Variational methods

Data assimilation is the process by which PDE-based models use measurements to produce an optimal representation of the state of the system. Different measurements bring different contributions to reducing uncertainty in the inference results. Quantifying the impact of observations is important for data pruning and for the configuration of sensor networks. This talk discusses several adjoint-based methodologies for formal observation impact assessment and for optimal design of sensor networks.

Asymptotic variational methods are aimed at describing the overall properties of an increasingly complicated system by computing an effective limit energy where some parameters are averaged out or greatly simplified. Such methods include Gamma-convergence and variational expansions. An extremely interesting field of application is that of discrete (lattice) systems, where the determination of the relevant parameters in the limit theory is part of the problem.

Variational methods are presented which allow to correlate pairs of
implicit shapes in 2D and 3D images, to morph pairs of explicit
surfaces, or
to analyse motion pattern in movies.
A particular focus is on joint methods.
Indeed, fundamental tasks in image processing are highly interdependent:
Registration of image morphology significantly benefits from previous
denoising and structure segmentation.
On the other hand, combined information of different image modalities
makes shape

Self-assembly, a process in which a disordered system of preexisting components forms an organized structure or pattern, is both ubiquitous in nature and important for the synthesis of many designer materials.
In this talk, we will address two variational paradigms for self-assembly from the point of view of analysis and computation.

The first variational model is a nonlocal perturbation (of Coulombic-type) to the well-known Ginzburg-Landau/Cahn-Hilliard free energy. The functional has a rich and complex energy landscape with many metastable states.

We consider the estimation of carbon flux at the ocean-atmosphere interface from the perspective of variational data assimilation. The flux is treated as a time-dependent Neumann boundary condition for the carbon mixing ratio, which evolves according to a linear diffusive transport equation. The variational problem has a natural, infinite-dimensional Bayesian formulation.

In this course we develop the methodology of “variational modeling” of energy-driven systems. This methodology applies to systems whose evolution (in time) is driven by the decrease of an energy, in a friction-dominated or strongly damped way. Recent developments have shown that a surprisingly large class of evolutionary systems is of this form, even though the energy and the friction mechanism may not be obvious.