Publication Date:
2023-03-31
Description:
Recent research has shown that piecewise smooth (PS) functions can be approximated by piecewise linear functions with second order error in the distance to
a given reference point. A semismooth Newton type algorithm based on successive application of these piecewise linearizations was subsequently developed
for the solution of PS equation systems. For local bijectivity of the linearization
at a root, a radius of quadratic convergence was explicitly calculated in terms
of local Lipschitz constants of the underlying PS function. In the present work
we relax the criterium of local bijectivity of the linearization to local openness.
For this purpose a weak implicit function theorem is proved via local mapping
degree theory. It is shown that there exist PS functions f:IR^2 --〉 IR^2 satisfying the weaker
criterium where every neighborhood of the root of f contains a point x such that
all elements of the Clarke Jacobian at x are singular. In such neighborhoods
the steps of classical semismooth Newton are not defined, which establishes
the new method as an independent algorithm. To further clarify the relation between a PS function and its piecewise linearization,
several statements about structure correspondences between the two are proved.
Moreover, the influence of the specific representation of the local piecewise linear models
on the robustness of our method is studied.
An example application from cardiovascular mathematics is given.
Language:
English
Type:
reportzib
,
doc-type:preprint
Format:
application/pdf
