Space-time adaptive approach to variational data assimilation using wavelets

This paper focuses on one of the main challenges of 4-dimensional variational data assimilation, namely the requirement to have a forward solution available when solving the adjoint problem. The issue is addressed by considering the time in the same fashion as the space variables, reformulating the mathematical model in the entire space-time domain, and solving the problem on a near optimal computational mesh that automatically adapts to spatio-temporal structures of the solution. The compressed form of the solution eliminates the need to save or recompute data for every time slice as it is typically done in traditional time marching approaches to 4-dimensional variational data assimilation. The reduction of the required computational degrees of freedom is achieved using the compression properties of multi-dimensional second generation wavelets. The simultaneous space-time discretization of both the forward and the adjoint models makes it possible to solve both models either concurrently or sequentially. In addition, the grid adaptation reduces the amount of saved data to the strict minimum for a given a priori controlled accuracy of the solution. The proposed approach is demonstrated for the advection diffusion problem in two space-time dimensions.