arXiv subjects

Gottfried Hastermann

Publications and source records attributed to Gottfried Hastermann.

4 recordsLinked to original sources

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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arXiv2024

Adaptive tempering schedules with approximative intermediate measures for filtering problems

Data assimilation algorithms integrate prior information from numerical model simulations with observed data. Ensemble-based filters, regarded as state-of-the-art, are widely employed for large-scale estimation tasks in disciplines such as geoscience and meteorology. Despite their inability to produce the true posterior distribution for nonlinear systems, their robustness and capacity for state tracking are noteworthy. In contrast, Particle filters yield the correct distribution in the ensemble limit but require substantially larger ensemble sizes than ensemble-based filters to maintain stability in higher-dimensional spaces. It is essential to transcend traditional Gaussian assumptions to achieve realistic quantification of uncertainties. One approach involves the hybridisation of filters, facilitated by tempering, to harness the complementary strengths of different filters. A new adaptive tempering method is proposed to tune the underlying schedule, aiming to systematically surpass the performance previously achieved. Although promising numerical results for certain filter combinations in toy examples exist in the literature, the tuning of hyperparameters presents a considerable challenge. A deeper understanding of these interactions is crucial for practical applications.

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arXiv2021

A one-step blended soundproof-compressible model with balanced data assimilation: theory and idealised tests

A challenge arising from the local Bayesian assimilation of data in an atmospheric flow simulation is the imbalances it may introduce. Acoustic fast-mode imbalances of the order of the slower dynamics can be negated by employing a blended numerical model with seamless access to the compressible and the soundproof pseudo-incompressible dynamics. Here, the blended modelling strategy by Benacchio et al., MWR, vol. 142 (2014) is upgraded in an advanced numerical framework and extended with a Bayesian local ensemble data assimilation method. Upon assimilation of data, the model configuration is switched to the pseudo-incompressible regime for one time-step. After that, the model configuration is switched back to the compressible model for the duration of the assimilation window. The switching between model regimes is repeated for each subsequent assimilation window. An improved blending strategy for the numerical model ensures that a single time-step in the pseudo-incompressible regime is sufficient to suppress imbalances coming from the initialisation and data assimilation. This improvement is based on three innovations: (i) the association of pressure fields computed at different stages of the numerical integration with actual time levels; (ii) a conversion of pressure-related variables between the model regimes derived from low Mach number asymptotics; and (iii) a judicious selection of the pressure variables used in converting numerical model states when a switch of models occurs. Idealised two-dimensional travelling vortex and buoyancy-driven bubble convection experiments show that acoustic imbalances arising from data assimilation can be eliminated by using this blended model, thereby achieving balanced analysis fields.

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arXiv2017

Balanced data assimilation for highly-oscillatory mechanical systems

Data assimilation algorithms are used to estimate the states of a dynamical system using partial and noisy observations. The ensemble Kalman filter has become a popular data assimilation scheme due to its simplicity and robustness for a wide range of application areas. Nevertheless, the ensemble Kalman filter also has limitations due to its inherent Gaussian and linearity assumptions. These limitations can manifest themselves in dynamically inconsistent state estimates. We investigate this issue in this paper for highly oscillatory Hamiltonian systems with a dynamical behavior which satisfies certain balance relations. We first demonstrate that the standard ensemble Kalman filter can lead to estimates which do not satisfy those balance relations, ultimately leading to filter divergence. We also propose two remedies for this phenomenon in terms of blended time-stepping schemes and ensemble-based penalty methods. The effect of these modifications to the standard ensemble Kalman filter are discussed and demonstrated numerically for two model scenarios. First, we consider balanced motion for highly oscillatory Hamiltonian systems and, second, we investigate thermally embedded highly oscillatory Hamiltonian systems. The first scenario is relevant for applications from meteorology while the second scenario is relevant for applications of data assimilation to molecular dynamics.

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