse problem is one where one estimates the refractive index of an object given the measurements of the scattered electromagnetic fields.
The inverse problem is nonlinear, ill-posed, and ill-conditioned. Such a problem has no unique solution, and shows several local minima if attacked by local optimization techniques. Further more, if I am given a solution, I have no way of checking whether it is the best solution (say in a least squares sense) or not. How would I go about trying to find out, or even start to think, whether this problem is in NP?
I'd appreciate any pointers to appropriate resources if you think the question is too basic for this forum. Thanks!