Research reports

Main content

Years: 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 

Multi-level Monte Carlo Finite Element method for parabolic stochastic partial differential equations

by A. Barth and A. Lang and Ch. Schwab

(Report number 2011-30)

Abstract
We analyze the convergence and complexity of multi-level Monte Carlo (MLMC) discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show, under regularity assumptions on the solution that are minimal under certain criteria, that the judicious combination of piecewise linear, continuous multi-level Finite Element discretizations in space and Euler--Maruyama discretizations in time yields mean square convergence of order one in space and of order $1/2$ in time to the expected value of the mild solution. The complexity of the multi-level estimator is shown to scale log-linearly with respect to the corresponding work to generate a single solution path on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh. Examples are provided for Lévy driven SPDEs as well as equations for randomly forced surface diffusions.

Keywords:

BibTeX
@Techreport{BLS11_95,
  author = {A. Barth and A. Lang and Ch. Schwab},
  title = {Multi-level Monte Carlo Finite Element method for parabolic stochastic partial differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-30},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-30.pdf },
  year = {2011}
}

Disclaimer
© The copyright for the following documents lies 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 accept any liability in this respect.

 
Page URL: https://www.math.ethz.ch/sam/research/reports.html
28.08.2016
© 2016 Eidgenössische Technische Hochschule Zürich