Full history recursive multilevel Picard approximations (MLP for short) are a method to approximate high-dimensional semilinear PDEs. The MLP method shows great performance in numerical simulations and is the only approximation scheme that has been proven to overcome the curse of dimensionality for a class of semilinear PDEs with general time horizon and Lipschitz nonlinearities. The MLP method was introduced in Partial differential equations and applications 2021 and in Proceedings of the Royal Society A 2020. Interested readers could start with Proceedings of the Royal Society A 2020.

Simulations:

The MLP method shows great performance in simulations; see ArXiv 2020 for 4 example PDEs. One of these simulated PDEs is the semilinear Black-Scholes PDE

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The following figure shows an approximative log-log-plot of the relative error of the MLP approximation algorithm against the computational effort of the algorithm in the case of this Black-Scholes PDE. The reference solution is an MLP approximation with higher accuracy. Theorem 1.1 in EJP 2020 guarantees that MLP approximations converge with rate 1/2- to the true solution of the above semilinear Black-Scholes PDE. The following figure empirically confirms this. Note that the computational effort grows only gradually in the dimension. 

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Machine learning methods might be less convincing. The following plot depicts the same simulations as above except that the reference solution is now a deep splitting approximation (from Beck, C., Becker, S., Cheridito, P., Jentzen, A., and Neufeld, A. Deep splitting method for parabolic PDEs. arXiv:1907.03452 (2019)). For more details on these plots see ArXiv 2020 .

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Research articles on MLP:

• Introduction of MLP with quadrature rules for time discretization and analysis of very smooth heat equations: Multilevel Picard iterations for solving smooth semilinear parabolic heat equations. ArXiv 2016. Partial differential equations and applications 2021. Authors: Weinan E, Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
• Simulations with Gauss-Legendre quadrature rule for time discretization show computational complexity 4+: On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations. ArXiv 2017. Journal of Scientific Computing 2019. Authors: Weinan E, Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
• Analysis of very smooth gradient-dependent heat equations:
  Multi-level Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities. ArXiv 2017 . Siam Journal on Numerical Analysis 2020. Authors: Martin Hutzenthaler, Thomas Kruse
• MLP with Monte Carlo for time discretization - Analysis of heat equations with globally Lipschitz nonlinearities: Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations. ArXiv 2018. Proceedings of the Royal Society A 2020. Authors: Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, Tuan Anh Nguyen, Philippe von Wurstemberger
• Semilinear heat equations can be approximated with Deep Neural Networks: A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations.  ArXiv 2019. SN PDEs and Applications 2020. Authors: Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, Tuan Anh Nguyen
• Nonlinear Black-Scholes equations: Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks. ArXiv 2019. EJP 2020. Authors: Martin Hutzenthaler, Arnulf Jentzen, Philippe von Wurstemberger
• Locally Lipschitz nonlinearities: Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full history recursive multilevel Picard approximations. ArXiv 2019. Journal of Numerical Analysis. Authors: Christian Beck, Fabian Hornung, Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
• Generalisation of MLP: Generalised multilevel Picard approximations. ArXiv 2019. Authors: Michael Giles, Arnulf Jentzen, Timo Welti
• Gradient-dependent nonlinearities: Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities. ArXiv 2019. FoCM. Authors: Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
• Elliptic PDEs: Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations. ArXiv 2020. Authors: Christian Beck, Lukas Gonon, Arnulf Jentzen
• Simulations with computational complexity 2+: Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations. ArXiv 2020. Commun. Comput. Phys 2020Authors: Sebastian Becker, Ramon Braunwarth, Martin Hutzenthaler, Arnulf Jentzen, Philippe von Wurstemberger
• Semilinear PDEs with globally Lipschitz coefficients: Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities. ArXiv 2020. Authors:  Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, Tuan Anh Nguyen
• ODEs with expectations: Full history recursive multilevel Picard approximations for ordinary differential equations with expectations.  ArXiv 2021 . Authors: Christian Beck, Martin Hutzenthaler, Arnulf Jentzen, Emilia Magnan
• HJB equations can be approximated with deep neural networks:  Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations. ArXiv 2021 Authors: Philipp Grohs, Lukas Herrmann
• McKean-Vlasov SDEs:  Multilevel Picard approximations for McKean-Vlasov stochastic differential equations. ArXiv 2021.  JMAA 2021. Authors: Martin Hutzenthaler, Thomas Kruse, Tuan Anh Nguyen
• BSDEs: Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations. ArXiv 2021. Authors: Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, Tuan Anh Nguyen
• Convergence in L^p:  Strong L^p-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations. ArXiv 2021. Authors: Martin Hutzenthaler, Arnulf Jentzen, Benno Kuckuck, Joshua Lee Padgett
• Non-globally Lipschitz coefficients: Forward SDE with locally monotone coefficients. ArXiv 2022.  https://doi.org/10.1016/j.apnum.2022.05.009. Authors: Martin Hutzenthaler, Tuan Anh Nguyen
• Decoupled FBSDEs: ArXiv 2022. Authors: Martin Hutzenthaler, Tuan Anh Nguyen
• Semilinear PIDEs:  Multilevel Picard approximations for high-dimensional semilinear partial integro-differential equations. ArXiv 2022. Authors: Ariel Neufeld, Sizhou Wu
• Semilinear PDEs can be approximated with Deep Neural Networks: A proof that ReLu DNNs overcome the curse of dimensionality in the numerical approximation of semilinear PDEs.  ArXiv 2022.  Authors: Martin Hutzenthaler, Tuan Anh Nguyen

Related research articles:

• Stochastic fixed point equations: On existence and uniqueness properties for solutions of stochastic fixed point equations. ArXiv 2019. DCDS-B 2021Authors Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen.
• Viscosity solutions of semilinear PDEs: On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations. ArXiv 2020.  Stochastics and Dynamics 2021Authors: Christian Beck, Martin Hutzenthaler, Arnulf Jentzen
• Overview of deep learning methods: An overview on deep learning-based approximation methods for partial differential equations ArXiv 2020. Authors: Christian Beck, Martin Hutzenthaler, Arnulf Jentzen, Benno Kuckuck
• Speed of convergence of Picard iterations: On the speed of convergence of Picard iterations of backward stochastic differential equations. ArXiv 2021. Authors: Martin Hutzenthaler, Thomas Kruse, Tuan Anh Nguyen

Source codes:

• Pseudo code
• C++ source code from ArXiv 2020 with which 4 example PDEs can be simulated.

Last update: June 13, 2022. Please send missing articles to martin.hutzenthaler AT uni-due.de