Sparse identification of nonlocal interaction kernels in nonlinear gradient flow equations via partial inversion
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| author | Carrillo, Jose A. Estrada-Rodriguez, Gissell Mikolas, Laszlo Tang, Sui |
| author_facet | Carrillo, Jose A. Estrada-Rodriguez, Gissell Mikolas, Laszlo Tang, Sui |
| contents | We address the inverse problem of identifying nonlocal interaction potentials in nonlinear aggregation-diffusion equations from noisy discrete trajectory data. Our approach involves formulating and solving a regularized variational problem, which requires minimizing a quadratic error functional across a set of hypothesis functions, further augmented by a sparsity-enhancing regularizer. We employ a partial inversion algorithm, akin to the CoSaMP [57] and subspace pursuit algorithms [31], to solve the Basis Pursuit problem. A key theoretical contribution is our novel stability estimate for the PDEs, validating the error functional ability in controlling the 2-Wasserstein distance between solutions generated using the true and estimated interaction potentials. Our work also includes an error analysis of estimators caused by discretization and observational errors in practical implementations. We demonstrate the effectiveness of the methods through various 1D and 2D examples showcasing collective behaviors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06355 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sparse identification of nonlocal interaction kernels in nonlinear gradient flow equations via partial inversion Carrillo, Jose A. Estrada-Rodriguez, Gissell Mikolas, Laszlo Tang, Sui Analysis of PDEs 35Q70, 70F17, 70-08, 65F22 We address the inverse problem of identifying nonlocal interaction potentials in nonlinear aggregation-diffusion equations from noisy discrete trajectory data. Our approach involves formulating and solving a regularized variational problem, which requires minimizing a quadratic error functional across a set of hypothesis functions, further augmented by a sparsity-enhancing regularizer. We employ a partial inversion algorithm, akin to the CoSaMP [57] and subspace pursuit algorithms [31], to solve the Basis Pursuit problem. A key theoretical contribution is our novel stability estimate for the PDEs, validating the error functional ability in controlling the 2-Wasserstein distance between solutions generated using the true and estimated interaction potentials. Our work also includes an error analysis of estimators caused by discretization and observational errors in practical implementations. We demonstrate the effectiveness of the methods through various 1D and 2D examples showcasing collective behaviors. |
| title | Sparse identification of nonlocal interaction kernels in nonlinear gradient flow equations via partial inversion |
| topic | Analysis of PDEs 35Q70, 70F17, 70-08, 65F22 |
| url | https://arxiv.org/abs/2402.06355 |