Convergence analysis of an optimal scaling algorithm for semilinear elliptic boundary value problems

Proof of convergence for iterative schemes for finding unstable solutions of semilinear elliptic boundary value problems is an arduous task. In perspective is a special iterative algorithm using the idea of scaling. In the form called Scaling Iterative Algorithm (SIA) based on normalizing each iterate’s function value to be 1 at a given interior point of the domain, it is found that SIA is computationally quite advantageous. Yet no convergence analysis is available. In this paper, we present a different idea of scaling which is an optimal scaling in the sense that the first integral is optimized. For this Optimal Scaling Iterative Algorithm (OSIA), we prove the convergence under certain assumptions on the nonlinearity and stipulated stepsize rule. 1: Department of Mathematics, Texas A&M University, College Station, TX 77843. E-mail addresses: gchen@math.tamu.edu and jzhou@math.tamu.edu. 2: Max Planck Institut für Quantanoptik, Garching, Germany. E-mail address: bge@mpq.mpg.de. 3: Supported in part by a TITF initiative from Texas A&M University. 1