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Sparse twisted tensor frame discretization of parametric transport operators

by P. Grohs and Ch. Schwab

(Report number 2011-41)

Abstract
We propose a novel family of frame discretizations for linear, high-dimensional parametric transport operators. Our approach is based on a least squares formulation in the phase space associated with the transport equation and by subsequent Galerkin discretization with a novel, sparse tensor product frame construction in the possibly high-dimensional phase space. The proposed twisted tensor product frame construction exploits invariance properties of the parameter space under certain group actions and accounts for propagation of singularities. Specically, invariance of the parametric transport operator under rotations of the transport direction. We prove convergence rates of the proposed least squares Galerkin discretizations associated with the twisted tensor frames in terms of the number of degrees of freedom. In particular, sparse versions of the twisted tensor frame constructions are proved to break the curse of dimensionality, also for solution classes with low regularity in isotropic Sobolev spaces due to propagating singularities, uniformly with respect to the propagation directions.

Keywords:

@Techreport{GS11_90,
  author = {P. Grohs and Ch. Schwab},
  title = {Sparse twisted tensor frame discretization of parametric transport operators},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-41},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-41.pdf },
  year = {2011}
}

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