Hierarchical regularization of solution ambiguity in underdetermined inverse and optimization problems

Estimating modeling parameters based on a prescribed optimization target requires to solve an inverse problem, which is commonly ill-posed. Consequently, either infinitely many or no solutions may exist, depending on whether the system is under- or overdetermined, and whether it is consistent or inconsistent. This paper focuses on scenarios where the solution is ambiguous and infinitely many combinations of possible parameter values can accurately achieve the optimization target. Selecting the most suitable solution requires incorporating additional constraints into the model, which is achieved by regularizing the inverse problem. However, common regularization approaches require the specification of a priori unknown regularization hyperparameters that are difficult and tedious to obtain, and can have a large impact on the result.

Here, a novel strategy to reduce the ambiguity of such inverse problems is presented, ensuring that the primary optimization target is always reached accurately. To further reduce the solution space, additional constraints are included, until the optimal modeling parameters are found. Importantly, the required regularization parameters have a direct physical meaning and can be derived sequentially, starting from an initial guess that can be obtained conveniently by solving the system without regularization.

By considering several illustrative examples, the applicability of the method is demonstrated, and its potential for various comparable inverse problems is highlighted.