Use of differentiable and nondifferentiable optimization algorithms for variational data assimilation with discontinuous cost functions

Cost functions formulated in four-dimensional variational data assimilation (4DVAR) 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 thes e cost functions with respect to the model's control variables at discontin uous points. Minimization of these cost functions using conventional differ entiable optimization algorithms may encounter difficulties. In this paper an idealized discontinuous model and an actual shallow convection parameter ization are used, both including on-off switches, to illustrate the perform ances of differentiable and nondifferentiable optimization algorithms. It w as found that (i) the differentiable optimization, such as the limited memo ry quasi-Newton (L-BFGS) algorithm, may still work well for minimizing a no ndifferentiable cost function, especially when the changes made in the fore cast model at switching points to the model state are not too large; (ii) f or a differentiable optimization algorithm to find the true minimum of a no nsmooth cost function, introducing a local smoothing that removes discontin uities may lead to more problems than solutions due to the insertion of art ificial stationary points; and (iii) a nondifferentiable optimization algor ithm is found to be able to find the true minima in cases where the differe ntiable minimization failed. For the case of strong smoothing, differentiab le minimization performance is much improved, as compared to the weak smoot hing cases.