We establish some properties of the bilateral Riemann–Liouville fractional derivative Ds. We set the notation, and study the associated Sobolev spaces of fractional order s, denoted by Ws,1(a,b), and the fractional bounded variation spaces of fractional order s, denoted by BVs(a,b). Examples, embeddings and compactness properties related to these spaces are addressed, aiming to set a functional framework suitable for fractional variational models for image analysis.
The estimation of ocean dynamics is a key challenge for applications ranging from climate modeling to ship routing. State-of-the-art methods relying on satellite-derived altimetry data can hardly resolve spatial scales below ∼100 km. In this work we investigate the relevance of AIS data streams as a new mean for the estimation of the surface current velocities. Using a physics-informed observation model, we propose to solve the associated the ill-posed inverse problem using a trainable variational formulation. The latter exploits variational auto-encoders coupled with neural ODE to represent sea surface dynamics. We report numerical experiments on a real AIS dataset off South Africa in a highly dynamical ocean region. They support the relevance of the proposed learning-based AIS-driven approach to significantly improve the reconstruction of sea surface currents compared with state-of-the-art methods, including altimetry-based ones.
Riemann–Liouville Fractional Sobolev and Bounded Variation Spaces
