Coarse-grained sensitivity for multiscale data assimilation

One of the keys to data assimilation for nonlinear multiscale system is how one can optimize the slow degrees of freedom after properly averaging out the fast degrees of freedom. This is also the case with data assimilation for atmosphere-ocean coupled system. From the statistical point of view, "averaging out" amounts to integrating out the partition function regarding the fast degrees of freedom to define an "effective" partition function. Geometrically, by coarse-graining and smoothing the rough surface of the original cost function, one is able to explore the control space going down the effective gradient to find the minimum of the effective cost function (action). Since similar problems have already been tackled in the context of quantum field theory, there exist relevant concepts like functional renormalization group or effective average action [Wetterich, 1993], which seem to be applicable to general multiscale phenomena. In this presentation, I propose a solution procedure for multiscale data assimilation problem by replacing the cost function with the effective average action and seeking its stationary point numerically through the use of coarse-grained sensitivities.