Regularization based on all-at-once formulations for inverse problems
For researchers working on parameter identification in inverse problems, this work provides theoretical and numerical insights into the computational benefits of all-at-once formulations, though the results are incremental and domain-specific.
The paper investigates all-at-once formulations for inverse problems, where model and observation equations are treated as a single system, and provides convergence results for variational, Newton-type, and gradient-based regularization methods. Numerical comparisons show that all-at-once approaches can offer computational advantages over reduced formulations, especially for iterative methods with nonlinear models.
Parameter identification problems typically consist of a model equation, e.g. a (system of) ordinary or partial differential equation(s), and the observation equation. In the conventional reduced setting, the model equation is eliminated via the parameter-to-state map. Alternatively, one might consider both sets of equations (model and observations) as one large system, to which some regularization method is applied. The choice of the formulation (reduced or all-at-once) can make a large difference computationally, depending on which regularization method is used: Whereas almost the same optimality system arises for the reduced and the all-at-once Tikhonov method, the situation is different for iterative methods, especially in the context of nonlinear models. In this paper we will exemplarily provide some convergence results for all-at-once versions of variational, Newton type and gradient based regularization methods. Moreover we will compare the implementation requirements for the respective all-at-one and reduced versions and provide some numerical comparison.