Study of well-posedness in the inverse problem of inferring ice-shelf viscosity

Ice shelves, a viscous gravity current floating above the ocean in Antarctica , play a critical role in mitigating ice discharge into the ocean. The flow motion of an ice shelf is governed by its effective viscosity, which dictates how the ice flows in response to stress; however viscosity cannot be directly measured. Using measured velocity and thickness enables the inversion of effective viscosity via solving an inverse problem. This can be approached using either classical PDE-constrained numerical methods, such as the control method, or through newly-developed physics-informed neural networks. However, one of the major challenges in inferring ice-shelf viscosity is the potential ill-posedness of this inverse problem, a topic often overlooked in the literature. In this presentation, we will discuss our theoretical and numerical investigation into the uniqueness and existence of solutions for the inverse problem of inferring ice-shelf viscosity. Our findings reveal the anisotropic properties of ice viscosity, shedding new light on their complex rheology.