arXiv subjects

Fanny Seizilles

Publications and source records attributed to Fanny Seizilles.

3 recordsLinked to original sources

arXiv2026

Inferring diffusivity from killed diffusion

We consider diffusion of independent molecules in an insulated Euclidean domain with unknown diffusivity parameter. At a random time and position, the molecules may bind and stop diffusing in dependence of a given `binding potential'. The binding process can be modeled by an additive random functional corresponding to the canonical construction of a `killed' diffusion Markov process. We study the problem of conducting inference on the infinite-dimensional diffusion parameter from a histogram plot of the `killing' positions of the process. We show first that these positions follow a Poisson point process whose intensity measure is determined by the solution of a certain Schrödinger equation. The inference problem can then be re-cast as a non-linear inverse problem for this PDE, which we show to be consistently solvable in a Bayesian way under natural conditions on the initial state of the diffusion, provided the binding potential is not too `aggressive'. In the course of our proofs we obtain novel posterior contraction rate results for high-dimensional Poisson count data that are of independent interest. A numerical illustration of the algorithm by standard MCMC methods is also provided.

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arXiv2026

Posterior contraction under misspecification and heteroscedasticity in non-linear inverse problems

In many practical and numerical inverse problems, the exact data log-likelihood is not fully accessible, motivating the use of surrogate models. We study heteroscedastic nonparametric nonlinear regression problems with Gaussian errors and establish contraction results for posterior distributions arising from a surrogate log-likelihood constructed from proxy error variances, an approximate forward map, and an appropriate Gaussian process prior. Under general assumptions on the approximation quality, we show that the resulting surrogate posterior is statistically reliable and contracts about the true parameter at rates comparable to those of the exact posterior. The analysis leverages consistency properties of the (penalised) MLE to effectively handle heteroscedastic noise and to control the impact of likelihood approximation errors. We apply the framework to PDE-constrained inverse problems for a reaction-diffusion equation and the two-dimensional Navier-Stokes equation. In the latter case, we consider misspecified viscosity and forcing terms as well as Oseen-type linearization models, highlighting the relevance of our results for numerical analysis applications.

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arXiv2023

The Bayesian approach to inverse Robin problems

In this paper we investigate the Bayesian approach to inverse Robin problems. These are problems for certain elliptic boundary value problems of determining a Robin coefficient on a hidden part of the boundary from Cauchy data on the observable part. Such a nonlinear inverse problem arises naturally in the initialisation of large-scale ice sheet models that are crucial in climate and sea-level predictions. We motivate the Bayesian approach for a prototypical Robin inverse problem by showing that the posterior mean converges in probability to the data-generating ground truth as the number of observations increases. Related to the stability theory for inverse Robin problems, we establish a logarithmic convergence rate for Sobolev-regular Robin coefficients, whereas for analytic coefficients we can attain an algebraic rate. The use of rescaled analytic Gaussian priors in posterior consistency for nonlinear inverse problems is new and may be of separate interest in other inverse problems. Our numerical results illustrate the convergence property in two observation settings.

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