SHANKAR VENKATARAMANI UNCERTAINTY QUANTIFICATION GROUP

My recent work has been on non-convex variational problems, which allow for the spontaneous generation of structure in equilibrium. A paradigm for these types of problems is a crumpled sheet of paper, where the structure of the crumpled sheet is described as a minimizer of the elastic energy of the sheet. However, as it is clear to anyone who has crumpled paper, the energetics do not fully describe the system. In particular, it is impossible to crumple two sheets of paper in the same way. This indicates that we have to go beyond energy minimizers, and try to understand the roles of the dynamics, as well as the stochasticity in determining the "real" structures one observes.

I am also interested in patter formation in extended dynamical systems, both in variational (energy-driven) and non-variational (forced systems with dissipation) settings. The idea here is that the extended systems can be described in terms of a few basic "modes" (elementary-excitations). One aspect of my work is to make this picture mathematically rigorous. Another aspect is to study the dynamics of the elementary excitations, especially in the presence of noise.

A new area of research is Homogenization of random media, and looking at corrections to the homogenized equations to better capture the effects of the stochasticity in the medium. along with collaborators in the UQ group, we are also trying to build on this approach to develop better estimators/predictors for problems in hydrology.

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