Nonsmooth optimization approaches to VDA of models with on/off parameterizations: Theoretical issues

作者：Zhu J^*; Kamachi M; Zhou GQ

来源：Advances in Atmospheric Sciences, 2002, 19(3): 405-424.

• variational data assimilation
• nonsmooth optimization
• set-valued adjoint model

Some variational data assimilation problems of time- and space-discrete models with on/off parameterizations; can be regarded as nonsmooth optimization problems. Some theoretical issues related to those problems is systematically addressed, One of the basic concept in nonsmooth optimization is subgradient, a generalized notation of a gradient of the cost function. First it is shown that the concept of subgradient leads to a clear definition of the adjoint variables in the conventional adjoint model at singular points caused by on/off switches, Using an illustrated example of a multi-layer diffusion model with the convective adjustment, it is proved that the solution of the conventional adjoint model can not be interpreted as Gateaux derivatives or directional derivatives, at singular points, but can be interpreted as a subgradient of the cost function.
Two existing smooth optimization approaches are then reviewed which are used in current data assimilation practice, The first approach is the conventional adjoint model plus smooth optimization algorithms. Some conditions under which the approach can converge to the minimal are discussed, Another approach is smoothing and regularization approach, which removes some thresholds in physical parameterizations.
Two nonsmooth optimization approaches are also reviewed, One is the subgradient method, which uses the conventional adjoint model. The method is convergent, but very slow. Another approach, the bundle methods are more efficient. The main idea of the bundle method is to use the minimal norm vector of subdifferential, which is the convex hull of all subgradients, as the descent director. However finding all subgradients is very difficult in general, Therefore bundle methods are modified to use only one subgradient that can be calculated by the conventional adjoint model. In order to develop an efficient bundle method, a set-valued adjoint model, as a generalization of the conventional adjoint model, is proposed, It is shown that the significance of the set-valued adjoint model is that at singular points, it can give all supporting subgradients. Therefore using the set-valued adjoint model, it is possible to develop a bundle method that may yield higher convergence scores.

• 出版日期2002
• 单位中国科学院大气物理研究所; 中国气象科学研究院