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Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

by M. Hutzenthaler and A. Jentzen and Th. Kruse and T. Nguyen and Ph. von Wurstemberger

(Report number 2019-46)

Abstract
For a long time it is well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows polynomially both in the dimension and in the reciprocal of the required accuracy.

Keywords: curse of dimensionality, high-dimensional PDEs, information based complexity, tractability of multivariate problems, high-dimensional semilinear BSDEs, multilevel Picard approximations, multilevel Monte Carlo method

@Techreport{HJKNv19_850,
  author = {M. Hutzenthaler and A. Jentzen and Th. Kruse and T. Nguyen and            Ph.  von Wurstemberger},
  title = {Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-46},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-46.pdf },
  year = {2019}
}

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