Randomised preconditioning for the forcing formulation of weak‐constraint 4D‐Var
Tóm tắt
There is growing awareness that errors in the model equations cannot be ignored in data assimilation methods such as four‐dimensional variational assimilation (4D‐Var). If allowed for, more information can be extracted from observations, longer time windows are possible, and the minimisation process is easier, at least in principle. Weak‐constraint 4D‐Var estimates the model error and minimises a series of quadratic cost functions, which can be achieved using the conjugate gradient (CG) method; minimising each cost function is called an inner loop. CG needs preconditioning to improve its performance. In previous work, limited‐memory preconditioners (LMPs) have been constructed using approximations of the eigenvalues and eigenvectors of the Hessian in the previous inner loop. If the Hessian changes significantly in consecutive inner loops, the LMP may be of limited usefulness. To circumvent this, we propose using randomised methods for low‐rank eigenvalue decomposition and use these approximations to construct LMPs cheaply using information from the current inner loop. Three randomised methods are compared. Numerical experiments in idealized systems show that the resulting LMPs perform better than the existing LMPs. Using these methods may allow more efficient and robust implementations of incremental weak‐constraint 4D‐Var.
Từ khóa
Tài liệu tham khảo
Butcher J.C., 1987, The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods
Daley R., 1993, Atmospheric Data Analysis
ECMWF(2020).Part II: Data Assimilation. No. 2 in IFS Documentation. European Centre for Medium Range Weather Forecasts. Available at.
Fisher M., 1998, Proceedings of the Seminar on Recent Developments in Numerical Methods for Atmospheric Modelling, 364
Lorenz E., 1996, Proceedings of the Seminar on Predictability, 1
Mogensen K. Alonso Balmaseda M.andWeaver A.(2012).The NEMOVAR ocean data assimilation system as implemented in the ECMWF ocean analysis for System 4. ECMWF Technical Memorandum. Reading UK: European Centre for Medium Range Weather Forecasts.
Morton K.W., 1994, Numerical Solution of Partial Differential Equations
Nakatsukasa Y.(2020). Fast and stable randomized low‐rank matrix approximation.
Nocedal J., 2006, Numerical Optimization
Rutishauser H., 1971, Handbook for Automatic Computation: Volume II: Linear Algebra. Die Grundlehren der mathematischen Wissenschaften, 284
Schnabel R.(1983).Quasi‐Newton methods using multiple secant equations. Technical Report CU‐CS‐247‐83 Boulder CO.
Tshimanga J.(2007).On a class of limited memory preconditioners for large‐scale nonlinear least‐squares problems (with application to variational ocean data assimilation). PhD Thesis Department of Mathematics University of Namur Belgium.