Optimization and eigenvalue computation with application to meteorology and oceanography. Progress report, August 1, 1992--July 31, 1993.

摘要

Our computational experience has shown that dynamic scaling techniques can have a dramatic effect in the efficiency of optimization algorithms, especially when the number of variables is very large. Conjugate gradient and limited memory methods do not have adequate information on the Hessian matrix and dynamic scaling of the variables can supply some of this missing information. In (1) we developed a full-rank scaling technique for variable metric methods that is globally and superlinearly convergent on general problems. We have also proposed a new technique for accelerating the optimization process in the solution of data assimilation problems. This is an area where optimization plays a crucial role, since the four dimensional assimilation approach consists of using a nonlinear optimization code to adjust the initial conditions of the model so that measurements distributed in space and time can be adequately predicted. Each iteration of the optimization method requires a full simulation of the model a process that typically takes 1 hour on the CRAY 90. We have found that by computing a few extreme eigenvalues of the Hessian matrix, and by incorporating this information in a limited memory code, the number of simulations can be reduced by at least 50%. This is highly significant and represents a substantial contribution to the area of variational assimilation of data. We have also developed an extension of the limited memory BFGS method that is capable of handling simple bounds on the variables, providing more freedom to the modeler (2). This work was possible by the development of compact matrix representations of limited memory matrices (3), which keep the computation cost of the iteration to a minimum. Indeed we have shown that the operations with limited memory BFGS matrices are optimal in terms of arithmetic operations, since they can be rewritten as the reverse automatic differentiation of an auxiliary function (4).