In recent years, novel ideas have been applied to several inverse problems in combination with machine learning approaches, to improve the inversion by optimally choosing different parameters of interest. A fruitful approach in this sense is bilevel optimization, where the inverse problems are considered as lower-level constraints, while on the upper-level a loss function based on a training set is used. When confronted with inverse problems with sparsity-based regularizers or nonlinear dynamics, however, the bilevel optimization problem structure becomes quite involved, and classical nonlinear or bilevel programming results cannot be directly utilized.
In this talk, I will present the general setting of bilevel inverse problems and discuss the main theoretical challenges, as well as the applicability to relevant practical instances. Partic- ular attention will be paid to imaging problems with sparsity-based regularizers and nonlinear inverse problems arising in variational data assimilation. The main difficulties related to the reformulation as single level problems, the derivation of optimality conditions and the numerical solution will be addressed.