Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning. (English) Zbl 1490.60202
Summary: In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the deep backward stochastic differential equations method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future.
MSC:
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 65C05 | Monte Carlo methods |
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 65M75 | Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs |
Keywords:
partial differential equations; high dimension; nonlinear Monte Carlo; deep learning; backward stochastic differential equationsReferences:
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