Reconstructing the kinetic chemotaxis kernel using macroscopic data: well-posedness and ill-posedness
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912824201052160 |
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| author | Hellmuth, Kathrin Klingenberg, Christian Li, Qin Tang, Min |
| author_facet | Hellmuth, Kathrin Klingenberg, Christian Li, Qin Tang, Min |
| contents | Bacterial motion is steered by external stimuli (chemotaxis), and the motion described on the mesoscopic scale is uniquely determined by a parameter $K$ that models velocity change response from the bacteria. This parameter is called chemotaxis kernel. In a practical setting, it is inferred by experimental data. We deploy a PDE-constrained optimization framework to perform this reconstruction using velocity-averaged, localized data taken in the interior of the domain. The problem can be well-posed or ill-posed depending on the data preparation and the experimental setup. In particular, we propose one specific design that guarantees numerical reconstructability and local convergence. This design is adapted to the discretization of $K$ in space and decouples the reconstruction of local values of $K$ into smaller cell problems, opening up parallelization opportunities. Numerical evidences support the theoretical findings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_05004 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Reconstructing the kinetic chemotaxis kernel using macroscopic data: well-posedness and ill-posedness Hellmuth, Kathrin Klingenberg, Christian Li, Qin Tang, Min Numerical Analysis Analysis of PDEs Optimization and Control Cell Behavior 35R30, 65M32, 92C17, 49M41, 49K40 Bacterial motion is steered by external stimuli (chemotaxis), and the motion described on the mesoscopic scale is uniquely determined by a parameter $K$ that models velocity change response from the bacteria. This parameter is called chemotaxis kernel. In a practical setting, it is inferred by experimental data. We deploy a PDE-constrained optimization framework to perform this reconstruction using velocity-averaged, localized data taken in the interior of the domain. The problem can be well-posed or ill-posed depending on the data preparation and the experimental setup. In particular, we propose one specific design that guarantees numerical reconstructability and local convergence. This design is adapted to the discretization of $K$ in space and decouples the reconstruction of local values of $K$ into smaller cell problems, opening up parallelization opportunities. Numerical evidences support the theoretical findings. |
| title | Reconstructing the kinetic chemotaxis kernel using macroscopic data: well-posedness and ill-posedness |
| topic | Numerical Analysis Analysis of PDEs Optimization and Control Cell Behavior 35R30, 65M32, 92C17, 49M41, 49K40 |
| url | https://arxiv.org/abs/2309.05004 |