Mathematics – Numerical Analysis
Scientific paper
2000-08-30
SISC Vol. 23 Number 4 pp. 1316-1 (2001)
Mathematics
Numerical Analysis
19 pages with 5 postscript figures
Scientific paper
10.1137/S1064827500377721
In this paper we study the application of the Sobolev gradients technique to the problem of minimizing several Schr\"odinger functionals related to timely and difficult nonlinear problems in Quantum Mechanics and Nonlinear Optics. We show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete Fourier basis and a Fast Fourier Transform. We show that the Sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators.
Garcia-Ripoll Juan Jose
Perez-Garcia Victor M.
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