Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

hp-DG-QTT solution of high-dimensional degenerate diffusion equations

by V. Kazeev and O. Reichmann and Ch. Schwab

(Report number 2012-11)

Abstract
We consider the discretization of degenerate, time-inhomogeneous Fokker-Planck equations for diffusion problems in high-dimensional domains. Well-posedness of the problem in time-weighted Bochner spaces is established. Analytic regularity of the time-dependence of the solution in countably normed, weighted Sobolev spaces is established. Time discretization by the hp-discontinuous We consider the discretization of degenerate, time-inhomogeneous Fokker-Planck equations for diffusion problems in high-dimensional domains. Well-posedness of the problem in time-weighted Bochner spaces is established. Analytic regularity of the time-dependence of the solution in countably normed, weighted Sobolev spaces is established. We consider the discretization of degenerate, time-inhomogeneous Fokker-Planck equations for diffusion problems in high-dimensional domains. Well-posedness of the problem in time-weighted Bochner spaces is established. Analytic regularity of the time-dependence of the solution in countably normed, weighted Sobolev spaces is established. Time discretization by the hp-discontinuous Galerkin method is shown to converge exponentially. The resulting elliptic spatial problems are discretized with the use of the tensor-product "hat" finite elements constructed on uniform or patch-wise uniform (Shishkin) meshes and are solved in the Quantized Tensor Train representation. For numerical experiments we consider compatible and incompatible initial data in up to 40 and 18 dimensions respectively on a workstation.

Keywords: Fokker-Planck equation, degenerate diffusion, Gevrey regularity, hp-discontinuous Galerkin, time stepping, low-rank representation, Tensor Train (TT), Quantized Tensor Train (QTT)

@Techreport{KRS12_454,
  author = {V. Kazeev and O. Reichmann and Ch. Schwab},
  title = {hp-DG-QTT solution of high-dimensional degenerate diffusion equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-11},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-11.pdf },
  year = {2012}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).