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  <doc>
    <id>939</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
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    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2006-11-02</completedDate>
    <publishedDate>2006-11-02</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory</title>
    <abstract language="eng">To approximate convolutions which occur in evolution equations with memory terms, a variable-stepsize algorithm is presented for which advancing $N$ steps requires only $O(N\log N)$ operations and $O(\log N)$ active memory, in place of $O(N^2)$ operations and $O(N)$ memory for a direct implementation. A basic feature of the fast algorithm is the reduction, via contour integral representations, to differential equations which are solved numerically with adaptive step sizes. Rather than the kernel itself, its Laplace transform is used in the algorithm. The algorithm is illustrated on three examples: a blow-up example originating from a Schrödinger equation with concentrated nonlinearity, chemical reactions with inhibited diffusion, and viscoelasticity with a fractional order constitutive law.</abstract>
    <identifier type="serial">06-45</identifier>
    <identifier type="opus3-id">939</identifier>
    <identifier type="doi">10.1137/060674168</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-9399</identifier>
    <enrichment key="SourceTitle">Appeared in: SIAM Journal on Scientific Computing Vol.30, No.2 (2008)</enrichment>
    <author>Maria Lopez-Fernandez</author>
    <author>Christian Lubich</author>
    <author>Achim Schädle</author>
    <series>
      <title>ZIB-Report</title>
      <number>06-45</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>convolution quadrature</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>adaptivity</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Volterra integral equations</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>numerical inverse Laplace transform</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>anomalous diffusion</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>fractional order visco</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="msc" number="65M99">None of the above, but in this section</collection>
    <collection role="msc" number="65R20">Integral equations</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="institutes" number="compnano">Computational Nano Optics</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/939/ZR-06-45.pdf</file>
    <file>https://opus4.kobv.de/opus4-zib/files/939/ZR-06-45.ps</file>
  </doc>
  <doc>
    <id>843</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2005-01-24</completedDate>
    <publishedDate>2005-01-24</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Fast and oblivious convolution quadrature</title>
    <abstract language="eng">We give an algorithm to compute $N$ steps of a convolution quadrature approximation to a continuous temporal convolution using only $O(N\, \log N)$ multiplications and $O(\log N)$ active memory. The method does not require evaluations of the convolution kernel, but instead $O(\log N)$ evaluations of its Laplace transform, which is assumed sectorial. The algorithm can be used for the stable numerical solution with quasi-optimal complexity of linear and nonlinear integral and integro-differential equations of convolution type. In a numerical example we apply it to solve a subdiffusion equation with transparent boundary conditions.</abstract>
    <identifier type="serial">05-09</identifier>
    <identifier type="opus3-id">843</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-8437</identifier>
    <enrichment key="SourceTitle">Appeared in:SIAM J. Sci. Comput. Vol. 28(2) (2005) 421-438</enrichment>
    <author>Achim Schädle</author>
    <author>Maria Lopez-Fernandez</author>
    <author>Christian Lubich</author>
    <series>
      <title>ZIB-Report</title>
      <number>05-09</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>convolution</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>numerical integration</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Runge-Kutta methods</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Volterra integral equation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>anomalous diffusion</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="msc" number="65R20">Integral equations</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="institutes" number="compnano">Computational Nano Optics</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/843/ZR-05-09.ps</file>
    <file>https://opus4.kobv.de/opus4-zib/files/843/ZR-05-09.pdf</file>
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