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

Multilevel Monte Carlo FEM for Elliptic PDEs with Besov Random Tree Priors

by Ch. Schwab and A. Stein

(Report number 2022-10)

Abstract
We develop a multilevel Monte Carlo (MLMC) FEM algorithm for linear, elliptic diffusion problems in polytopal domain \(\mathcal D\subset \mathbb R^d\), with Besov-tree random coefficients. This is to say that the logarithms of the diffusion coefficients are sampled from so-called Besov-tree priors, which have recently been proposed to model data for fractal phenomena in science and engineering. Numerical analysis of the fully discrete FEM for the elliptic PDE includes quadrature approximation and must account for a) nonuniform pathwise upper and lower coefficient bounds, and for b) low path-regularity of the Besov-tree coefficients. Admissible non-parametric random coefficients correspond to random functions exhibiting singularities on random fractals with tunable fractal dimension, but involve no a-priori specification of the fractal geometry of singular supports of sample paths. Optimal complexity and convergence rate estimates for quantities of interest and for their second moments are proved. A convergence analysis for MLMC-FEM is performed which yields choices of the algorithmic steering parameters for efficient implementation. A complexity (``error vs work'') analysis of the MLMC-FEM approximations is provided.

Keywords: Besov prior, Galton-Watson tree, Fractal models, Elliptic PDE with random coefficient, Finite element discretization, Multilevel Monte Carlo

@Techreport{SS22_998,
  author = {Ch. Schwab and A. Stein},
  title = {Multilevel Monte Carlo FEM for Elliptic PDEs with Besov Random Tree Priors},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-10},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-10.pdf },
  year = {2022}
}

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).