arXiv subjects

Maia Tienstra

Publications and source records attributed to Maia Tienstra.

2 recordsLinked to original sources

arXiv2025

Early Stopping for Ensemble Kalman-Bucy Inversion

Bayesian linear inverse problems aim to recover an unknown signal from noisy observations, incorporating prior knowledge. This paper analyses a data-dependent method to choose the scale parameter of a Gaussian prior. The method we study arises from early stopping methods, which have been successfully applied to a range of problems, such as statistical inverse problems, in the frequentist setting. These results are extended to the Bayesian setting. We study the use of a discrepancy-based stopping rule in the setting of random noise, which allows for adaptation. Our proposed stopping rule results in optimal rates for the reparameterized problem under certain conditions on the prior covariance operator. We furthermore derive for which class of signals this method is adaptive. It is also shown that the associated posterior contracts at the same rate as the MAP estimator and provides a conservative measure of uncertainty. We implement the proposed stopping rule using the continuous-time ensemble Kalman--Bucy filter (EnKBF). The fictitious time parameter replaces the scale parameter, and the ensemble size is appropriately adjusted in order not to lose the statistical optimality of the computed estimator. With this Monte Carlo algorithm, we extend our results numerically to a nonlinear problem.

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arXiv2025

Hyperparameter Selection via Early Stopping for Bayesian Semilinear PDEs

We provide a data-driven method for choosing the scale of a Gaussian prior for non-linear Bayesian inverse problems arising from semilinear PDEs. Following \cite{koers2024}, the non-linear model for the parameter $f$ is reparametrized as a linear model for $v = \mathbb{L}u_f$, in which the prior scale can be selected by early stopping at the discrepancy principle \cite{tienstra2025}. We extend the transfer of frequentist guarantees from the linearised problem to the original one beyond Lipschitz solution maps, to maps admitting an arbitrary modulus of continuity, and we provide a general condition on the non-linearity under which such a map exists and is Lipschitz near the truth. The resulting posterior for $f$ contracts adaptively over a range of Sobolev smoothness, and its credible sets have asymptotic frequentist coverage one. We demonstrate our theory in detail for the stationary Schr\"odinger equation, for which we also provide numerical experiments. We further show how the results apply to the stationary Allen-Cahn equation and to a parabolic Allen-Cahn equation, and we discuss Darcy flow as an edge case outside the semilinear setting. The proposed method thus provides a data-driven way to tune Gaussian priors via early stopping, which is computationally efficient and statistically near-optimal for non-linear problems.

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