Use of Differentiable and Nondifferentiable Optimization Algorithms for Variational Data Assimilation with Discontinuous Cost Functions

S. Zhang, X. Zou , J. Ahlquist, I. M. Navon, J. G. Sela

Cost functions formulated in four-dimensional variational data assimilation (4D-Var) are nonsmooth in the presence of discontinuous physical processes (i.e., the presence of \on-off" switches in NWP models). The adjoint model integration produces values of subgradients, instead of gradients, of these cost functions with respect to the model's control variables at discontinuous points. Minimization of these cost functions using conventional differentiable optimization algorithms may encounter difficulties. In this paper we use an idealized discontinuous model and an actual shallow convection parameterization, both including \on-off" switches, to illustrate the performances of differentiable and nondifferentiable optimization algorithms. It was found that (i) the differentiable optimization, such as the limited memory quasi-Newton (L-BFGS) algorithm, may still work well for minimizing a nondifferentiable cost function, especially when the changes made in the forecast model at switching points to the model state are not too large; (ii) for a differentiable optimization algorithm to find the true minimum of a nonsmooth cost function, introducing a local smoothing that removes discontinuities may cause more problems than solutions due to the insertion of artificial stationary points; and (iii) a nondifferentiable optimization algorithm is found to be able to find the true minima in cases where the differentiable minimization failed. For the case of strong smoothing, differentiable minimization performance is much improved, as compared to the weak smoothing cases.