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  <doc>
    <id>7041</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
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    <publishedDate>2018-09-15</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">An Open Newton Method for Piecewise Smooth Systems</title>
    <abstract language="eng">Recent research has shown that piecewise smooth (PS) functions can be approximated by piecewise linear functions with second order error in the distance to&#13;
a given reference point. A semismooth Newton type algorithm based on successive application of these piecewise linearizations was subsequently developed&#13;
for the solution of PS equation systems. For local bijectivity of the linearization&#13;
at a root, a radius of quadratic convergence was explicitly calculated in terms&#13;
of local Lipschitz constants of the underlying PS function. In the present work&#13;
we relax the criterium of local bijectivity of the linearization to local openness.&#13;
For this purpose a weak implicit function theorem is proved via local mapping&#13;
degree theory. It is shown that there exist PS functions f:IR^2 --&gt; IR^2 satisfying the weaker&#13;
criterium where every neighborhood of the root of f contains a point x such that&#13;
all elements of the Clarke Jacobian at x are singular. In such neighborhoods&#13;
the steps of classical semismooth Newton are not defined, which establishes&#13;
the new method as an independent algorithm. To further clarify the relation between a PS function and its piecewise linearization,&#13;
several statements about structure correspondences between the two are proved. &#13;
Moreover, the influence of the specific representation of the local piecewise linear models&#13;
on the robustness of our method is studied.&#13;
 An example application from cardiovascular mathematics is given.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-70418</identifier>
    <author>Manuel Radons</author>
    <submitter>Tom Streubel</submitter>
    <author>Lutz Lehmann</author>
    <author>Tom Streubel</author>
    <author>Andreas Griewank</author>
    <series>
      <title>ZIB-Report</title>
      <number>18-43</number>
    </series>
    <collection role="institutes" number="optimization">Mathematical Optimization</collection>
    <collection role="projects" number="MODAL-GasLab">MODAL-GasLab</collection>
    <collection role="projects" number="MODAL-Gesamt">MODAL-Gesamt</collection>
    <collection role="institutes" number="enernet">Energy Network Optimization</collection>
    <collection role="institutes" number="aopt">Applied Optimization</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/7041/Open_Newton.pdf</file>
  </doc>
</export-example>
