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Documents authored by Dahlke, Stephan

Document
DOI: 10.4230/DagRep.1.1.108
Sparse Representations and Efficient Sensing of Data (Dagstuhl Seminar 11051)

Authors: Stephan Dahlke, Michael Elad, Yonina Eldar, Gitta Kutyniok, and Gerd Teschke

Published in: Dagstuhl Reports, Volume 1, Issue 1 (2011)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 11051 ``Sparse Representations and Efficient Sensing of Data''. The scope of the seminar was twofold. First, we wanted to elaborate the state of the art in the field of sparse data representation and corresponding efficient data sensing methods. Second, we planned to explore and analyze the impact of methods in computational science disciplines that serve these fields, and the possible resources allocated for industrial applications.
Cite as

Stephan Dahlke, Michael Elad, Yonina Eldar, Gitta Kutyniok, and Gerd Teschke. Sparse Representations and Efficient Sensing of Data (Dagstuhl Seminar 11051). In Dagstuhl Reports, Volume 1, Issue 1, pp. 108-127, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@Article{dahlke_et_al:DagRep.1.1.108,
  author =	{Dahlke, Stephan and Elad, Michael and Eldar, Yonina and Kutyniok, Gitta and Teschke, Gerd},
  title =	{{Sparse Representations and Efficient Sensing of Data (Dagstuhl Seminar 11051)}},
  pages =	{108--127},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2011},
  volume =	{1},
  number =	{1},
  editor =	{Dahlke, Stephan and Elad, Michael and Eldar, Yonina and Kutyniok, Gitta and Teschke, Gerd},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.1.1.108},
  URN =		{urn:nbn:de:0030-drops-31507},
  doi =		{10.4230/DagRep.1.1.108},
  annote =	{Keywords: Efficient signal sensing schemes, sparse signal representations, efficient signal reconstruction algorithms, impact of the methods in neighboring research fields and applications}
}
Document
DOI: 10.4230/DagSemProc.08492.9
The Continuous Shearlet Transform in Arbitrary Space Dimensions

Authors: Stephan Dahlke, Gabriele Steidl, and Gerd Teschke

Published in: Dagstuhl Seminar Proceedings, Volume 8492, Structured Decompositions and Efficient Algorithms (2009)

Abstract
This note is concerned with the generalization of the continuous shearlet transform to higher dimensions. Similar to the two-dimensional case, our approach is based on translations, anisotropic dilations and specific shear matrices. We show that the associated integral transform again originates from a square-integrable representation of a specific group, the full $n$-variate shearlet group. Moreover, we verify that by applying the coorbit theory, canonical scales of smoothness spaces and associated Banach frames can be derived. We also indicate how our transform can be used to characterize singularities in signals.
Cite as

Stephan Dahlke, Gabriele Steidl, and Gerd Teschke. The Continuous Shearlet Transform in Arbitrary Space Dimensions. In Structured Decompositions and Efficient Algorithms. Dagstuhl Seminar Proceedings, Volume 8492, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{dahlke_et_al:DagSemProc.08492.9,
  author =	{Dahlke, Stephan and Steidl, Gabriele and Teschke, Gerd},
  title =	{{The Continuous Shearlet  Transform in Arbitrary Space Dimensions}},
  booktitle =	{Structured Decompositions and Efficient Algorithms},
  pages =	{1--7},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{8492},
  editor =	{Stephan Dahlke and Ingrid Daubechies and Michal Elad and Gitta Kutyniok and Gerd Teschke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08492.9},
  URN =		{urn:nbn:de:0030-drops-19216},
  doi =		{10.4230/DagSemProc.08492.9},
  annote =	{Keywords: }
}
Document
DOI: 10.4230/DagSemProc.08492.1
08492 Abstracts Collection – Structured Decompositions and Efficient Algorithms

Authors: Stephan Dahlke, Ingrid Daubechies, Michael Elad, Gitta Kutyniok, and Gerd Teschke

Published in: Dagstuhl Seminar Proceedings, Volume 8492, Structured Decompositions and Efficient Algorithms (2009)

