We present a method that exploits self-consistent simulation of coarse-grained and fine-grained models, in order to analyse properties of physical systems. The method uses the coarse-grained model to obtain a first estimate of the quantity of interest, before computing a correction by analysing properties of the fine system. We illustrate the method by applying it to the Asakura-Oosawa (AO) model of colloid-polymer mixtures. We show that the liquid-vapour critical point in that system is affected by three-body interactions which are neglected in the corresponding coarse-grained model. We analyse the size of this effect and the nature of the three-body interactions. We also analyse the accuracy of the method, as a function of the associated computational effort. arXiv:1907.07912v1 [cond-mat.stat-mech]

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Epidemiological forecasts are beset by uncertainties about the underlying epidemiological processes, and the surveillance process through which data are acquired. We present a Bayesian inference methodology that quantifies these uncertainties, for epidemics that are modelled by (possibly) non-stationary, continuous-time, Markov population processes. The efficiency of the method derives from a functional central limit theorem approximation of the likelihood, valid for large populations. We demonstrate the methodology by analysing the early stages of the COVID-19 pandemic in the UK, based on age-structured data for the number of deaths. This includes maximum a posteriori estimates, Markov chain Monte Carlo sampling of the posterior, computation of the model evidence, and the determination of parameter sensitivities via the Fisher information matrix. Our methodology is implemented in PyRoss, an open-source platform for analysis of epidemiological compartment models.

Low rank tensor approximations have been employed successfully, for example, to build surrogate models that can be used to speed up large-scale inference problems in high dimensions. The success of this depends critically on the rank that is necessary to represent or approximate the underlying distribution. In this paper, we develop a-priori rank bounds for approximations in the functional Tensor-Train representation for the case of a Gaussian (normally distributed) model. We show that under suitable conditions on the precision matrix, we can represent the Gaussian density to high accuracy without suffering from an exponential growth of complexity as the dimension increases. Our results provide evidence of the suitability and limitations of low rank tensor methods in a simple but important model case. Numerical experiments confirm that the rank bounds capture the qualitative behavior of the rank structure when varying the parameters of the precision matrix and the accuracy of the approximation.

Low-rank tensor approximations have shown great potential for uncertainty quantification in high dimensions, for example, to build surrogate models that can be used to speed up largescale inference problems (Eigel et al., Inverse Problems 34, 2018; Dolgov et al., Statistics & Computing 30, 2020). The feasibility and efficiency of such approaches depends critically on the rank that is necessary to represent or approximate the underlying distribution. In this paper, a-priori rank bounds for approximations in the functional tensor-train representation for the case of Gaussian models are developed. It is shown that under suitable conditions on the precision matrix, the Gaussian density can be approximated to high accuracy without suffering from an exponential growth of complexity as the dimension increases. These results provide a rigorous justification of the suitability and the limitations of low-rank tensor methods in a simple but important model case. Numerical experiments confirm that the rank bounds capture the qualitative behavior of the rank structure when varying the parameters of the precision matrix and the accuracy of the approximation. Finally, the practical relevance of the theoretical results is demonstrated in the context of a Bayesian filtering problem.