Multilevel Picard approximations for highdimensional semilinear secondorder PDEs with Lipschitz nonlinearities
Abstract
The recently introduced fullhistory recursive multilevel Picard (MLP) approximation methods have turned out to be quite successful in the numerical approximation of solutions of highdimensional nonlinear PDEs. In particular, there are mathematical convergence results in the literature which prove that MLP approximation methods do overcome the curse of dimensionality in the numerical approximation of nonlinear secondorder PDEs in the sense that the number of computational operations of the proposed MLP approximation method grows at most polynomially in both the reciprocal $1/\epsilon$ of the prescribed approximation accuracy $\epsilon>0$ and the PDE dimension $d\in \mathbb{N}=\{1,2,3, \ldots\}$. However, in each of the convergence results for MLP approximation methods in the literature it is assumed that the coefficient functions in front of the secondorder differential operator are affine linear. In particular, until today there is no result in the scientific literature which proves that any semilinear secondorder PDE with a general time horizon and a non affine linear coefficient function in front of the secondorder differential operator can be approximated without the curse of dimensionality. It is the key contribution of this article to overcome this obstacle and to propose and analyze a new type of MLP approximation method for semilinear secondorder PDEs with possibly nonlinear coefficient functions in front of the secondorder differential operators. In particular, the main result of this article proves that this new MLP approximation method does indeed overcome the curse of dimensionality in the numerical approximation of semilinear secondorder PDEs.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 arXiv:
 arXiv:2009.02484
 Bibcode:
 2020arXiv200902484H
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Probability