A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910144176062464 |
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| author | Bågmark, Kasper Andersson, Adam Larsson, Stig Rydin, Filip |
| author_facet | Bågmark, Kasper Andersson, Adam Larsson, Stig Rydin, Filip |
| contents | A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established, theoretically under a parabolic Hörmander condition, and empirically in numerical examples. In a prediction step, between the noisy and partial measurements at discrete times, the scheme approximates the Fokker--Planck equation with a deep splitting scheme, followed by an exact update through Bayes' formula. This results in a classical prediction-update filtering algorithm that operates online for new observation sequences post-training. The algorithm employs a sampling-based Feynman--Kac approach, designed to mitigate the curse of dimensionality. As a corollary we obtain the convergence rate for the approximation of the Fokker--Planck equation alone, disconnected from the filtering problem. The convergence analysis is complemented by a nonlinear $10$-dimensional numerical example demonstrating the robustness of the method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14585 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting Bågmark, Kasper Andersson, Adam Larsson, Stig Rydin, Filip Numerical Analysis Probability Computation Machine Learning 60G25, 60G35, 62F15, 62G07, 62M20, 65C30, 65M75, 68T07 A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established, theoretically under a parabolic Hörmander condition, and empirically in numerical examples. In a prediction step, between the noisy and partial measurements at discrete times, the scheme approximates the Fokker--Planck equation with a deep splitting scheme, followed by an exact update through Bayes' formula. This results in a classical prediction-update filtering algorithm that operates online for new observation sequences post-training. The algorithm employs a sampling-based Feynman--Kac approach, designed to mitigate the curse of dimensionality. As a corollary we obtain the convergence rate for the approximation of the Fokker--Planck equation alone, disconnected from the filtering problem. The convergence analysis is complemented by a nonlinear $10$-dimensional numerical example demonstrating the robustness of the method. |
| title | A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting |
| topic | Numerical Analysis Probability Computation Machine Learning 60G25, 60G35, 62F15, 62G07, 62M20, 65C30, 65M75, 68T07 |
| url | https://arxiv.org/abs/2409.14585 |