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**Limited-memory BFGS diagonal preconditioners for a data assimilation problem in meteorology.**
*(English)*
Zbl 0991.65051

Summary: This paper uses two simple variational data assimilation problems with the 1D viscous Burgers’ equation on a periodic domain to investigate the impact of various diagonal-preconditioner update and scaling strategies, both on the limited-memory BFGS (Broyden, Fletcher, Goldfarb and Shanno) inverse Hessian approximation and on the minimization performance. These simple problems share some characteristics with the large-scale variational data assimilation problems commonly dealt with in meteorology and oceanography.

The update formulae studied are those proposed by J. C. Gilbert and C. Lemaréchal [Math. Program. 45, No. 3 (B), 407–435 (1989; Zbl 0694.90086)] and the quasi-Cauchy formula of M. Zhu, J. L. Nazareth and H. Wolkowicz [SIAM J. Optim. 9, No. 4, 1192–1204 (1999; Zbl 1013.90137)]. Which information should be used for updating the diagonal preconditioner, the one to be forgotten or the most recent one, is considered first. Then, following the former authors, a scaling of the diagonal preconditioner is introduced for the corresponding formulae in order to improve the minimization performance. The large negative impact of such a scaling on the quality of the L-BFGS inverse Hessian approximation led us to propose an alternate updating and scaling strategy, that provides a good inverse Hessian approximation and gives the best minimization performance for the problems considered. With this approach the quality of the inverse Hessian approximation improves steadily during the minimization process. Moreover, this quality and the L-BFGS minimization performance improves when the amount of stored information is increased.

The update formulae studied are those proposed by J. C. Gilbert and C. Lemaréchal [Math. Program. 45, No. 3 (B), 407–435 (1989; Zbl 0694.90086)] and the quasi-Cauchy formula of M. Zhu, J. L. Nazareth and H. Wolkowicz [SIAM J. Optim. 9, No. 4, 1192–1204 (1999; Zbl 1013.90137)]. Which information should be used for updating the diagonal preconditioner, the one to be forgotten or the most recent one, is considered first. Then, following the former authors, a scaling of the diagonal preconditioner is introduced for the corresponding formulae in order to improve the minimization performance. The large negative impact of such a scaling on the quality of the L-BFGS inverse Hessian approximation led us to propose an alternate updating and scaling strategy, that provides a good inverse Hessian approximation and gives the best minimization performance for the problems considered. With this approach the quality of the inverse Hessian approximation improves steadily during the minimization process. Moreover, this quality and the L-BFGS minimization performance improves when the amount of stored information is increased.

### MSC:

65K05 | Numerical mathematical programming methods |

90C55 | Methods of successive quadratic programming type |

86A10 | Meteorology and atmospheric physics |