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Anisotropic Multiscale Systems on Bounded Domains

by P. Grohs and G. Kutyniok and J. Ma and P. Petersen

(Report number 2015-30)

Abstract
In this paper we provide a construction of multiscale systems on a bounded domain \(\Omega \subset \mathbb{R}^2\) coined boundary shearlet systems, which satisfy several properties advantageous for applications to imaging science and numerical analysis of partial differential equations. More precisely, we construct boundary shearlet systems that form frames for \(L^2(\Omega)\) with controllable frame bounds and admit optimally sparse approximations for functions, which are smooth apart from a curve-like discontinuit\pp{y}. Indeed, the constructed systems allow for boundary conditions, and characterize Sobolev spaces over \(\Omega\) by their analysis coefficients. Finally, we demonstrate numerically that these systems also constitute a Gelfand frame for \((H^s(\Omega), L^2(\Omega), H^{-s}(\Omega))\) for \(s \in \mathbb{N}\).

Keywords: Shearlets, Sobolev Frames on Domains, Gelfand Frames, Adaptive Algorithms for PDEs

@Techreport{GKMP15_620,
  author = {P. Grohs and G. Kutyniok and J. Ma and P. Petersen},
  title = {Anisotropic Multiscale Systems on Bounded Domains},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-30},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-30.pdf },
  year = {2015}
}

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