What are the techniques on inverse problems to discover the distribution of parameters from a discrete set of values? For instance, I know that my domain where the PDE is defined is made up of materials A, B and C, but I do not know how they are distributed inside the domain. Working with discrete optimization algorithms seems unfeasible. What are the techniques to solve such a problem?
Edit: I will describe the problem for just two materials, as it seems that the strategy is different than for multimaterials, according to Clason's comments (bang-bang vs multibang)
$$ \underset{u \in U}{\text{min}} \frac{1}{2} \lVert F(u) - y^{\delta} \rVert_Y \\ $$ where $$ U = \{ u \in X : u \in \{ u_1, u_2 \} ~\text{pointwise}\}\\ F: X \rightarrow Y $$ $X$ and $Y$ are function spaces and $y^{\delta} \in Y$