Title:  Validated continuation for infinite dimensional problems

Abstract:
One of the most efficient methods for determining solutions of a
continuous parameterized family of equations $F(u,p)=0$ is to use
predictor-corrector continuation techniques. In the case of functional
differential equations like PDEs or delay equations, this procedure must
be applied to some finite dimensional approximation which of course raises
the question of the validity of the output. We introduce a new technique
that combines the information obtained from the predictor-corrector steps
with ideas from rigorous computations and verifies that the numerically
produced zero for the finite dimensional system can be used to explicitly
define a set which contains a unique solution for the original infinite
dimensional equation. The cost of this new validated continuation is less
than twice the cost of the standard continuation method alone.