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Documents authored by Novak, Erich

Document
DOI: 10.4230/DagSemProc.04401.1
04401 Abstracts Collection – Algorithms and Complexity for Continuous

Authors: Thomas Müller-Gronbach, Erich Novak, Knut Petras, and Joseph F. Traub

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

Abstract
From 26.09.04 to 01.10.04, the Dagstuhl Seminar ``Algorithms and Complexity for Continuous Problems'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. 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

Thomas Müller-Gronbach, Erich Novak, Knut Petras, and Joseph F. Traub. 04401 Abstracts Collection – Algorithms and Complexity for Continuous. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 4401, pp. 1-21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{mullergronbach_et_al:DagSemProc.04401.1,
  author =	{M\"{u}ller-Gronbach, Thomas and Novak, Erich and Petras, Knut and Traub, Joseph F.},
  title =	{{04401 Abstracts Collection – Algorithms and Complexity for Continuous}},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  pages =	{1--21},
  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-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.04401.1},
  URN =		{urn:nbn:de:0030-drops-1545},
  doi =		{10.4230/DagSemProc.04401.1},
  annote =	{Keywords: Complexity and regularization of ill-posed problems , nonlinear approximation , tractability of high-dimensional numerical problems quantum computing , stochastic computation and quantization , global optimization , differential and integral equations}
}
Document
DOI: 10.4230/DagSemProc.04401.2
04401 Summary – Algorithms and Complexity for Continuous Problems

Authors: Thomas Müller-Gronbach, Erich Novak, Knut Petras, and Joseph F. Traub

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

Abstract
The goal of this workshop was to bring together researchers from different communities working on computational aspects of continuous problems. Continuous computational problems arise in many areas of science and engineering. Examples include path and multivariate integration, function approximation, optimization, as well as differential, integral, and operator equations. Understanding the complexity of such problems and constructing efficient algorithms is both important and challenging. The workshop was of a very interdisciplinary nature with invitees from, e.g., computer science, numerical analysis, discrete, applied, and pure mathematics, physics, statistics, and scientific computation. Many of the lectures were presented by Ph.D. students. Compared to earlier meetings, several very active research areas received more emphasis. These include Quantum Computing, Complexity and Tractability of high-dimensional problems, Stochastic Computation, and Quantization, which was an entirely new field for this workshop. Due to strong connections between the topics treated at this workshop many of the participants initiated new cooperations and research projects. For more details, see the pdf-file with the same title.
Cite as

Thomas Müller-Gronbach, Erich Novak, Knut Petras, and Joseph F. Traub. 04401 Summary – Algorithms and Complexity for Continuous Problems. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 4401, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{mullergronbach_et_al:DagSemProc.04401.2,
  author =	{M\"{u}ller-Gronbach, Thomas and Novak, Erich and Petras, Knut and Traub, Joseph F.},
  title =	{{04401 Summary – Algorithms and Complexity for Continuous Problems}},
  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-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.04401.2},
  URN =		{urn:nbn:de:0030-drops-1530},
  doi =		{10.4230/DagSemProc.04401.2},
  annote =	{Keywords: Complexity and Regularization of Ill-Posed Problems , Non-Linear Approximation , Tractability of High-Dimensional Numerical Problems Quasi-Monte Carlo Methods , Quantum Computing , Stochastic Computation and Quantization , Global Optimization , Differential and Integral Equation}
}
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-dev.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-dev.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}
}
Document
DOI: 10.4230/DagSemRep.50
Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 9242)

Authors: Erich Novak, Steve Smale, and Joseph F. Traub

Published in: Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Abstract
Cite as

Erich Novak, Steve Smale, and Joseph F. Traub. Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 9242). Dagstuhl Seminar Report 50, pp. 1-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (1992)

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@TechReport{novak_et_al:DagSemRep.50,
  author =	{Novak, Erich and Smale, Steve and Traub, Joseph F.},
  title =	{{Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 9242)}},
  pages =	{1--24},
  ISSN =	{1619-0203},
  year =	{1992},
  type = 	{Dagstuhl Seminar Report},
  number =	{50},
  institution =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemRep.50},
  URN =		{urn:nbn:de:0030-drops-149383},
  doi =		{10.4230/DagSemRep.50},
}
Document
DOI: 10.4230/DagSemRep.11
Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 9116)

Authors: Erich Novak, Josef F. Traub, and Henryk Wozniakowski

Published in: Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Abstract
Cite as

Erich Novak, Josef F. Traub, and Henryk Wozniakowski. Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 9116). Dagstuhl Seminar Report 11, pp. 1-28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (1991)

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@TechReport{novak_et_al:DagSemRep.11,
  author =	{Novak, Erich and Traub, Josef F. and Wozniakowski, Henryk},
  title =	{{Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 9116)}},
  pages =	{1--28},
  ISSN =	{1619-0203},
  year =	{1991},
  type = 	{Dagstuhl Seminar Report},
  number =	{11},
  institution =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemRep.11},
  URN =		{urn:nbn:de:0030-drops-148998},
  doi =		{10.4230/DagSemRep.11},
}
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