Abstract
From 30.11. to 05.12.2008, the Dagstuhl Seminar 08492 ``Structured Decompositions and Efficient Algorithms '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
Cite as

Stephan Dahlke, Ingrid Daubechies, Michael Elad, Gitta Kutyniok, and Gerd Teschke. 08492 Abstracts Collection – Structured Decompositions and Efficient Algorithms. In Structured Decompositions and Efficient Algorithms. Dagstuhl Seminar Proceedings, Volume 8492, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{dahlke_et_al:DagSemProc.08492.1,
  author =	{Dahlke, Stephan and Daubechies, Ingrid and Elad, Michael and Kutyniok, Gitta and Teschke, Gerd},
  title =	{{08492 Abstracts Collection – Structured Decompositions and Efficient Algorithms}},
  booktitle =	{Structured Decompositions and Efficient Algorithms},
  pages =	{1--18},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{8492},
  editor =	{Stephan Dahlke and Ingrid Daubechies and Michal Elad and Gitta Kutyniok and Gerd Teschke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08492.1},
  URN =		{urn:nbn:de:0030-drops-18860},
  doi =		{10.4230/DagSemProc.08492.1},
  annote =	{Keywords: Sparse signal representation, optimal signal reconstruction, approximation, compression}
}
Document
DOI: 10.4230/DagSemProc.08492.2
08492 Executive Summary – Structured Decompositions and Efficient Algorithms

Authors: Stephan Dahlke, Ingrid Daubechies, Michael Elad, Gitta Kutyniok, and Gerd Teschke

Published in: Dagstuhl Seminar Proceedings, Volume 8492, Structured Decompositions and Efficient Algorithms (2009)

Abstract
New emerging technologies such as high-precision sensors or new MRI machines drive us towards a challenging quest for new, more effective, and more daring mathematical models and algorithms. Therefore, in the last few years researchers have started to investigate different methods to efficiently represent or extract relevant information from complex, high dimensional and/or multimodal data. Efficiently in this context means a representation that is linked to the features or characteristics of interest, thereby typically providing a sparse expansion of such. Besides the construction of new and advanced ansatz systems the central question is how to design algorithms that are able to treat complex and high dimensional data and that efficiently perform a suitable approximation of the signal. One of the main challenges is to design new sparse approximation algorithms that would ideally combine, with an adjustable tradeoff, two properties: a provably good `quality' of the resulting decomposition under mild assumptions on the analyzed sparse signal, and numerically efficient design.
Cite as

Stephan Dahlke, Ingrid Daubechies, Michael Elad, Gitta Kutyniok, and Gerd Teschke. 08492 Executive Summary – Structured Decompositions and Efficient Algorithms. In Structured Decompositions and Efficient Algorithms. Dagstuhl Seminar Proceedings, Volume 8492, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{dahlke_et_al:DagSemProc.08492.2,
  author =	{Dahlke, Stephan and Daubechies, Ingrid and Elad, Michael and Kutyniok, Gitta and Teschke, Gerd},
  title =	{{08492 Executive Summary – Structured Decompositions and Efficient Algorithms }},
  booktitle =	{Structured Decompositions and Efficient Algorithms},
  pages =	{1--5},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{8492},
  editor =	{Stephan Dahlke and Ingrid Daubechies and Michal Elad and Gitta Kutyniok and Gerd Teschke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08492.2},
  URN =		{urn:nbn:de:0030-drops-18851},
  doi =		{10.4230/DagSemProc.08492.2},
  annote =	{Keywords: Sparse signal representation, optimal signal reconstruction, approximation, compression}
}
Document
DOI: 10.4230/DagSemProc.06391.1
06391 Abstracts Collection – Algorithms and Complexity for Continuous Problems

Authors: Stephan Dahlke, Klaus Ritter, Ian H. Sloan, and Joseph F. Traub

Published in: Dagstuhl Seminar Proceedings, Volume 6391, Algorithms and Complexity for Continuous Problems (2007)

Abstract
From 24.09.06 to 29.09.06, the Dagstuhl Seminar 06391 ``Algorithms and Complexity for Continuous Problems'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
Cite as

