Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities
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Ariel Neufeld
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Sizhou Wu
Abstract
In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence and complexity analysis of our algorithm. To obtain our main results, we consider a particular stochastic fixed-point equation (SFPE) motivated by the Feynman–Kac representation and the Bismut–Elworthy–Li formula. We show that the PDE under consideration has a unique viscosity solution which coincides with the first component of the unique solution of the stochastic fixed-point equation. Moreover, the gradient of the unique viscosity solution of the PDE exists and coincides with the second component of the unique solution of the stochastic fixed-point equation. Furthermore, we also provide a numerical example in up to 300 dimensions to demonstrate the practical applicability of our multilevel Picard algorithm.
Funding source: Nanyang Technological University
Award Identifier / Grant number: Nanyang Assistant Professorship Grant (NAP Grant) Machine Learning based Algorithms in Finance and Insurance
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Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Use of Large Language Models, AI and Machine Learning Tools: None declared.
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Conflict of interest: The authors state no conflict of interest.
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Research funding: 1. The Nanyang Assistant Professorship Grant (NAP Grant) Machine Learning based Algorithms in Finance and Insurance. 2. The Fundamental Research Funds for the Central Universities (China).
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Data availability: The code for the MLP algorithm can be found here: https://github.com/SizhouWu/MLP_Gradient_Nonlinearity.
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