For a function F:XβY between real Banach spaces, we show how
continuation methods to solve F(u)=g may improve from basic understanding
of the critical set C of F. The algorithm aims at special points with a
large number of preimages, which in turn may be used as initial conditions for
standard continuation methods applied to the solution of the desired equation.
A geometric model based on the sets C and Fβ1(F(C)) substantiate our
choice of curves cβX with abundant intersections with C.
We consider three classes of examples. First we handle functions F:R2βR2, for which the reasoning behind the techniques is visualizable. The second
set of examples, between spaces of dimension 15, is obtained by discretizing a
nonlinear Sturm-Liouville problem for which special points admit a high number
of solutions. Finally, we handle a semilinear elliptic operator, by computing
the six solutions of an equation of the form βΞβf(u)=g studied by
Solimini.Comment: 23 pages, 14 figure