含有开关参数化物理过程的模式变分资料同化的非光滑优化方法:几个理论问题

摘要

时间、空间上离散的模式如果含有带开关的参数化物理过程,某些对应的变分同化问题可以看成为非光滑优化问题.本文系统地讨论了与此相关的几个理论问题.首先,利用次梯度这个非光滑优化中的基本概念可以对通常的伴随方程的解进行清楚的解释和定义.利用一个带对流调整的多层扩散模式,演示了通常的伴随方程的解在奇异点处不是由代价函数的Gateaux导数构成,而是代价函数的一个次梯度.其次,在非光滑优化的框架下对现有的解决这类问题的方法进行了评述.这些方法包括:(1)利用光滑优化方法和通常的伴随模式,并且讨论了这个方法的收敛条件;(2)正则化方法.这个方法是把模式中不光滑的项用光滑的项近似,以把问题变为光滑优化问题.(3)次梯度方法.这个方法利用通常的伴随模式来计算一个次梯度,是收敛的算法,不过收敛速度很慢.(4)非光滑优化中的Bundle方法.这个方法比次梯度方法收敛得快,不过需要利用所有的次梯度信息.由于计算所有的次梯度十分困难,现有的Bundle方法大都进行了一些改变,仅需要用户能够计算出一个次梯度.不过这个改变是无奈的,多个次梯度的信息可以提高Bundle方法的效率.本文最主要的工作是提出了集值伴随方程的概念.集值伴随模式可以在奇异点处计算出所有支撑次梯度,因此利用集值伴随方程有可能提出基于Bundle方法的更好算法.%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.