# Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities

@article{Hutzenthaler2020MultilevelPA, title={Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities}, author={Martin Hutzenthaler and Arnulf Jentzen and Thomas Kruse and Tuan Anh Nguyen}, journal={ArXiv}, year={2020}, volume={abs/2009.02484} }

The recently introduced full-history recursive multilevel Picard (MLP) approximation methods have turned out to be quite successful in the numerical approximation of solutions of high-dimensional 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 second-order PDEs in the sense that the number of computational operations of the…

## 12 Citations

### Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning

- Computer ScienceNonlinearity
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It is demonstrated 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.

### Full history recursive multilevel Picard approximations for ordinary differential equations with expectations

- Mathematics, Computer ScienceArXiv
- 2021

This work shows for every δ > 0 that the proposed MLP approximation algorithm requires only a computational effort of order ε to achieve a root-mean-square error of size ε.

### An overview on deep learning-based approximation methods for partial differential equations

- Computer Science, MathematicsDiscrete and Continuous Dynamical Systems - B
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An introduction to this area of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ideas of their proofs is provided.

### Multilevel Picard approximations for high-dimensional decoupled forward-backward stochastic differential equations

- MathematicsArXiv
- 2022

Backward stochastic differential equations (BSDEs) appear in numeruous applications. Classical approximation methods suffer from the curse of dimensionality and deep learning-based approximation…

### Deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear partial differential equations

- Mathematics, Computer ScienceArXiv
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We prove that deep neural networks are capable of approximating solutions of semilinear Kolmogorov PDE in the case of gradient-independent, Lipschitz-continuous nonlinearities, while the required…

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- Mathematics, Computer ScienceApplied Numerical Mathematics
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- 2022

We present an algorithm for the numerical solution of nonlinear parabolic partial diﬀerential equations. This algorithm extends the classical Feynman-Kac formula to fully nonlinear partial…

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- Computer Science, MathematicsJournal of Numerical Mathematics
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This article introduces a new Monte Carlo-type numerical approximation method for high-dimensional BSDEs and proves that it does indeed overcome the curse of dimensionality in the approximative computation of solution paths ofBSDEs.

### Gradient boosting-based numerical methods for high-dimensional backward stochastic differential equations

- Computer Science, MathematicsAppl. Math. Comput.
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### On the speed of convergence of Picard iterations of backward stochastic differential equations

- MathematicsProbability, Uncertainty and Quantitative Risk
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It is a well-established fact in the scientiﬁc literature that Picard iterations of backward stochastic diﬀerential equations with globally Lipschitz continuous nonlinearity converge at least…

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