Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915155916357632 |
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| author | Neufeld, Ariel Wu, Sizhou |
| author_facet | Neufeld, Ariel Wu, Sizhou |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_12545 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities Neufeld, Ariel Wu, Sizhou Numerical Analysis Analysis of PDEs Probability 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. |
| title | Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities |
| topic | Numerical Analysis Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2310.12545 |