  Parabolic partial differential equations (PDEs) and backward stochastic
differential equations (BSDEs) are key ingredients in a number of models in
physics and financial engineering. In particular, parabolic PDEs and BSDEs are
fundamental tools in the state-of-the-art pricing and hedging of financial
derivatives. The PDEs and BSDEs appearing in such applications are often
high-dimensional and nonlinear. Since explicit solutions of such PDEs and BSDEs
are typically not available, it is a very active topic of research to solve
such PDEs and BSDEs approximately. In the recent article [E, W., Hutzenthaler,
M., Jentzen, A., and Kruse, T. Linear scaling algorithms for solving
high-dimensional nonlinear parabolic differential equations. arXiv:1607.03295
(2017)] we proposed a family of approximation methods based on Picard
approximations and multilevel Monte Carlo methods and showed under suitable
regularity assumptions on the exact solution for semilinear heat equations that
the computational complexity is bounded by $O( d \, \epsilon^{-(4+\delta)})$
for any $\delta\in(0,\infty)$, where $d$ is the dimensionality of the problem
and $\epsilon\in(0,\infty)$ is the prescribed accuracy. In this paper, we test
the applicability of this algorithm on a variety of $100$-dimensional nonlinear
PDEs that arise in physics and finance by means of numerical simulations
presenting approximation accuracy against runtime. The simulation results for
these 100-dimensional example PDEs are very satisfactory in terms of accuracy
and speed. In addition, we also provide a review of other approximation methods
for nonlinear PDEs and BSDEs from the literature.
