An open Newton method for piecewise smooth systems

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. In the present work we relax the criterion 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:R2→R2 satisfying the weaker criterion where every neighbourhood of the root of f contains a point x such that all elements of the Clarke Jacobian at x are singular. In such neighbourhoods 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.