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Extraction of Quantifiable Information from Complex Systems pp 25–51Cite as

Efficient Resolution of Anisotropic Structures

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 102))

Abstract

We highlight some results obtained in the DFG-SPP project “Adaptive Anisotropic Discretization Concepts”. We focus on new developments concerning the sparse representation of possibly high-dimensional functions exhibiting strong anisotropic features and low regularity in isotropic Sobolev or Besov scales. Specifically, we focus on the solution of transport equations which exhibit propagation of singularities where, additionally, high-dimensionality enters when the convection field, and hence the solutions, depend on parameters varying over some compact set. Important constituents of our approach are directionally adaptive discretization concepts motivated by compactly supported shearlet systems, and well-conditioned stable variational formulations that support trial spaces with anisotropic refinements with arbitrary directionalities. We prove that they provide tight error-residual relations which are used to contrive rigorously founded adaptive refinement schemes which converge in L 2. Moreover, in the context of parameter dependent problems we discuss two approaches serving different purposes and working under different regularity assumptions. For “frequent query problems”, making essential use of the novel well-conditioned variational formulations, a new Reduced Basis Method is outlined which exhibits a certain rate-optimal performance for indefinite, unsymmetric or singularly perturbed problems. For the radiative transfer problem with scattering a sparse tensor method is presented which mitigates or even overcomes the curse of dimensionality under suitable (so far still isotropic) regularity assumptions. Numerical examples for both methods illustrate the theoretical findings.

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Author information

Authors and Affiliations

  1. RWTH Aachen, Templergraben 55, 52056, Aachen, Germany

    Wolfgang Dahmen

  2. School of Applied Mathematics, Central University of Finance and Economics, 39 South College Road, Haidian District Beijing, 100081, P.R. China

    Chunyan Huang

  3. Technical University of Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Gitta Kutyniok & Wang-Q Lim

  4. ETH Zürich, HG G 57.1, Rämistr. 101, 8092, Zürich, Switzerland

    Christoph Schwab

  5. Texas A & M University, Mailstop 3368, College Station, TX, 77843-3368, USA

    Gerrit Welper

Authors
  1. Wolfgang Dahmen
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  2. Chunyan Huang
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  3. Gitta Kutyniok
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  4. Wang-Q Lim
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  5. Christoph Schwab
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  6. Gerrit Welper
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Corresponding author

Correspondence to Wolfgang Dahmen .

Editor information

Editors and Affiliations

  1. FB 12 Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Stephan Dahlke

  2. Inst. Geometrie & Praktische Mathemathik, RWTH Aachen University, Aachen, Germany

    Wolfgang Dahmen

  3. Universität Bonn Institut für Numerische Simulation, Bonn, Germany

    Michael Griebel

  4. Max-Planck-Inst. f. Mathem. in d. Naturwissenschaften, Leipzig, Sachsen, Germany

    Wolfgang Hackbusch

  5. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Klaus Ritter

  6. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Reinhold Schneider

  7. Seminar für Angewandte Mathematik, ETH Zürich, Zürich, Switzerland

    Christoph Schwab

  8. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Harry Yserentant

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Dahmen, W., Huang, C., Kutyniok, G., Lim, WQ., Schwab, C., Welper, G. (2014). Efficient Resolution of Anisotropic Structures. In: Dahlke, S., et al. Extraction of Quantifiable Information from Complex Systems. Lecture Notes in Computational Science and Engineering, vol 102. Springer, Cham. https://doi.org/10.1007/978-3-319-08159-5_2

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