On gradient stability in nonlinear PDE models and inference in interacting particle systems
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918291085197312 |
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| author | Castre, Aurélien Nickl, Richard |
| author_facet | Castre, Aurélien Nickl, Richard |
| contents | We consider general parameter to solution maps $θ\mapsto \mathcal G(θ)$ of non-linear partial differential equations and describe an approach based on a Banach space version of the implicit function theorem to verify the gradient stability condition of Nickl&Wang (JEMS 2024) for the underlying non-linear inverse problem, providing also injectivity estimates and corresponding statistical identifiability results. We illustrate our methods in two examples involving a non-linear reaction diffusion system as well as a McKean--Vlasov interacting particle model, both with periodic boundary conditions. We apply our results to prove the polynomial time convergence of a Langevin-type algorithm sampling the posterior measure of the interaction potential arising from a discrete aggregate measurement of the interacting particle system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10326 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On gradient stability in nonlinear PDE models and inference in interacting particle systems Castre, Aurélien Nickl, Richard Statistics Theory Numerical Analysis Analysis of PDEs Probability We consider general parameter to solution maps $θ\mapsto \mathcal G(θ)$ of non-linear partial differential equations and describe an approach based on a Banach space version of the implicit function theorem to verify the gradient stability condition of Nickl&Wang (JEMS 2024) for the underlying non-linear inverse problem, providing also injectivity estimates and corresponding statistical identifiability results. We illustrate our methods in two examples involving a non-linear reaction diffusion system as well as a McKean--Vlasov interacting particle model, both with periodic boundary conditions. We apply our results to prove the polynomial time convergence of a Langevin-type algorithm sampling the posterior measure of the interaction potential arising from a discrete aggregate measurement of the interacting particle system. |
| title | On gradient stability in nonlinear PDE models and inference in interacting particle systems |
| topic | Statistics Theory Numerical Analysis Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2601.10326 |