Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Numerical simulations for full history recursive multilevel Picard
  approximations for systems of high-dimensional partial differential equations   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 $\epsilon > 0$ grows at most polynomially in both the reciprocal
$1/\epsilon$ of the required accuracy and the dimension $d \in \mathbb{N}$ 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. The presented numerical simulation results suggest in
the case of each of these example PDEs that the proposed MLP approximation
method produces very accurate results in short runtimes and, in particular, the
presented numerical simulation results indicate that the proposed MLP
approximation scheme significantly outperforms certain deep learning based
approximation methods for high-dimensional semilinear PDEs.

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The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin