Variational data assimilation is a minimization problem of the cost function.Main characteristics of the problem are that the cost function is quadratic or nearly-quadratic and Hessian matrix of the cost function is sparse,symmetric and positive-definite.Iterative methods are suitable for solving this problem,but the calculation of the cost function and its gradient is very expensive,especially in four-dimensional variational data assimilation (4DVar).To find an optimal solution and achieve acceptable convergence rate,it is necessary to precondition the minimization algorithm.GRAPES global 4DVar system is developed for the operational use in China Meteorological Administration (CMA).It solves the minimization using L-BFGS algorithm,which is well known as a practical algorithm for variational data assimilation and originated from the works of Nocedal and Liu et al.It uses the information from the previous m iterations to compute the BFGS matrix which is an approximation to the inverse of Hessian matrix.GRAPES global 4DVar system adopts the incremental approach.In the incremental 4DVar,the inner loop minimization is solved several times with multi-outer-loop updates to find a more accurate solution of the nonlinear problem.It's possible to use information from previous minimization to precondition the next minimization.It is also related to the so-called warm-start of L-BFGS.Preconditioned L-BFGS is introduced and impacts of the preconditioning of L-BFGS on the convergence rate in 4DVar experiments of real observations are evaluated.Firstly,a case study is performed with four inner loop minimizations and 50 iterations during each inner loop minimization.Since the preconditioning works from the second inner loop minimization,nonlinear observation terms in the cost function are compared during 50-200 iterations.Results show the preconditioning of L-BFGS is effective,especially during the second inner loop minimization which uses the information from the first inner loop minimization.The scheme which uses information from the previous day to precondition the 4DVar minimization at the next day is also investigated.Given the small change of Hessian matrix between 6 and 24 hours,it may also be positive to precondition the 4DVar minimization using information from the previous day.Analysis-forecast cycling experiments are also carried out in May 2013.The performance of the preconditioned L-BFGS is consistent,leading to quicker convergence of 4DVar minimization.It is encouraging that the 4DVar run-time is reduced significantly,which is vital to the operational use of GRAPES global 4DVar system in the future.%在进行多次外循环更新的增量分析框架下,前一次极小化迭代过程中产生的信息可提供给下一次极小化做预调节.该文在GRAPES全球四维变分同化系统中对极小化算法——L-BFGS算法实施了这种预调节,通过全观测的个例试验和批量试验进行评估,发现进行预调节后L-BFGS算法的收敛效率得到明显提高,而且在1个月的循环试验中表现十分稳定.该工作可以帮助GRAPES全球四维变分同化系统有效减少极小化的迭代次数,有利于满足业务化运行的时效要求.另外,间隔6h和间隔24 h的两次4DVar分析对应的海森矩阵变化不大,因此,前一时刻极小化过程产生的信息提供给后一时刻的极小化进行预调节也有一定效果.
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