|Preprint-No.:||< 388 >||Published in:||March 2014||PDF-File:||IGPM388.pdf|
|Title:||Efficient Resolution of Anisotropic Structures|
|Authors:||Wolfgang Dahmen, Chunyan Huang, Gitta Kutyniok, Wang-Q Lim, Christoph Schwab, Gerrit Welper|
We highlight some recent new delevelopments 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 L2 . 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.
|Keywords:||Shearlets, anisotropic meshes, parametric transport equations, Petrov-Galerkin formulations, δ-proximality, high-dimensional problems, adaptivity, reduced basis methods, sparse tensor interpolation and approximation|
|Publication:||Lecture Notes in Computational Science and Engineering|
102, 25-51 (2014)