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**Published in:** Dagstuhl Reports, Volume 9, Issue 8 (2020)

From 18.08. to 23.08.2019, the Dagstuhl Seminar 19341 Algorithms and Complexity for Continuous Problems was held in the International Conference and Research Center (LZI), 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 can be found in this report. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.

Dmitriy Bilyk, Aicke Hinrichs, Frances Y. Kuo, and Klaus Ritter. Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 19341). In Dagstuhl Reports, Volume 9, Issue 8, pp. 26-48, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{bilyk_et_al:DagRep.9.8.26, author = {Bilyk, Dmitriy and Hinrichs, Aicke and Kuo, Frances Y. and Ritter, Klaus}, title = {{Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 19341)}}, pages = {26--48}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2019}, volume = {9}, number = {8}, editor = {Bilyk, Dmitriy and Hinrichs, Aicke and Kuo, Frances Y. and Ritter, Klaus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.9.8.26}, URN = {urn:nbn:de:0030-drops-116835}, doi = {10.4230/DagRep.9.8.26}, annote = {Keywords: applied harmonic analysis, computational stochastics, computing and complexity in infinite dimensions, discrepancy theory, tractability analysis} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9391, Algorithms and Complexity for Continuous Problems (2009)

Prices of path dependent options may be modeled as expectations of functions of an infinite sequence of real variables. This talk presents recent work on bounding the error of such expectations using quasi-Monte Carlo algorithms. The expectation is approximated by an average of $n$ samples, and the functional of an infinite number of variables is approximated by a function of only $d$ variables. A multilevel algorithm employing a sum of sample averages, each with different truncated dimensions, $d_l$, and different sample sizes, $n_l$, yields faster convergence than a single level algorithm. This talk presents results in the worst-case error setting.

Fred J. Hickernell, Thomas Müller-Gronbach, Ben Niu, and Klaus Ritter. Evaluating Expectations of Functionals of Brownian Motions: a Multilevel Idea. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 9391, pp. 1-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{hickernell_et_al:DagSemProc.09391.3, author = {Hickernell, Fred J. and M\"{u}ller-Gronbach, Thomas and Niu, Ben and Ritter, Klaus}, title = {{Evaluating Expectations of Functionals of Brownian Motions: a Multilevel Idea}}, booktitle = {Algorithms and Complexity for Continuous Problems}, pages = {1--19}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2009}, volume = {9391}, editor = {Thomas M\"{u}ller-Gronbach and Leszek Plaskota 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.09391.3}, URN = {urn:nbn:de:0030-drops-22987}, doi = {10.4230/DagSemProc.09391.3}, annote = {Keywords: Brownian motions, multilevel, option pricing, worst-case error} }

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

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.

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} }

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

We study algorithms for approximation of the mild solution
of stochastic heat equations on the spatial domain ]0,1[^d.
The error of an algorithm is defined in L_2-sense.
We derive lower bounds for the error of every algorithm
that uses a total of N evaluations of one-dimensional components
of the driving Wiener process W. For equations with additive
noise we derive matching upper bounds and we construct
asymptotically optimal algorithms. The error bounds depend on
N and d, and on the decay of eigenvalues of the covariance of W
in the case of nuclear noise. In the latter case the use of
non-uniform time discretizations is crucial.

Klaus Ritter and Thomas Müller-Gronbach. Lower Bounds and Non-Uniform Time Discretization for Approximation of Stochastic Heat Equations. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 4401, pp. 1-37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{ritter_et_al:DagSemProc.04401.6, author = {Ritter, Klaus and M\"{u}ller-Gronbach, Thomas}, title = {{Lower Bounds and Non-Uniform Time Discretization for Approximation of Stochastic Heat Equations}}, booktitle = {Algorithms and Complexity for Continuous Problems}, pages = {1--37}, 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.04401.6}, URN = {urn:nbn:de:0030-drops-1518}, doi = {10.4230/DagSemProc.04401.6}, annote = {Keywords: Stochastic heat equation , Non-uniform time discretization , minimal errors , upper and lower bounds} }

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

We consider linear parabolic initial value
problems of second order in several dimensions.
The initial condition is supposed to be fixed
and we investigate the comutational complexity if
the coefficients of the parabolic equations
may vary in certain function spaces. Using the
parametrix method (or Neumann series), we prove
that lower bounds for the error of numerical
methods are related to lower bounds for
integration problems. On the other hand,
approximating the Neumann series with Smolyak's
method, we show that the problem is not much
harder than a certain approximation problem. For
HÃƒÂ¶lder classes on compact sets, e.g., lower and
upper bounds are close together, such that we have
an almost optimal method.

Knut Petras and Klaus Ritter. On the Complexity of Parabolic Initial Value Problems with Variable Drift. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 4401, pp. 1-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{petras_et_al:DagSemProc.04401.10, author = {Petras, Knut and Ritter, Klaus}, title = {{On the Complexity of Parabolic Initial Value Problems with Variable Drift}}, booktitle = {Algorithms and Complexity for Continuous Problems}, pages = {1--24}, 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.04401.10}, URN = {urn:nbn:de:0030-drops-1495}, doi = {10.4230/DagSemProc.04401.10}, annote = {Keywords: Partial differential equations , parabolic problems , Smolyak method , optimal methods} }

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

Leszek Plakote, Klaus Ritter, Ian H. Sloan, and Joseph F. Traub. Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 02401). Dagstuhl Seminar Report 356, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2003)

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@TechReport{plakote_et_al:DagSemRep.356, author = {Plakote, Leszek and Ritter, Klaus and Sloan, Ian H. and Traub, Joseph F.}, title = {{Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 02401)}}, pages = {1--18}, ISSN = {1619-0203}, year = {2003}, type = {Dagstuhl Seminar Report}, number = {356}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.356}, URN = {urn:nbn:de:0030-drops-152361}, doi = {10.4230/DagSemRep.356}, }

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

Peter Mathé, Klaus Ritter, and Joseph F. Traub. Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 98201). Dagstuhl Seminar Report 211, pp. 1-25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (1998)

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@TechReport{mathe_et_al:DagSemRep.211, author = {Math\'{e}, Peter and Ritter, Klaus and Traub, Joseph F.}, title = {{Algorithms and Complexity for Continuous Problems (Dagstuhl Seminar 98201)}}, pages = {1--25}, ISSN = {1619-0203}, year = {1998}, type = {Dagstuhl Seminar Report}, number = {211}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.211}, URN = {urn:nbn:de:0030-drops-150972}, doi = {10.4230/DagSemRep.211}, }

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