Implicit and adaptive inverse preconditioned gradient methods for nonlinear problems

Abstract

Numerical schemes for approximating the inverse of a given matrix, using an ordinary differential equation (ODE) model, were recently developed. We extend that approach to solve nonlinear minimization problems, within the framework of a preconditioned gradient method. The main idea is to develop an automatic and implicit scheme to approximate directly the preconditioned search direction at every iteration, without an a priori knowledge of the Hessian of the objective function, and involving only a reduced and controlled amount of storage and computational cost. The new scheme allows us to obtain asymptotically the Newton's direction by improving the accuracy in the ODE solver associated with the implicit scheme. We will present extensive and encouraging numerical results on some well-known test problems, on the problem of computing the square root of a given symmetric and positive definite matrix, and also on a nonlinear Poisson type equation.