A note on SINGLE-ITERATION SOBOLEV DESCENT FOR LINEAR INITIAL VALUE
AbstractMahavier and Montgomery construct a Sobolev space for approximate
solution of linear initial value problems in a finite difference setting
in SINGLE-ITERATION SOBOLEV DESCENT FOR LINEAR INITIAL VALUE PROBLEMS,
Mahavier, Montgomery, MJMS, 2013. Their Sobolev space is constructed so
that gradient-descent converges to a solution in a single iteration,
demonstrating the existence of a best Sobolev gradient for finite
difference approximation of solutions of linear initial value problems.
They then ask if there is a broader class of problems for which
convergence in a single iteration in an appropriate Sobolev space
occurs. We use their results to show the existence of single-step
iteration to solution in a lower dimensional Sobolev space for their
examples and then a class of problems for single-step convergence.