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

Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations

by W. E and M. Hutzenthaler and A. Jentzen and T. Kruse

(Report number 2017-43)

Abstract
We introduce, for the first time, a family of algorithms for solving general high-dimensional nonlinear parabolic partial differential equations with a polynomial complexity in both the dimensionality and the reciprocal of the accuracy requirement. The algorithm is obtained through a delicate combination of the Feynman-Kac and the Bismut-Elworthy-Li formulas, and an approximate decomposition of the Picard fixed-point iteration with multi-level accuracy. The algorithm has been tested on a variety of nonlinear partial differential equations that arise in physics and finance, with very satisfactory results. Analytical tools needed for the analysis of such algorithms, including a nonlinear Feynman-Kac formula, a new class of semi-norms and their recursive inequalities, are also introduced. They allow us to prove that for semi-linear heat equations, the computational complexity of the proposed algorithm is bounded by \(O(d\,\varepsilon^{-(4+\delta)})\) for any \(\delta > 0\), where \(d\) is the dimensionality of the problem and \(\varepsilon\in(0,\infty)\) is the prescribed accuracy.

Keywords:

@Techreport{EHJK17_739,
  author = {W. E and M. Hutzenthaler and A. Jentzen and T. Kruse},
  title = {Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-43},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-43.pdf },
  year = {2017}
}

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).