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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}
}

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