arXiv subjects

F. Hoffmann

Publications and source records attributed to F. Hoffmann.

6 recordsLinked to original sources

arXiv2025

Statistical accuracy of the ensemble Kalman filter in the near-linear setting

Estimating the state of a dynamical system from partial and noisy observations is a ubiquitous problem in a large number of applications, such as probabilistic weather forecasting and prediction of epidemics. Particle filters are a widely adopted approach to the problem and provide provably accurate approximations of the statistics of the state, but they perform poorly in high dimensions because of weight collapse. The ensemble Kalman filter does not suffer from this issue, as it relies on an interacting particle system with equal weights. Despite its wide adoption in the geophysical sciences, mathematical analysis of the accuracy of this filter is predominantly confined to the setting of linear dynamical models and linear observations operators, and analysis beyond the linear Gaussian setting is still in its infancy. In this short note, we provide an accessible overview of recent work in which the authors take first steps to analyze the accuracy of the filter beyond the linear Gaussian setting.

math.ST↗

arXiv2024

Does small-scale turbulence matter for ice growth in mixed-phase clouds?

Representing the glaciation of mixed-phase clouds in terms of the Wegener-Bergeron-Findeisen process is a challenge for many weather and climate models, which tend to overestimate this process because cloud dynamics and microphysics are not accurately represented. As turbulence is essential for the transport of water vapour from evaporating liquid droplets to ice crystals, we developed a statistical model using established closures to assess the role of small-scale turbulence. The model successfully captures results of direct numerical simulations, and we use it to assess the role of small-scale turbulence. We find that small-scale turbulence broadens the droplet-size distribution somewhat, but it does not significantly affect the glaciation time on submetre scales. However, our analysis indicates that turbulence on larger spatial scales is likely to affect ice growth. While the model must be amended to describe larger scales, the present work facilitates a path forward to understanding the role of turbulence in the Wegener-Bergeron-Findeisen process.

physics.flu-dyn↗

arXiv2024

Statistical Accuracy of Approximate Filtering Methods

Estimating the statistics of the state of a dynamical system, from partial and noisy observations, is both mathematically challenging and finds wide application. Furthermore, the applications are of great societal importance, including problems such as probabilistic weather forecasting and prediction of epidemics. Particle filters provide a well-founded approach to the problem, leading to provably accurate approximations of the statistics. However these methods perform poorly in high dimensions. In 1994 the idea of ensemble Kalman filtering was introduced by Evensen, leading to a methodology that has been widely adopted in the geophysical sciences and also finds application to quite general inverse problems. However, ensemble Kalman filters have defied rigorous analysis of their statistical accuracy, except in the linear Gaussian setting. In this article we describe recent work which takes first steps to analyze the statistical accuracy of ensemble Kalman filters beyond the linear Gaussian setting. The subject is inherently technical, as it involves the evolution of probability measures according to a nonlinear and nonautonomous dynamical system; and the approximation of this evolution. It can nonetheless be presented in a fairly accessible fashion, understandable with basic knowledge of dynamical systems, numerical analysis and probability.

math.NA↗

arXiv2022

The Mean Field Ensemble Kalman Filter: Near-Gaussian Setting

The ensemble Kalman filter is widely used in applications because, for high dimensional filtering problems, it has a robustness that is not shared for example by the particle filter; in particular it does not suffer from weight collapse. However, there is no theory which quantifies its accuracy as an approximation of the true filtering distribution, except in the Gaussian setting. To address this issue we provide the first analysis of the accuracy of the ensemble Kalman filter beyond the Gaussian setting. We prove two types of results: the first type comprise a stability estimate controlling the error made by the ensemble Kalman filter in terms of the difference between the true filtering distribution and a nearby Gaussian; and the second type use this stability result to show that, in a neighbourhood of Gaussian problems, the ensemble Kalman filter makes a small error, in comparison with the true filtering distribution. Our analysis is developed for the mean field ensemble Kalman filter. We rewrite the update equations for this filter, and for the true filtering distribution, in terms of maps on probability measures. We introduce a weighted total variation metric to estimate the distance between the two filters and we prove various stability estimates for the maps defining the evolution of the two filters, in this metric. Using these stability estimates we prove results of the first and second types, in the weighted total variation metric. We also provide a generalization of these results to the Gaussian projected filter, which can be viewed as a mean field description of the unscented Kalman filter.

math.OC↗

arXiv2021

Consensus Based Sampling

We propose a novel method for sampling and optimization tasks based on a stochastic interacting particle system. We explain how this method can be used for the following two goals: (i) generating approximate samples from a given target distribution; (ii) optimizing a given objective function. The approach is derivative-free and affine invariant, and is therefore well-suited for solving inverse problems defined by complex forward models: (i) allows generation of samples from the Bayesian posterior and (ii) allows determination of the maximum a posteriori estimator. We investigate the properties of the proposed family of methods in terms of various parameter choices, both analytically and by means of numerical simulations. The analysis and numerical simulation establish that the method has potential for general purpose optimization tasks over Euclidean space; contraction properties of the algorithm are established under suitable conditions, and computational experiments demonstrate wide basins of attraction for various specific problems. The analysis and experiments also demonstrate the potential for the sampling methodology in regimes in which the target distribution is unimodal and close to Gaussian; indeed we prove that the method recovers a Laplace approximation to the measure in certain parametric regimes and provide numerical evidence that this Laplace approximation attracts a large set of initial conditions in a number of examples.

math.DS↗

arXiv2011

Spin-orbit coupling effect in (Ga,Mn)As films: anisotropic exchange interactions and magnetocrystalline anisotropy

The magneto-crystalline anisotropy (MCA) of (Ga,Mn)As films has been studied on the basis of ab-initio electronic structure theory by performing magnetic torque calculations. An appreciable contribution to the in-plane uniaxial anisotropy can be attributed to an extended region adjacent to the surface. Calculations of the exchange tensor allow to ascribe a significant part to the MCA to the exchange anisotropy, caused either by a tetragonal distortion of the lattice or by the presence of the surface or interface.

cond-mat.mtrl-sci↗