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  <doc>
    <id>6861</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2018-05-09</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Computational aspects of the Generalized Trapezoidal Rule</title>
    <abstract language="eng">In this article we analyze a generalized trapezoidal rule for initial value problems with piecewise smooth right hand side F:IR^n -&gt; IR^n.&#13;
When applied to such a problem, the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a non-differentiability of F. In such a situation the investigated generalized trapezoidal rule achieves a higher convergence order than the classical method. While the asymptotic behavior of the generalized method was investigated in a previous work, in the present article we develop the algorithmic structure for efficient implementation strategies&#13;
and estimate the actual computational cost of the latter.&#13;
Moreover, energy preservation of the generalized trapezoidal rule is proved for Hamiltonian systems with piecewise linear right hand side.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-68615</identifier>
    <enrichment key="SourceTitle">submitted to Optimization Methods and Software</enrichment>
    <author>Richard Hasenfelder</author>
    <submitter>Tom Streubel</submitter>
    <author>Lutz Lehmann</author>
    <author>Manuel Radons</author>
    <author>Tom Streubel</author>
    <author>Christian Strohm</author>
    <author>Andreas Griewank</author>
    <series>
      <title>ZIB-Report</title>
      <number>18-23</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Algorithmic Differentiation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Automatic Differentiation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Lipschitz Continuity</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Piecewise Linearization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Nonsmooth</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Trapezoidal Rule</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Implementation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Computational Cost</value>
    </subject>
    <collection role="msc" number="65-XX">NUMERICAL ANALYSIS</collection>
    <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/6861/ODE_imp_latest.pdf</file>
  </doc>
  <doc>
    <id>7041</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <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>
  <doc>
    <id>6463</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2017-07-20</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Integrating Lipschitzian Dynamical Systems using Piecewise Algorithmic Differentiation</title>
    <abstract language="eng">In this article we analyze a generalized trapezoidal rule for initial value problems with piecewise smooth right hand side \(F:R^n \to R^n\) based on a generalization of algorithmic differentiation. When applied to such a problem, the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a nondifferentiability of \(F\). The advantage of the proposed generalized trapezoidal rule is threefold: Firstly, we can achieve a higher convergence order than with the classical method. Moreover, the method is energy preserving for piecewise linear Hamiltonian systems. Finally, in analogy to the classical case we derive a third order interpolation polynomial for the numerical trajectory. In the smooth case the generalized rule reduces to the classical one. Hence, it is a proper extension of the classical theory. An error estimator is given and numerical results are presented.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-64639</identifier>
    <identifier type="doi">10.1080/10556788.2017.1378653</identifier>
    <enrichment key="SourceTitle">published at Optimization Methods and Software</enrichment>
    <author>Andreas Griewank</author>
    <submitter>Tom Streubel</submitter>
    <author>Richard Hasenfelder</author>
    <author>Manuel Radons</author>
    <author>Lutz Lehmann</author>
    <author>Tom Streubel</author>
    <series>
      <title>ZIB-Report</title>
      <number>17-44</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Automatic Differentiation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Lipschitz Continuity</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Piecewise Linearization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Nonsmooth</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Trapezoidal Rule</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Energy Preservation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Dense Output</value>
    </subject>
    <collection role="msc" number="65-XX">NUMERICAL ANALYSIS</collection>
    <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/6463/ODE_paper_ZibReport.pdf</file>
  </doc>
  <doc>
    <id>6164</id>
    <completedYear/>
    <publishedYear>2016</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2016-12-28</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Piecewise linear secant approximation via Algorithmic Piecewise Differentiation</title>
    <abstract language="eng">It is shown how piecewise differentiable functions \(F: R^n → R^m\) that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points x̌ and x̂. We show that the discrepancy between function and model at any point x is of the bilinear order O(||x − x̌|| ||x − x̂||). This is a little surprising since x ∈ R^n may vary over the whole Euclidean space, and we utilize only two function samples F̌ = F(x̌) and F̂ = F(x̂), as well as the intermediates computed during their evaluation. As an application of the piecewise linearization procedure we devise a generalized Newton’s method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equaling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-61642</identifier>
    <identifier type="doi">10.1080/10556788.2017.1387256</identifier>
    <enrichment key="SourceTitle">published at Optimization Methods and Software</enrichment>
    <author>Andreas Griewank</author>
    <submitter>Tom Streubel</submitter>
    <author>Tom Streubel</author>
    <author>Lutz Lehmann</author>
    <author>Richard Hasenfelder</author>
    <author>Manuel Radons</author>
    <series>
      <title>ZIB-Report</title>
      <number>16-54</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Automatic differentiation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Computational graph</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Lipschitz continuity</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Generalized Hermite interpolation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>ADOL-C</value>
    </subject>
    <collection role="ccs" number="G.">Mathematics of Computing</collection>
    <collection role="msc" number="65-XX">NUMERICAL ANALYSIS</collection>
    <collection role="msc" number="68-XX">COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section -04 in that area)</collection>
    <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/6164/newton_secant_approx_paper.pdf</file>
  </doc>
</export-example>
