For example, you approximate structure functions of finite simple graphs in cases where only cut sets of the systems are known. The inverse problem means to build possible scenarios in underdetermined system. Gröbner bases provides a way to express a set of structure functions -- however how can you know that the set of structure functions is a sufficient approximation about possible graphs corresponding to the cut sets? This is an example inverse problem where algebraic geometry and approximations for things such as the topological space on the system are useful. So
Which works on inverse problems have a focus on approximation theory and algebraic geometry?