The Preconditioning of Minimization Algorithm in GRAPES Global Four-dimensional Variational Data Assimilation System

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.