@Article{CiCP-28-5,
author = {Sebastian, Becker and Ramon, Braunwarth and Martin, Hutzenthaler and Jentzen, Arnulf and von, Wurstemberger, Philippe},
title = {Numerical Simulations for Full History Recursive Multilevel Picard Approximations for Systems of High-Dimensional Partial Differential Equations},
journal = {Communications in Computational Physics},
year = {2020},
volume = {28},
number = {5},
pages = {2109--2138},
abstract = {<p style="text-align: justify;">One of the most challenging issues in applied mathematics is to develop
and analyze algorithms which are able to approximately compute solutions of high-dimensional nonlinear partial differential equations (PDEs). In particular, it is very
hard to develop approximation algorithms which do not suffer under the curse of dimensionality in the sense that the number of computational operations needed by the
algorithm to compute an approximation of accuracy $ε$&gt;0 grows at most polynomially
in both the reciprocal 1/$ε$ of the required accuracy and the dimension $d∈\mathbb{N}$&nbsp;of the PDE.
Recently, a new approximation method, the so-called full history recursive multilevel Picard (MLP) approximation method, has been introduced and, until today, this approximation scheme is the only approximation method in the scientific literature which has
been proven to overcome the curse of dimensionality in the numerical approximation
of semilinear PDEs with general time horizons. It is a key contribution of this article
to extend the MLP approximation method to systems of semilinear PDEs and to numerically test it on several example PDEs. More specifically, we apply the proposed
MLP approximation method in the case of Allen-Cahn PDEs, Sine-Gordon-type PDEs,
systems of coupled semilinear heat PDEs, and semilinear Black-Scholes PDEs in up to
1000 dimensions. We also compare the performance of the proposed MLP approximation algorithm with a deep learning based approximation method from the scientific
literature.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2020-0130},
url = {https://global-sci.com/article/79750/numerical-simulations-for-full-history-recursive-multilevel-picard-approximations-for-systems-of-high-dimensional-partial-differential-equations}
}