Multilevel Geometric Optimization for Regularised Constrained Linear Inverse Problems
This work addresses optimization challenges in inverse problems for researchers in computational mathematics and engineering, presenting an incremental improvement by extending multigrid methods to Riemannian structures.
The paper tackles the problem of solving box-constrained linear inverse problems by introducing a geometric multilevel optimization approach that incorporates constraints smoothly, resulting in faster updates at fine discretization levels while preserving feasibility.
We present a geometric multilevel optimization approach that smoothly incorporates box constraints. Given a box constrained optimization problem, we consider a hierarchy of models with varying discretization levels. Finer models are accurate but expensive to compute, while coarser models are less accurate but cheaper to compute. When working at the fine level, multilevel optimisation computes the search direction based on a coarser model which speeds up updates at the fine level. Moreover, exploiting geometry induced by the hierarchy the feasibility of the updates is preserved. In particular, our approach extends classical components of multigrid methods like restriction and prolongation to the Riemannian structure of our constraints.