Numerical optimization for the calculus of variations by gradients on non-Hilbert Sobolev spaces using conjugate gradients and normalized differential equations of steepest descent

摘要

The purpose of this paper is to illustrate the application of numerical optimization methods for nonquadratic functionals defined on non-Hilbert Sobolev spaces. These methods use a gradient defined on a norm-reflexive and hence strictly convex normed linear space. This gradient is defined by Michael Golomb and Richard A. Tapia in [M. Golomb,R.A. Tapia, The metric gradient in normed linear spaces, Numer. Math. 20 (1972) 115-124].It is also the same gradient described by Jean-Paul Penot in [J.P. Penot, On the convergence of descent algorithms, Comput. Optim. Appl. 23 (3) (2002) 279-284]. In this paper we shall restrict our attention to variational problems with zero boundary values. Nonzero boundary value problems can be converted to zero boundary value problems by anappropriate transformation of the dependent variables. The original functional changes by such a transformation. The connection to the calculus of variations is: The notion of a relative minimum for the Sobolev norm for p positive and large and with only first derivatives and function values is related to the classical weak relative minimum in the calculus of variations. The motivation for minimizing nonquadratic functionals on these non-Hilbert Sobolev spaces is twofold. First, a norm equivalent to this Sobolev norm approaches the norm used for weak relative minimums in the calculus of variations as papproaches infinity. Secondly, the Sobolev norm is both norm-reflexive and strictly convex so that the gradient for a non-Hilbert Sobolev space consists of a singleton set; hence,the gradient exists and is unique in this non-Hilbert normed linear space. Two gradient minimization methods are presented here. They are the conjugate gradient methods and an approach that uses differential equations of steepest descent. The Hilbert space conjugate gradient method of James Daniel in [J. Daniel, The Approximate Minimization of Functionals, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1971], is one conjugate gradient method extended to a conjugate gradient procedure for a non-Hilbert normed linear space.