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

journal of scientific computing

On multilevel Picard numerical approximations for high-dimensional  nonlinear parabolic partial differential equations and high-dimensional  nonlinear backward stochastic differential equations