We introduce a novel framework for efficient sampling from complex, unnormalised target distributions by exploiting multiscale dynamics. Traditional score-based sampling methods either rely on learned approximations of the score function or involve computationally expensive nested Markov chain Monte Carlo (MCMC) loops. In contrast, the proposed approach leverages stochastic averaging within a slow-fast system of stochastic differential equations (SDEs) to estimate intermediate scores along a diffusion path without training or inner-loop MCMC. Two algorithms are developed under this framework: *MultALMC*, which uses multiscale annealed Langevin dynamics, and *MultCDiff*, based on multiscale controlled diffusions for the reverse-time Ornstein-Uhlenbeck process. Both overdamped and underdamped variants are considered, with theoretical guarantees of convergence to the desired diffusion path. The framework is extended to handle heavy-tailed target distributions using Student’s t-based noise models and tailored fast-process dynamics. Empirical results across synthetic and real-world benchmarks, including multimodal and high-dimensional distributions, demonstrate that the proposed methods are competitive with existing samplers in terms of accuracy and efficiency, without the need for learned models.

Diffusion models have had a profound impact on many application areas, including those where data are intrinsically infinite-dimensional, such as images or time series. The standard approach is first to discretize and then to apply diffusion models to the discretized data. While such approaches are practically appealing, the performance of the resulting algorithms typically deteriorates as discretization parameters are refined. In this paper, we instead directly formulate diffusion-based generative models in infinite dimensions and apply them to the generative modelling of functions. We prove that our formulations are well posed in the infinite-dimensional setting and provide dimension-independent distance bounds from the sample to the target measure. Using our theory, we also develop guidelines for the design of infinite-dimensional diffusion models. For image distributions, these guidelines are in line with current canonical choices. For other distributions, however, we can improve upon these canonical choices. We demonstrate these results both theoretically and empirically, by applying the algorithms to data distributions on manifolds and to distributions arising in Bayesian inverse problems or simulation-based inference.

A machine learning technique is implemented for retrieving space-borne Global Navigation Satellite System Reflectometry (GNSS-R) wind speed. Conventional approaches commonly fit a function in a predefined form to matchup data in a least-squares (LS) sense, mapping GNSS-R observations to wind speed. In this study, a feedforward neural network is trained for TechDemoSat-1 (TDS-1) wind speed inversion. The input variables, along with the derived bistatic radar cross-section σ0, are selected after investigating the wind speed dependence and the model performance. When compared to an LS-based approach, the derived model shows a significant improvement of 20% in the root mean square error (RMSE). The proposed neural network demonstrates an ability to model a variety of effects degrading the retrieval accuracy such as the different levels of the effective isotropic radiated power (EIRP) of GPS satellites. For example, the derived Mean Absolute Error (MAE) of the satellite with SVN 34 is decreased by 32% using the machine-learning-based approach.

Understanding the decision process underlying gaze control is an important question in cognitive neuroscience with applications in diverse fields ranging from psychology to computer vision. The decision for choosing an upcoming saccade target can be framed as a selection process between two states: Should the observer further inspect the information near the current gaze position (local attention) or continue with exploration of other patches of the given scene (global attention)? Here we propose and investigate a mathematical model motivated by switching between these two attentional states during scene viewing. The model is derived from a minimal set of assumptions that generates realistic eye movement behavior. We implemented a Bayesian approach for model parameter inference based on the model's likelihood function. In order to simplify the inference, we applied data augmentation methods that allowed the use of conjugate priors and the construction of an efficient Gibbs sampler. This approach turned out to be numerically efficient and permitted fitting interindividual differences in saccade statistics. Thus, the main contribution of our modeling approach is two-fold; first, we propose a new model for saccade generation in scene viewing. Second, we demonstrate the use of novel methods from Bayesian inference in the field of scan path modeling.

After developing a wind speed retrieval algorithm, derived winds from measurements of UK TechDemoSat-1 (TDS-1), from May 2015 to July 2017, are compared to wind products of Advanced Scatterometer showing a reliable performance, especially during rain events. However, a rain signature in GNSS-R observations, a decrease in the value of the bistatic radar cross section at low winds, is demonstrated, which can potentially enable the technique to detect precipitation over oceans induced by low-to-moderate winds. This phenomenon is investigated and finally characterized as the rain splash effect altering the ocean surface roughness. To improve the quality of derived winds, a machine learning technique is implemented for the wind speed inversion as a geophysical model function. The trained feedforward neural network shows a significant improvement of 17% in the wind speed RMSE compared to the LS approach. In the end, one can conclude that space-borne ocean monitoring is evolving existing products with a potential for novel geophysical applications.