Vectorization of conjugate-gradient methods for large-scale minimization in meteorology

During the last few years, conjugate-gradient methods have been found to be the best available tool for large-scale minimization of nonlinear functions occurring in geophysical applications. While vectorization techniques have been applied to linear conjugate-gradient methods designed to solve symmetric linear systems of algebraic equations, arising mainly from discretization of elliptic partial differential equations, due to their suitability for vector or parallel processing, no such effort was undertaken for the nonlinear conjugate-gradient method for large-scale unconstrained minimization. Computational results are presented here using a robust memoryless quasi-Newton-like conjugate-gradient algorithm by Shanno and Phua applied to a set of large-scale meteorological problems. These results point to the vectorization of the conjugate-gradient code inducing a significant speed-up in the function and gradient evaluation for the nonlinear conjugate-gradient method, resulting in a sizable reduction in the CPU time for minimizing nonlinear functions of 10⁴ to 10⁵ variables. This is particularly true for many real-life problems where the gradient and function evaluation take the bulk of the computational effort. It is concluded that vector computers are advantageous for largescale numerical optimization problems where local minima of nonlinear functions are to be found using the nonlinear conjugate-gradient method.