An Open Newton Method for Piecewise Smooth Functions
For researchers solving piecewise smooth equation systems, this work provides a more robust Newton-type method that handles singular Jacobians, though the improvement is incremental over existing semismooth Newton methods.
This paper extends the semismooth Newton method for piecewise smooth functions by relaxing the local bijectivity condition to local openness, proving quadratic convergence under the weaker criterion. It demonstrates cases where classical semismooth Newton fails but the new method succeeds, with an application in cardiovascular mathematics.
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 $f:\mathbb R^2\rightarrow\mathbb R^2$ 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.