  Strong Constraint 4D Variational (SC-4DVAR) is a data assimilation method
that is widely used in climate and weather applications. SC-4DVAR involves
solving a minimization problem to compute the maximum a posteriori estimate,
which we tackle using the Gauss-Newton method. The computation of the descent
direction is expensive since it involves the solution of a large-scale and
potentially ill-conditioned linear system, solved using the preconditioned
conjugate gradient (PCG) method. To address this cost, we efficiently construct
scalable preconditioners using three different randomization techniques, which
all rely on a certain low-rank structure involving the Gauss-Newton Hessian.
The proposed techniques come with theoretical (probabilistic) guarantees on the
condition number, and at the same time, are amenable to parallelization. We
also develop an adaptive approach to estimate the sketch size and to choose
between the reuse or recomputation of the preconditioner. We demonstrate the
performance and effectiveness of our methodology on two representative model
problems -- the Burgers and barotropic vorticity equation -- showing a drastic
reduction in both the number of PCG iterations and the number of Gauss-Newton
Hessian products (after including the preconditioner construction cost).