Stephan Dahlke, Klaus Ritter, Ian H. Sloan, and Joseph F. Traub. 06391 Abstracts Collection – Algorithms and Complexity for Continuous Problems. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 6391, pp. 1-21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{dahlke_et_al:DagSemProc.06391.1,
  author =	{Dahlke, Stephan and Ritter, Klaus and Sloan, Ian H. and Traub, Joseph F.},
  title =	{{06391 Abstracts Collection – Algorithms and Complexity for Continuous Problems}},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  pages =	{1--21},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{6391},
  editor =	{Stephan Dahlke and Klaus Ritter and Ian H. Sloan and Joseph F. Traub},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06391.1},
  URN =		{urn:nbn:de:0030-drops-8782},
  doi =		{10.4230/DagSemProc.06391.1},
  annote =	{Keywords: Computational complexity, partial information, high-dimensional problems, operator equations, non-linear approximation, quantum computation, stochastic computation, ill posed-problems}
}
Document
DOI: 10.4230/DagSemProc.04401.12
Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings

Authors: Erich Novak, Stephan Dahlke, and Winfried Sickel

Published in: Dagstuhl Seminar Proceedings, Volume 4401, Algorithms and Complexity for Continuous Problems (2005)

Abstract
We study the optimal approximation of the solution of an operator equation Au=f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs on an arbitrary bounded Lipschitz domain. Here we prove that approximation by linear mappings is as good as the best n-term approximation with respect to an optimal Riesz basis. Our results are concerned with approximation, not with computation. Our goal is to understand better the possibilities of nonlinear approximation.
Cite as

Erich Novak, Stephan Dahlke, and Winfried Sickel. Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 4401, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{novak_et_al:DagSemProc.04401.12,
  author =	{Novak, Erich and Dahlke, Stephan and Sickel, Winfried},
  title =	{{Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings}},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4401},
  editor =	{Thomas M\"{u}ller-Gronbach and Erich Novak and Knut Petras and Joseph F. Traub},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04401.12},
  URN =		{urn:nbn:de:0030-drops-1471},
  doi =		{10.4230/DagSemProc.04401.12},
  annote =	{Keywords: elliptic operator equation , worst case error , linear approximation method , nonlinear approximation method , best n-term approximation Bernstein widths , manifold widths}
}
Document
DOI: 10.4230/DagSemProc.04401.13
Optimal Approximation of Elliptic Problems II: Wavelet Methods

Authors: Stephan Dahlke, Erich Novak, and Winfried Sickel

Published in: Dagstuhl Seminar Proceedings, Volume 4401, Algorithms and Complexity for Continuous Problems (2005)

Abstract
This talk is concerned with optimal approximations of the solutions of elliptic boundary value problems. After briefly recalling the fundamental concepts of optimality, we shall especially discuss best n-term approximation schemes based on wavelets. We shall mainly be concerned with the Poisson equation in Lipschitz domains. It turns out that wavelet schemes are suboptimal in general, but nevertheless they are superior to the usual uniform approximation methods. Moreover, for specific domains, i.e., for polygonal domains, wavelet methods are in fact optimal. These results are based on regularity estimates of the exact solution in a specific scale of Besov spaces.
Cite as

Stephan Dahlke, Erich Novak, and Winfried Sickel. Optimal Approximation of Elliptic Problems II: Wavelet Methods. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 4401, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{dahlke_et_al:DagSemProc.04401.13,
  author =	{Dahlke, Stephan and Novak, Erich and Sickel, Winfried},
  title =	{{Optimal Approximation of Elliptic Problems II: Wavelet Methods}},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  pages =	{1--4},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4401},
  editor =	{Thomas M\"{u}ller-Gronbach and Erich Novak and Knut Petras and Joseph F. Traub},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04401.13},
  URN =		{urn:nbn:de:0030-drops-1381},
  doi =		{10.4230/DagSemProc.04401.13},
  annote =	{Keywords: Elliptic operator equations , worst case error , best n-term approximation , wavelets , Besov regularity}
}
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